Abstract group 1728.18843: $D_{18}:(C_6\times D_4)$

• Label: 1728.18843
• Order: \( 2^{6} \cdot 3^{3} \)
• Exponent: \( 2^{2} \cdot 3^{2} \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \cdot 3 \)
• $\card{Z(G)}$: \( 2^{2} \)
• $\card{\Aut(G)}$: \( 2^{10} \cdot 3^{4} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{6} \cdot 3 \)
• Perm deg.: $17$
• Trans deg.: $72$
• Rank: $4$

Group information

Description:	$D_{18}:(C_6\times D_4)$
Order:	\(1728\)\(\medspace = 2^{6} \cdot 3^{3} \)
Exponent:	\(36\)\(\medspace = 2^{2} \cdot 3^{2} \)
Automorphism group:	$D_9:C_3\times C_2^6:C_3.D_4$, of order \(82944\)\(\medspace = 2^{10} \cdot 3^{4} \)
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	199	8	120	512	18	312	342	216	1728
Conjugacy classes	1	21	3	6	52	3	15	30	9	140
Divisions	1	21	2	6	31	2	9	20	6	98
Autjugacy classes	1	6	3	2	15	3	5	9	3	47

Dimension	1	2	4	6	8	12
Irr. complex chars.	48	60	18	8	0	6	140
Irr. rational chars.	16	36	26	8	6	6	98

Minimal presentations

Permutation degree:	$17$
Transitive degree:	$72$
Rank:	$4$
Inequivalent generating quadruples:	$31449600$

Minimal degrees of faithful linear representations

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

Constructions

Presentation:	${\langle a, b, c, d, e \mid a^{2}=b^{2}=c^{4}=d^{6}=e^{18}=[a,d]=[a,e]=[b,d]= \!\cdots\! \rangle}$
Permutation group:	Degree $17$ $\langle(2,4)(3,8)(5,6)(7,9), (10,11)(12,16)(13,14)(15,17), (12,16)(15,17), (10,12) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 0 \\ 0 & 25 \end{array}\right), \left(\begin{array}{rr} 1 & 18 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 17 & 0 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 19 & 18 \\ 18 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 18 \\ 18 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 9 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 19 & 0 \\ 0 & 19 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/36\Z)$
Direct product:	$(C_2^2\wr C_2)$ $\, \times\, $ $(C_9:C_6)$
Semidirect product:	$(D_4\times D_{18})$ $\,\rtimes\,$ $C_6$	$(D_{18}:D_4)$ $\,\rtimes\,$ $C_6$	$(D_{18}.D_4)$ $\,\rtimes\,$ $C_6$	$(D_{18}:C_6)$ $\,\rtimes\,$ $D_4$	all 56
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$D_{18}$ . $(C_6\times D_4)$	$(C_6^2:D_4)$ . $S_3$	$C_6^2$ . $(S_3\times D_4)$	$(C_4:C_6^2)$ . $D_6$	all 35
Aut. group:	$\Aut(C_{36}:C_4)$	$\Aut(C_{36}:C_4)$	$\Aut(C_{18}.D_4)$	$\Aut(C_4\times D_{18})$	all 12

Elements of the group are displayed as matrices in $\GL_{2}(\Z/{36}\Z)$.

Homology

Abelianization:	$C_{2}^{3} \times C_{6} \simeq C_{2}^{4} \times C_{3}$
Schur multiplier:	$C_{2}^{7}$
Commutator length:	$1$

Subgroups

There are 13098 subgroups in 2051 conjugacy classes, 288 normal (37 characteristic).

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

Special subgroups

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

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

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Classes of subgroups up to automorphism

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Normal subgroups

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Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 1728: $D_{18}:(C_6\times D_4)$

Order 864: $C_2\times D_{36}:C_6$ x 3, $(D_4\times C_{18}):C_6$ x 3, $C_6^2.D_{12}$ x 3, $(C_2\times D_{18}):C_{12}$ x 3, $C_2^3\times C_{18}:C_6$, $(D_4\times C_{18}):C_6$, $(C_2^3\times C_{18}):C_6$

Order 576: $C_6^2.C_2^4$, $C_2^4:D_{18}$

Order 432: $C_6^2.D_6$ x 19, $D_{36}:C_6$ x 12, $C_6^2.D_6$ x 6, $D_{18}:C_{12}$ x 6, $C_{12}.C_6^2$ x 3, $D_{36}:C_6$ x 3, $D_{18}:C_{12}$ x 3, $C_6^2.C_{12}$ x 3, $C_6^2.D_6$ x 3, $C_2^2\times C_{18}:C_6$

Order 288: $C_6^2:C_2^3$ x 3, $C_2^2:D_{36}$ x 3, $D_{18}.D_4$ x 3, $C_6^2:D_4$ x 3, $C_6^2:D_4$ x 3, $C_6^2.C_2^3$ x 3, $D_4\times D_{18}$ x 3, $D_{18}:D_4$ x 3, $C_2^4:C_{18}$ x 2, $C_2^3\times D_{18}$, $C_6^2:D_4$, $C_6^2:D_4$, $C_2^4:D_9$, $C_6^2:C_2^3$

