Abstract group 1728.30973: $D_4\times D_6\times C_{18}$

• Label: 1728.30973
• Order: \( 2^{6} \cdot 3^{3} \)
• Exponent: \( 2^{2} \cdot 3^{2} \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{5} \cdot 3^{2} \)
• $\card{Z(G)}$: \( 2^{3} \cdot 3^{2} \)
• $\card{\Aut(G)}$: \( 2^{15} \cdot 3^{3} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{12} \cdot 3^{2} \)
• Perm deg.: $20$
• Trans deg.: $288$
• Rank: $5$

Group information

Description:	$D_4\times D_6\times C_{18}$
Order:	\(1728\)\(\medspace = 2^{6} \cdot 3^{3} \)
Exponent:	\(36\)\(\medspace = 2^{2} \cdot 3^{2} \)
Automorphism group:	$C_2^5.C_2^6.C_3^3.C_2^4$, of order \(884736\)\(\medspace = 2^{15} \cdot 3^{3} \)
Composition factors:	$C_2$ x 6, $C_3$ x 3
Derived length:	$2$

This group is nonabelian, supersolvable (hence solvable and monomial), and metabelian.

Group statistics

Order	1	2	3	4	6	9	12	18	36
Elements	1	95	8	32	328	18	112	846	288	1728
Conjugacy classes	1	31	5	8	107	12	28	276	72	540
Divisions	1	31	3	8	61	2	16	46	12	180
Autjugacy classes	1	5	3	2	11	2	4	8	3	39

Dimension	1	2	4	6	8	12	24
Irr. complex chars.	288	216	36	0	0	0	0	540
Irr. rational chars.	32	56	28	32	4	24	4	180

Minimal presentations

Permutation degree:	$20$
Transitive degree:	$288$
Rank:	$5$
Inequivalent generating 5-tuples:	$5104360800$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	not computed	none
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	$\langle a, b, c, d, e \mid a^{2}=b^{2}=c^{2}=d^{12}=e^{18}=[a,b]=[a,c]=[a,d]=[a,e]=[b,c]=[b,e]=[c,e]=[d,e]=1, d^{b}=d^{5}, d^{c}=d^{7} \rangle$
Permutation group:	Degree $20$ $\langle(1,2)(3,4)(6,7), (1,2)(3,4)(17,18)(19,20), (1,2)(3,4)(17,19)(18,20), (3,4) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 1 & 38 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 39 & 38 \\ 38 & 39 \end{array}\right), \left(\begin{array}{rr} 49 & 0 \\ 0 & 49 \end{array}\right), \left(\begin{array}{rr} 45 & 0 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 39 & 0 \\ 0 & 39 \end{array}\right), \left(\begin{array}{rr} 61 & 0 \\ 0 & 61 \end{array}\right), \left(\begin{array}{rr} 20 & 57 \\ 57 & 20 \end{array}\right), \left(\begin{array}{rr} 37 & 0 \\ 0 & 37 \end{array}\right), \left(\begin{array}{rr} 57 & 68 \\ 64 & 57 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/76\Z)$
Direct product:	$C_2$ ${}^2$ $\, \times\, $ $C_9$ $\, \times\, $ $S_3$ $\, \times\, $ $D_4$
Semidirect product:	$(S_3\times C_{36})$ $\,\rtimes\,$ $C_2^3$	$(S_3\times C_{18})$ $\,\rtimes\,$ $C_2^4$	$C_2^4$ $\,\rtimes\,$ $(S_3\times C_{18})$	$(D_6\times C_{36})$ $\,\rtimes\,$ $C_2^2$	all 68
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_6^2:D_4)$ . $C_6$	$(D_4\times C_6^2)$ . $C_6$	$(C_6^2:D_4)$ . $C_6$	$C_6^2$ . $(C_6\times D_4)$	all 66
Aut. group:	$\Aut(C_{24}:C_{38})$	$\Aut(C_6:C_{152})$	$\Aut(C_{12}.C_{76})$	$\Aut(D_{12}:C_{38})$	all 23

Elements of the group are displayed as matrices in $\GL_{2}(\Z/{76}\Z)$.

Homology

Abelianization:	$C_{2}^{4} \times C_{18} \simeq C_{2}^{5} \times C_{9}$
Schur multiplier:	$C_{2}^{10}$
Commutator length:	$1$

Subgroups

There are 10928 subgroups in 4322 conjugacy classes, 1509 normal (45 characteristic).

Characteristic subgroups are shown in this color. Normal (but not characteristic) subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_2^2\times C_{18}$	$G/Z \simeq$ $C_2\times D_6$
Commutator:	$G' \simeq$ $C_6$	$G/G' \simeq$ $C_2^4\times C_{18}$
Frattini:	$\Phi \simeq$ $C_6$	$G/\Phi \simeq$ $C_6^2:C_2^3$
Fitting:	$\operatorname{Fit} \simeq$ $C_6\times D_4\times C_{18}$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $D_4\times D_6\times C_{18}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2\times C_6^2$	$G/\operatorname{soc} \simeq$ $C_2^2\times C_6$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $D_4\times C_2^3$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3\times C_9$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

No diagram available: subgroups only stored up to automorphism

Classes of subgroups up to automorphism

There are too many subgroups to show. See a full page version of the diagram.

Normal subgroups

No diagram available: subgroups only stored up to automorphism

Normal subgroups up to automorphism

There are too many subgroups to show. See a full page version of the diagram.

For the default diagram, subgroups are sorted vertically by the number of prime divisors (counted with multiplicity) in their orders.
To see subgroups sorted vertically by order instead, check this box.

Sorry, your browser does not support the subgroup diagram.

Subgroup information Click on a subgroup in the diagram to see information about it.

See a full page version of the diagram

See a full page version of the diagram

See a full page version of the diagram

See a full page version of the diagram

Classes of subgroups up to conjugation

Order 1728: $D_4\times D_6\times C_{18}$

Order 864: $S_3\times D_4\times C_{18}$ x 24, $C_2^2\times D_6\times C_{18}$ x 2, $C_6:D_4\times C_{18}$ x 2, $C_6\times D_4\times C_{18}$, $C_2\times C_{18}\times D_{12}$, $C_2\times D_6\times C_{36}$

Order 576: $C_6^2:C_2^4$, $C_{36}:C_2^4$

Order 432: $C_{36}:D_6$ x 64, $C_2\times D_6\times C_{18}$ x 43, $C_2\times D_6:C_{18}$ x 24, $C_{12}.C_6^2$ x 12, $C_{18}\times D_{12}$ x 12, $D_6\times C_{36}$ x 12, $C_2^2\times C_6\times C_{18}$ x 2, $C_2\times C_6\times C_{36}$, $C_6^2.C_{12}$

Order 288: $C_2^4:C_{18}$ x 29, $C_6^2:C_2^3$ x 24, $C_2^4\times C_{18}$ x 2, $C_6^2:C_2^3$ x 2, $C_6^2:D_4$ x 2, $C_6^2:D_4$, $C_6^2.C_2^3$, $C_2^3\times C_{36}$, $D_4\times C_6^2$

Order 216: $D_6\times C_{18}$ x 162, $D_6:C_{18}$ x 32, $C_2\times C_6\times C_{18}$ x 21, $C_{12}:C_{18}$ x 16, $C_9\times D_{12}$ x 16, $S_3\times C_{36}$ x 16, $C_6\times C_{36}$ x 6, $C_6:C_{36}$ x 6

Order 192: $C_{12}:C_2^4$, $C_{12}:C_2^4$

Order 144: $D_4\times C_{18}$ x 124, $C_{12}:D_6$ x 64, $C_2^3\times C_{18}$ x 47, $C_6^2:C_2^2$ x 43, $C_6^2:C_2^2$ x 24, $C_2^2\times C_{36}$ x 15, $C_4:C_6^2$ x 12, $C_6\times D_{12}$ x 12, $C_{12}\times D_6$ x 12, $C_2^2\times C_6^2$ x 2, $C_2\times C_6\times C_{12}$, $C_6^2:C_4$

Order 108: $S_3\times C_{18}$ x 124, $C_6\times C_{18}$ x 39, $C_3:C_{36}$ x 4, $C_3\times C_{36}$ x 4

Order 96: $C_{12}:C_2^3$ x 30, $D_4\times D_6$ x 24, $C_2^4\times C_6$ x 2, $C_2^3\times D_6$ x 2, $C_2^3:D_6$ x 2, $C_2^3\times C_{12}$, $C_2^2\times D_{12}$, $C_{12}:C_2^3$

Order 72: $C_2^2\times C_{18}$ x 204, $C_6\times D_6$ x 162, $D_4\times C_9$ x 80, $C_2\times C_{36}$ x 34, $C_6\wr C_2$ x 32, $C_2\times C_6^2$ x 21, $D_4\times C_3^2$ x 16, $C_3\times D_{12}$ x 16, $S_3\times C_{12}$ x 16, $C_6\times C_{12}$ x 6, $C_6:C_{12}$ x 6