Order 216: $D_{18}:C_6$ x 85, $D_{18}:C_6$ x 12, $C_6.C_6^2$ x 10, $C_{36}:C_6$ x 6, $C_{36}:C_6$ x 6, $C_{36}:C_6$ x 6, $C_{18}:C_{12}$ x 3, $C_{18}:C_{12}$ x 3

Order 192: $C_2^5:C_6$, $C_2^4:D_6$

Order 144: $C_6^2:C_2^2$ x 19, $C_2^2\times D_{18}$ x 19, $D_4\times D_9$ x 12, $C_{12}:D_6$ x 12, $D_6:C_{12}$ x 6, $D_4\times C_{18}$ x 6, $C_{18}:D_4$ x 6, $C_2^2:C_{36}$ x 6, $C_6^2:C_2^2$ x 6, $D_{18}:C_4$ x 6, $C_6^2:C_4$ x 3, $C_2\times D_{36}$ x 3, $C_4\times D_{18}$ x 3, $C_{18}.D_4$ x 3, $C_4:C_6^2$ x 3, $C_6\times D_{12}$ x 3, $C_{12}\times D_6$ x 3, $C_6^2:C_4$ x 3, $C_2^3\times C_{18}$ x 2, $C_2^2\times C_6^2$

Order 108: $C_{18}:C_6$ x 73, $C_{18}:C_6$ x 24, $C_9:C_{12}$ x 3, $C_9:C_{12}$ x 3

Order 96: $C_2^4:C_6$ x 9, $C_2^2:D_{12}$ x 3, $D_6.D_4$ x 3, $C_{12}:C_2^3$ x 3, $D_4\times D_6$ x 3, $C_2^3:C_{12}$ x 3, $D_6:D_4$ x 3, $C_2^4\times C_6$, $C_2^3\times D_6$, $C_2^4:S_3$

Order 72: $C_6\times D_6$ x 85, $C_2\times D_{18}$ x 85, $C_2^2\times C_{18}$ x 20, $C_9:D_4$ x 12, $C_6\wr C_2$ x 12, $D_4\times C_9$ x 12, $C_2\times C_6^2$ x 10, $C_2\times C_{36}$ x 6, $D_{36}$ x 6, $C_4\times D_9$ x 6, $D_4\times C_3^2$ x 6, $C_3\times D_{12}$ x 6, $S_3\times C_{12}$ x 6, $C_{18}:C_4$ x 3, $C_6\times C_{12}$ x 3, $C_6:C_{12}$ x 3

Order 64: $C_2^3:D_4$

Order 54: $C_9:C_6$ x 11, $C_9:C_6$ x 10

Order 48: $C_6\times D_4$ x 27, $C_2^3\times C_6$ x 21, $C_2^2\times D_6$ x 19, $C_2^2:C_{12}$ x 15, $S_3\times D_4$ x 12, $C_6:D_4$ x 6, $D_6:C_4$ x 6, $C_2^2\times C_{12}$ x 3, $C_2\times D_{12}$ x 3, $C_4\times D_6$ x 3, $C_6.D_4$ x 3

Order 36: $D_{18}$ x 73, $C_6\times S_3$ x 73, $C_2\times C_{18}$ x 48, $C_6^2$ x 24, $C_{36}$ x 6, $C_3\times C_{12}$ x 3, $C_3:C_{12}$ x 3, $C_9:C_4$ x 3

Order 32: $C_2^2\wr C_2$ x 8, $C_2^2\times D_4$ x 3, $C_2^3:C_4$ x 3, $C_2^5$

Order 27: $C_9:C_3$

Order 24: $C_2^2\times C_6$ x 105, $C_2\times D_6$ x 85, $C_3\times D_4$ x 30, $C_2\times C_{12}$ x 15, $C_3:D_4$ x 12, $D_{12}$ x 6, $C_4\times S_3$ x 6, $C_6:C_4$ x 3

Order 18: $C_{18}$ x 20, $C_3\times S_3$ x 11, $D_9$ x 11, $C_3\times C_6$ x 10

Order 16: $C_2\times D_4$ x 24, $C_2^4$ x 20, $C_2^2:C_4$ x 12, $C_2^2\times C_4$ x 3

Order 12: $C_2\times C_6$ x 121, $D_6$ x 73, $C_{12}$ x 9, $C_3:C_4$ x 3

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

Order 8: $C_2^3$ x 95, $D_4$ x 24, $C_2\times C_4$ x 12

Order 6: $C_6$ x 31, $S_3$ x 11

Order 4: $C_2^2$ x 97, $C_4$ x 6

Order 3: $C_3$ x 2

Order 2: $C_2$ x 21

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 1728: $D_{18}:(C_6\times D_4)$