Order 64: $D_4\times C_2^3$

Order 54: $S_3\times C_9$ x 16, $C_3\times C_{18}$ x 15

Order 48: $C_6\times D_4$ x 136, $S_3\times D_4$ x 64, $C_2^3\times C_6$ x 49, $C_2^2\times D_6$ x 43, $C_6:D_4$ x 24, $C_2^2\times C_{12}$ x 16, $C_2\times D_{12}$ x 12, $C_4\times D_6$ x 12, $C_6.C_2^3$

Order 36: $C_2\times C_{18}$ x 202, $C_6\times S_3$ x 124, $C_6^2$ x 39, $C_{36}$ x 12, $C_3\times C_{12}$ x 4, $C_3:C_{12}$ x 4

Order 32: $C_2^2\times D_4$ x 28, $C_2^5$ x 2, $C_2^3\times C_4$

Order 27: $C_3\times C_9$

Order 24: $C_2^2\times C_6$ x 225, $C_2\times D_6$ x 162, $C_3\times D_4$ x 96, $C_2\times C_{12}$ x 40, $C_3:D_4$ x 32, $D_{12}$ x 16, $C_4\times S_3$ x 16, $C_6:C_4$ x 6

Order 18: $C_{18}$ x 46, $C_3\times S_3$ x 16, $C_3\times C_6$ x 15

Order 16: $C_2\times D_4$ x 112, $C_2^4$ x 45, $C_2^2\times C_4$ x 14

Order 12: $C_2\times C_6$ x 241, $D_6$ x 124, $C_{12}$ x 16, $C_3:C_4$ x 4

Order 9: $C_9$ x 2, $C_3^2$

Order 8: $C_2^3$ x 183, $D_4$ x 64, $C_2\times C_4$ x 28

Order 6: $C_6$ x 61, $S_3$ x 16

Order 4: $C_2^2$ x 163, $C_4$ x 8

Order 3: $C_3$ x 3

Order 2: $C_2$ x 31

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 1728: $D_4\times D_6\times C_{18}$

Order 864: $C_2^2\times D_6\times C_{18}$, $C_6\times D_4\times C_{18}$, $C_6:D_4\times C_{18}$, $S_3\times D_4\times C_{18}$, $C_2\times C_{18}\times D_{12}$, $C_2\times D_6\times C_{36}$

Order 576: $C_6^2:C_2^4$, $C_{36}:C_2^4$

Order 432: $C_2\times D_6\times C_{18}$ x 4, $C_2^2\times C_6\times C_{18}$, $C_{12}.C_6^2$, $C_2\times C_6\times C_{36}$, $C_2\times D_6:C_{18}$, $C_6^2.C_{12}$, $C_{36}:D_6$, $C_{18}\times D_{12}$, $D_6\times C_{36}$

Order 288: $C_2^4:C_{18}$ x 5, $C_6^2:C_2^3$, $C_6^2:D_4$, $C_6^2.C_2^3$, $C_2^4\times C_{18}$, $C_2^3\times C_{36}$, $C_6^2:C_2^3$, $D_4\times C_6^2$, $C_6^2:D_4$

Order 216: $D_6\times C_{18}$ x 6, $C_2\times C_6\times C_{18}$ x 3, $C_{12}:C_{18}$, $C_6\times C_{36}$, $D_6:C_{18}$, $C_6:C_{36}$, $C_9\times D_{12}$, $S_3\times C_{36}$

Order 192: $C_{12}:C_2^4$, $C_{12}:C_2^4$

Order 144: $C_2^3\times C_{18}$ x 6, $D_4\times C_{18}$ x 5, $C_2^2\times C_{36}$ x 4, $C_6^2:C_2^2$ x 4, $C_2^2\times C_6^2$, $C_4:C_6^2$, $C_2\times C_6\times C_{12}$, $C_6^2:C_2^2$, $C_6^2:C_4$, $C_{12}:D_6$, $C_6\times D_{12}$, $C_{12}\times D_6$

Order 108: $S_3\times C_{18}$ x 5, $C_6\times C_{18}$ x 4, $C_3:C_{36}$, $C_3\times C_{36}$

Order 96: $C_{12}:C_2^3$ x 6, $C_2^4\times C_6$, $C_2^3\times D_6$, $C_2^3\times C_{12}$, $C_2^3:D_6$, $D_4\times D_6$, $C_2^2\times D_{12}$, $C_{12}:C_2^3$

Order 72: $C_2^2\times C_{18}$ x 12, $C_6\times D_6$ x 6, $C_2\times C_{36}$ x 4, $D_4\times C_9$ x 4, $C_2\times C_6^2$ x 3, $D_4\times C_3^2$, $C_6\times C_{12}$, $C_6\wr C_2$, $C_6:C_{12}$, $C_3\times D_{12}$, $S_3\times C_{12}$

Order 64: $D_4\times C_2^3$

Order 54: $C_3\times C_{18}$ x 3, $S_3\times C_9$ x 2

Order 48: $C_2^3\times C_6$ x 7, $C_6\times D_4$ x 6, $C_2^2\times C_{12}$ x 5, $C_2^2\times D_6$ x 4, $C_6:D_4$, $C_6.C_2^3$, $S_3\times D_4$, $C_2\times D_{12}$, $C_4\times D_6$

Order 36: $C_2\times C_{18}$ x 13, $C_6\times S_3$ x 5, $C_6^2$ x 4, $C_{36}$ x 3, $C_3\times C_{12}$, $C_3:C_{12}$

Order 32: $C_2^2\times D_4$ x 4, $C_2^5$, $C_2^3\times C_4$

Order 27: $C_3\times C_9$

Order 24: $C_2^2\times C_6$ x 15, $C_2\times D_6$ x 6, $C_2\times C_{12}$ x 5, $C_3\times D_4$ x 5, $C_3:D_4$, $C_6:C_4$, $D_{12}$, $C_4\times S_3$

Order 18: $C_{18}$ x 8, $C_3\times C_6$ x 3, $C_3\times S_3$ x 2

Order 16: $C_2^4$ x 5, $C_2\times D_4$ x 4, $C_2^2\times C_4$ x 3

Order 12: $C_2\times C_6$ x 17, $D_6$ x 5, $C_{12}$ x 4, $C_3:C_4$

Order 9: $C_9$ x 2, $C_3^2$

Order 8: $C_2^3$ x 9, $D_4$ x 3, $C_2\times C_4$ x 3

Order 6: $C_6$ x 11, $S_3$ x 2

Order 4: $C_2^2$ x 9, $C_4$ x 2

Order 3: $C_3$ x 3

Order 2: $C_2$ x 5

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1728: $D_4\times D_6\times C_{18}$ ($C_1$)

Order 864: $S_3\times D_4\times C_{18}$ ($C_2$) x 24, $C_2^2\times D_6\times C_{18}$ ($C_2$) x 2, $C_6:D_4\times C_{18}$ ($C_2$) x 2, $C_6\times D_4\times C_{18}$ ($C_2$), $C_2\times C_{18}\times D_{12}$ ($C_2$), $C_2\times D_6\times C_{36}$ ($C_2$)

Order 576: $C_6^2:C_2^4$ ($C_3$)

Order 432: $C_{36}:D_6$ ($C_2^2$) x 64, $C_2\times D_6\times C_{18}$ ($C_2^2$) x 27, $C_2\times D_6:C_{18}$ ($C_2^2$) x 24, $C_{12}.C_6^2$ ($C_2^2$) x 12, $C_{18}\times D_{12}$ ($C_2^2$) x 12, $D_6\times C_{36}$ ($C_2^2$) x 12, $C_2^2\times C_6\times C_{18}$ ($C_2^2$) x 2, $C_2\times C_6\times C_{36}$ ($C_2^2$), $C_6^2.C_{12}$ ($C_2^2$)

Order 288: $C_6^2:C_2^3$ ($C_6$) x 24, $C_6^2:C_2^3$ ($C_6$) x 2, $C_6^2:D_4$ ($C_6$) x 2, $C_6^2:D_4$ ($C_6$), $C_6^2.C_2^3$ ($C_6$), $C_2^4:C_{18}$ ($S_3$), $D_4\times C_6^2$ ($C_6$)

Order 216: $D_6\times C_{18}$ ($C_2^3$) x 50, $D_6:C_{18}$ ($C_2^3$) x 32, $C_{12}:C_{18}$ ($C_2^3$) x 16, $C_9\times D_{12}$ ($C_2^3$) x 16, $S_3\times C_{36}$ ($C_2^3$) x 16, $C_2\times C_6\times C_{18}$ ($C_2^3$) x 13, $D_6\times C_{18}$ ($D_4$) x 8, $C_6\times C_{36}$ ($C_2^3$) x 6, $C_6:C_{36}$ ($C_2^3$) x 6

Order 192: $C_{12}:C_2^4$ ($C_9$)

Order 144: $C_{12}:D_6$ ($C_2\times C_6$) x 64, $C_6^2:C_2^2$ ($C_2\times C_6$) x 27, $C_6^2:C_2^2$ ($C_2\times C_6$) x 24, $D_4\times C_{18}$ ($D_6$) x 12, $C_4:C_6^2$ ($C_2\times C_6$) x 12, $C_6\times D_{12}$ ($C_2\times C_6$) x 12, $C_{12}\times D_6$ ($C_2\times C_6$) x 12, $C_2^2\times C_6^2$ ($C_2\times C_6$) x 2, $C_2^3\times C_{18}$ ($D_6$) x 2, $C_2^2\times C_{36}$ ($D_6$), $C_2\times C_6\times C_{12}$ ($C_2\times C_6$), $C_6^2:C_4$ ($C_2\times C_6$)

Order 108: $S_3\times C_{18}$ ($C_2\times D_4$) x 24, $S_3\times C_{18}$ ($C_2^4$) x 12, $C_6\times C_{18}$ ($C_2^4$) x 11, $C_3:C_{36}$ ($C_2^4$) x 4, $C_6\times C_{18}$ ($C_2\times D_4$) x 4, $C_3\times C_{36}$ ($C_2^4$) x 4

Order 96: $D_4\times D_6$ ($C_{18}$) x 24, $C_2^3\times D_6$ ($C_{18}$) x 2, $C_2^3:D_6$ ($C_{18}$) x 2, $C_{12}:C_2^3$ ($C_3\times S_3$), $C_{12}:C_2^3$ ($C_{18}$), $C_2^2\times D_{12}$ ($C_{18}$), $C_{12}:C_2^3$ ($C_{18}$)

Order 72: $C_6\times D_6$ ($C_2^2\times C_6$) x 50, $C_6\wr C_2$ ($C_2^2\times C_6$) x 32, $D_4\times C_3^2$ ($C_2^2\times C_6$) x 16, $C_3\times D_{12}$ ($C_2^2\times C_6$) x 16, $S_3\times C_{12}$ ($C_2^2\times C_6$) x 16, $D_4\times C_9$ ($C_2\times D_6$) x 16, $C_2\times C_6^2$ ($C_2^2\times C_6$) x 13, $C_2^2\times C_{18}$ ($C_2\times D_6$) x 13, $C_6\times D_6$ ($C_3\times D_4$) x 8, $C_2\times C_{36}$ ($C_2\times D_6$) x 6, $C_6\times C_{12}$ ($C_2^2\times C_6$) x 6, $C_6:C_{12}$ ($C_2^2\times C_6$) x 6

Order 54: $S_3\times C_9$ ($C_2^2\times D_4$) x 8, $C_3\times C_{18}$ ($C_2^2\times D_4$) x 6, $C_3\times C_{18}$ ($C_2^5$)

Order 48: $S_3\times D_4$ ($C_2\times C_{18}$) x 64, $C_2^2\times D_6$ ($C_2\times C_{18}$) x 27, $C_6:D_4$ ($C_2\times C_{18}$) x 24, $C_6\times D_4$ ($C_2\times C_{18}$) x 12, $C_6\times D_4$ ($C_6\times S_3$) x 12, $C_2\times D_{12}$ ($C_2\times C_{18}$) x 12, $C_4\times D_6$ ($C_2\times C_{18}$) x 12, $C_2^3\times C_6$ ($C_2\times C_{18}$) x 2, $C_2^3\times C_6$ ($C_6\times S_3$) x 2, $C_2^2\times C_{12}$ ($C_2\times C_{18}$), $C_2^2\times C_{12}$ ($C_6\times S_3$), $C_6.C_2^3$ ($C_2\times C_{18}$)

Order 36: $C_6\times S_3$ ($C_6\times D_4$) x 24, $C_6\times S_3$ ($C_2^3\times C_6$) x 12, $C_2\times C_{18}$ ($C_2^2\times D_6$) x 11, $C_6^2$ ($C_2^3\times C_6$) x 11, $C_3\times C_{12}$ ($C_2^3\times C_6$) x 4, $C_3:C_{12}$ ($C_2^3\times C_6$) x 4, $C_2\times C_{18}$ ($S_3\times D_4$) x 4, $C_{36}$ ($C_2^2\times D_6$) x 4, $C_6^2$ ($C_6\times D_4$) x 4

Order 32: $C_2^2\times D_4$ ($S_3\times C_9$)

Order 27: $C_3\times C_9$ ($D_4\times C_2^3$)

Order 24: $C_2\times D_6$ ($C_2^2\times C_{18}$) x 50, $C_3:D_4$ ($C_2^2\times C_{18}$) x 32, $D_{12}$ ($C_2^2\times C_{18}$) x 16, $C_4\times S_3$ ($C_2^2\times C_{18}$) x 16, $C_3\times D_4$ ($C_6\times D_6$) x 16, $C_3\times D_4$ ($C_2^2\times C_{18}$) x 16, $C_2^2\times C_6$ ($C_6\times D_6$) x 13, $C_2^2\times C_6$ ($C_2^2\times C_{18}$) x 13, $C_2\times D_6$ ($D_4\times C_9$) x 8, $C_2\times C_{12}$ ($C_6\times D_6$) x 6, $C_2\times C_{12}$ ($C_2^2\times C_{18}$) x 6, $C_6:C_4$ ($C_2^2\times C_{18}$) x 6

Order 18: $C_3\times S_3$ ($C_{12}:C_2^3$) x 8, $C_3\times C_6$ ($C_{12}:C_2^3$) x 6, $C_{18}$ ($D_4\times D_6$) x 6, $C_3\times C_6$ ($C_2^4\times C_6$), $C_{18}$ ($C_2^3\times D_6$)

Order 16: $C_2\times D_4$ ($S_3\times C_{18}$) x 12, $C_2^4$ ($S_3\times C_{18}$) x 2, $C_2^2\times C_4$ ($S_3\times C_{18}$)

Order 12: $D_6$ ($D_4\times C_{18}$) x 24, $D_6$ ($C_2^3\times C_{18}$) x 12, $C_2\times C_6$ ($C_6^2:C_2^2$) x 11, $C_2\times C_6$ ($C_2^3\times C_{18}$) x 11, $C_2\times C_6$ ($D_4\times C_{18}$) x 4, $C_2\times C_6$ ($C_{12}:D_6$) x 4, $C_{12}$ ($C_6^2:C_2^2$) x 4, $C_{12}$ ($C_2^3\times C_{18}$) x 4, $C_3:C_4$ ($C_2^3\times C_{18}$) x 4

Order 9: $C_3^2$ ($C_{12}:C_2^4$), $C_9$ ($C_{12}:C_2^4$)

Order 8: $D_4$ ($D_6\times C_{18}$) x 16, $C_2^3$ ($D_6\times C_{18}$) x 13, $C_2\times C_4$ ($D_6\times C_{18}$) x 6

Order 6: $S_3$ ($C_2^4:C_{18}$) x 8, $C_6$ ($C_6^2:C_2^3$) x 6, $C_6$ ($C_2^4:C_{18}$) x 6, $C_6$ ($C_2^4\times C_{18}$), $C_6$ ($C_6^2:C_2^3$)

Order 4: $C_2^2$ ($C_2\times D_6\times C_{18}$) x 11, $C_2^2$ ($C_{36}:D_6$) x 4, $C_4$ ($C_2\times D_6\times C_{18}$) x 4

Order 3: $C_3$ ($C_6^2:C_2^4$), $C_3$ ($C_{36}:C_2^4$)

Order 2: $C_2$ ($S_3\times D_4\times C_{18}$) x 6, $C_2$ ($C_2^2\times D_6\times C_{18}$)

Order 1: $C_1$ ($D_4\times D_6\times C_{18}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1728: $D_4\times D_6\times C_{18}$ ($C_1$)

Order 864: $C_2^2\times D_6\times C_{18}$ ($C_2$), $C_6\times D_4\times C_{18}$ ($C_2$), $C_6:D_4\times C_{18}$ ($C_2$), $S_3\times D_4\times C_{18}$ ($C_2$), $C_2\times C_{18}\times D_{12}$ ($C_2$), $C_2\times D_6\times C_{36}$ ($C_2$)

Order 576: $C_6^2:C_2^4$ ($C_3$)

Order 432: $C_2\times D_6\times C_{18}$ ($C_2^2$) x 3, $C_2^2\times C_6\times C_{18}$ ($C_2^2$), $C_{12}.C_6^2$ ($C_2^2$), $C_2\times C_6\times C_{36}$ ($C_2^2$), $C_2\times D_6:C_{18}$ ($C_2^2$), $C_6^2.C_{12}$ ($C_2^2$), $C_{36}:D_6$ ($C_2^2$), $C_{18}\times D_{12}$ ($C_2^2$), $D_6\times C_{36}$ ($C_2^2$)

Order 288: $C_6^2:C_2^3$ ($C_6$), $C_6^2:D_4$ ($C_6$), $C_6^2.C_2^3$ ($C_6$), $C_2^4:C_{18}$ ($S_3$), $C_6^2:C_2^3$ ($C_6$), $D_4\times C_6^2$ ($C_6$), $C_6^2:D_4$ ($C_6$)

Order 216: $D_6\times C_{18}$ ($C_2^3$) x 3, $C_2\times C_6\times C_{18}$ ($C_2^3$) x 2, $C_{12}:C_{18}$ ($C_2^3$), $C_6\times C_{36}$ ($C_2^3$), $D_6:C_{18}$ ($C_2^3$), $C_6:C_{36}$ ($C_2^3$), $C_9\times D_{12}$ ($C_2^3$), $S_3\times C_{36}$ ($C_2^3$), $D_6\times C_{18}$ ($D_4$)

Order 192: $C_{12}:C_2^4$ ($C_9$)

Order 144: $C_6^2:C_2^2$ ($C_2\times C_6$) x 3, $D_4\times C_{18}$ ($D_6$), $C_2^2\times C_{36}$ ($D_6$), $C_2^2\times C_6^2$ ($C_2\times C_6$), $C_4:C_6^2$ ($C_2\times C_6$), $C_2\times C_6\times C_{12}$ ($C_2\times C_6$), $C_6^2:C_2^2$ ($C_2\times C_6$), $C_6^2:C_4$ ($C_2\times C_6$), $C_{12}:D_6$ ($C_2\times C_6$), $C_6\times D_{12}$ ($C_2\times C_6$), $C_{12}\times D_6$ ($C_2\times C_6$), $C_2^3\times C_{18}$ ($D_6$)

Order 108: $C_6\times C_{18}$ ($C_2^4$) x 2, $S_3\times C_{18}$ ($C_2^4$) x 2, $C_3:C_{36}$ ($C_2^4$), $C_6\times C_{18}$ ($C_2\times D_4$), $S_3\times C_{18}$ ($C_2\times D_4$), $C_3\times C_{36}$ ($C_2^4$)

Order 96: $C_2^3\times D_6$ ($C_{18}$), $C_{12}:C_2^3$ ($C_3\times S_3$), $C_{12}:C_2^3$ ($C_{18}$), $C_2^3:D_6$ ($C_{18}$), $D_4\times D_6$ ($C_{18}$), $C_2^2\times D_{12}$ ($C_{18}$), $C_{12}:C_2^3$ ($C_{18}$)

Order 72: $C_6\times D_6$ ($C_2^2\times C_6$) x 3, $C_2\times C_6^2$ ($C_2^2\times C_6$) x 2, $C_2^2\times C_{18}$ ($C_2\times D_6$) x 2, $C_2\times C_{36}$ ($C_2\times D_6$), $C_6\times D_6$ ($C_3\times D_4$), $D_4\times C_3^2$ ($C_2^2\times C_6$), $C_6\times C_{12}$ ($C_2^2\times C_6$), $C_6\wr C_2$ ($C_2^2\times C_6$), $C_6:C_{12}$ ($C_2^2\times C_6$), $C_3\times D_{12}$ ($C_2^2\times C_6$), $S_3\times C_{12}$ ($C_2^2\times C_6$), $D_4\times C_9$ ($C_2\times D_6$)

Order 54: $C_3\times C_{18}$ ($C_2^5$), $C_3\times C_{18}$ ($C_2^2\times D_4$), $S_3\times C_9$ ($C_2^2\times D_4$)

Order 48: $C_2^2\times D_6$ ($C_2\times C_{18}$) x 3, $C_2^3\times C_6$ ($C_2\times C_{18}$), $C_2^3\times C_6$ ($C_6\times S_3$), $C_6\times D_4$ ($C_2\times C_{18}$), $C_6\times D_4$ ($C_6\times S_3$), $C_2^2\times C_{12}$ ($C_2\times C_{18}$), $C_2^2\times C_{12}$ ($C_6\times S_3$), $C_6:D_4$ ($C_2\times C_{18}$), $C_6.C_2^3$ ($C_2\times C_{18}$), $S_3\times D_4$ ($C_2\times C_{18}$), $C_2\times D_{12}$ ($C_2\times C_{18}$), $C_4\times D_6$ ($C_2\times C_{18}$)

Order 36: $C_2\times C_{18}$ ($C_2^2\times D_6$) x 2, $C_6^2$ ($C_2^3\times C_6$) x 2, $C_6\times S_3$ ($C_2^3\times C_6$) x 2, $C_3\times C_{12}$ ($C_2^3\times C_6$), $C_3:C_{12}$ ($C_2^3\times C_6$), $C_2\times C_{18}$ ($S_3\times D_4$), $C_{36}$ ($C_2^2\times D_6$), $C_6^2$ ($C_6\times D_4$), $C_6\times S_3$ ($C_6\times D_4$)

Order 32: $C_2^2\times D_4$ ($S_3\times C_9$)

Order 27: $C_3\times C_9$ ($D_4\times C_2^3$)

Order 24: $C_2\times D_6$ ($C_2^2\times C_{18}$) x 3, $C_2^2\times C_6$ ($C_6\times D_6$) x 2, $C_2^2\times C_6$ ($C_2^2\times C_{18}$) x 2, $C_2\times C_{12}$ ($C_6\times D_6$), $C_2\times C_{12}$ ($C_2^2\times C_{18}$), $C_3:D_4$ ($C_2^2\times C_{18}$), $C_6:C_4$ ($C_2^2\times C_{18}$), $D_{12}$ ($C_2^2\times C_{18}$), $C_4\times S_3$ ($C_2^2\times C_{18}$), $C_2\times D_6$ ($D_4\times C_9$), $C_3\times D_4$ ($C_6\times D_6$), $C_3\times D_4$ ($C_2^2\times C_{18}$)

Order 18: $C_3\times C_6$ ($C_2^4\times C_6$), $C_3\times C_6$ ($C_{12}:C_2^3$), $C_3\times S_3$ ($C_{12}:C_2^3$), $C_{18}$ ($C_2^3\times D_6$), $C_{18}$ ($D_4\times D_6$)

Order 16: $C_2^4$ ($S_3\times C_{18}$), $C_2\times D_4$ ($S_3\times C_{18}$), $C_2^2\times C_4$ ($S_3\times C_{18}$)

Order 12: $C_2\times C_6$ ($C_6^2:C_2^2$) x 2, $C_2\times C_6$ ($C_2^3\times C_{18}$) x 2, $D_6$ ($C_2^3\times C_{18}$) x 2, $C_2\times C_6$ ($D_4\times C_{18}$), $C_2\times C_6$ ($C_{12}:D_6$), $D_6$ ($D_4\times C_{18}$), $C_{12}$ ($C_6^2:C_2^2$), $C_{12}$ ($C_2^3\times C_{18}$), $C_3:C_4$ ($C_2^3\times C_{18}$)

Order 9: $C_3^2$ ($C_{12}:C_2^4$), $C_9$ ($C_{12}:C_2^4$)

Order 8: $C_2^3$ ($D_6\times C_{18}$) x 2, $D_4$ ($D_6\times C_{18}$), $C_2\times C_4$ ($D_6\times C_{18}$)

Order 6: $C_6$ ($C_6^2:C_2^3$), $C_6$ ($C_2^4\times C_{18}$), $C_6$ ($C_2^4:C_{18}$), $C_6$ ($C_6^2:C_2^3$), $S_3$ ($C_2^4:C_{18}$)

Order 4: $C_2^2$ ($C_2\times D_6\times C_{18}$) x 2, $C_2^2$ ($C_{36}:D_6$), $C_4$ ($C_2\times D_6\times C_{18}$)

Order 3: $C_3$ ($C_6^2:C_2^4$), $C_3$ ($C_{36}:C_2^4$)

Order 2: $C_2$ ($C_2^2\times D_6\times C_{18}$), $C_2$ ($S_3\times D_4\times C_{18}$)

Order 1: $C_1$ ($D_4\times D_6\times C_{18}$)

Series

Derived series

$D_4\times D_6\times C_{18}$

$\rhd$

$C_6$

$\rhd$

$C_1$

Chief series

$D_4\times D_6\times C_{18}$

$\rhd$

$C_2^2\times D_6\times C_{18}$

$\rhd$

$C_2\times D_6\times C_{18}$

$\rhd$

$C_2\times C_6\times C_{18}$

$\rhd$

$C_6\times C_{18}$

$\rhd$

$C_3\times C_{18}$

$\rhd$

$C_{18}$

$\rhd$

$C_9$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$D_4\times D_6\times C_{18}$

$\rhd$

$C_6$

$\rhd$

$C_3$

Upper central series

$C_1$

$\lhd$

$C_2^2\times C_{18}$

$\lhd$

$C_2^4:C_{18}$

Character theory

Complex character table

The $540 \times 540$ character table is not available for this group.

Rational character table

See the $180 \times 180$ rational character table (warning: may be slow to load).