Order 864: $C_2^3\times C_{18}:C_6$, $C_2\times D_{36}:C_6$, $(D_4\times C_{18}):C_6$, $(C_2^3\times C_{18}):C_6$, $(D_4\times C_{18}):C_6$, $C_6^2.D_{12}$, $(C_2\times D_{18}):C_{12}$

Order 576: $C_6^2.C_2^4$, $C_2^4:D_{18}$

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

Order 288: $C_2^4:C_{18}$ x 2, $C_6^2:C_2^3$, $C_2^2:D_{36}$, $D_{18}.D_4$, $C_2^3\times D_{18}$, $C_6^2:D_4$, $C_6^2:D_4$, $C_6^2:D_4$, $C_6^2:D_4$, $C_6^2.C_2^3$, $D_4\times D_{18}$, $C_2^4:D_9$, $D_{18}:D_4$, $C_6^2:C_2^3$

Order 216: $D_{18}:C_6$ x 9, $C_6.C_6^2$ x 3, $D_{18}:C_6$ x 2, $C_{36}:C_6$, $C_{18}:C_{12}$, $C_{18}:C_{12}$, $C_{36}:C_6$, $C_{36}:C_6$

Order 192: $C_2^5:C_6$, $C_2^4:D_6$

Order 144: $C_6^2:C_2^2$ x 4, $C_2^2\times D_{18}$ x 4, $D_4\times C_{18}$ x 2, $C_{18}:D_4$ x 2, $C_2^2:C_{36}$ x 2, $C_6^2:C_2^2$ x 2, $C_2^3\times C_{18}$ x 2, $C_6^2:C_4$, $D_6:C_{12}$, $D_4\times D_9$, $C_2\times D_{36}$, $C_4\times D_{18}$, $C_2^2\times C_6^2$, $C_{18}.D_4$, $C_4:C_6^2$, $C_{12}:D_6$, $C_6\times D_{12}$, $C_{12}\times D_6$, $D_{18}:C_4$, $C_6^2:C_4$

Order 108: $C_{18}:C_6$ x 7, $C_{18}:C_6$ x 5, $C_9:C_{12}$, $C_9:C_{12}$

Order 96: $C_2^4:C_6$ x 5, $C_2^2:D_{12}$, $D_6.D_4$, $C_2^4\times C_6$, $C_2^3\times D_6$, $C_{12}:C_2^3$, $D_4\times D_6$, $C_2^3:C_{12}$, $C_2^4:S_3$, $D_6:D_4$

Order 72: $C_6\times D_6$ x 9, $C_2\times D_{18}$ x 9, $C_2^2\times C_{18}$ x 6, $C_2\times C_6^2$ x 3, $C_2\times C_{36}$ x 2, $C_9:D_4$ x 2, $C_6\wr C_2$ x 2, $D_4\times C_9$ x 2, $C_{18}:C_4$, $D_{36}$, $C_4\times D_9$, $D_4\times C_3^2$, $C_6\times C_{12}$, $C_6:C_{12}$, $C_3\times D_{12}$, $S_3\times C_{12}$

Order 64: $C_2^3:D_4$

Order 54: $C_9:C_6$ x 3, $C_9:C_6$ x 3

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

Order 36: $C_2\times C_{18}$ x 10, $D_{18}$ x 7, $C_6\times S_3$ x 7, $C_6^2$ x 5, $C_{36}$ x 2, $C_3\times C_{12}$, $C_3:C_{12}$, $C_9:C_4$

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

Order 27: $C_9:C_3$

Order 24: $C_2^2\times C_6$ x 15, $C_2\times D_6$ x 9, $C_3\times D_4$ x 5, $C_2\times C_{12}$ x 4, $C_3:D_4$ x 2, $C_6:C_4$, $D_{12}$, $C_4\times S_3$

Order 18: $C_{18}$ x 6, $C_3\times C_6$ x 3, $C_3\times S_3$ x 3, $D_9$ x 3

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

Order 12: $C_2\times C_6$ x 17, $D_6$ x 7, $C_{12}$ x 3, $C_3:C_4$

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

Order 8: $C_2^3$ x 12, $D_4$ x 4, $C_2\times C_4$ x 3

Order 6: $C_6$ x 9, $S_3$ x 3

Order 4: $C_2^2$ x 12, $C_4$ x 2

Order 3: $C_3$ x 2

Order 2: $C_2$ x 6

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1728: $D_{18}:(C_6\times D_4)$ ($C_1$)

Order 864: $C_2\times D_{36}:C_6$ ($C_2$) x 3, $(D_4\times C_{18}):C_6$ ($C_2$) x 3, $C_6^2.D_{12}$ ($C_2$) x 3, $(C_2\times D_{18}):C_{12}$ ($C_2$) x 3, $C_2^3\times C_{18}:C_6$ ($C_2$), $(D_4\times C_{18}):C_6$ ($C_2$), $(C_2^3\times C_{18}):C_6$ ($C_2$)

Order 576: $C_2^4:D_{18}$ ($C_3$)

Order 432: $C_6^2.D_6$ ($C_2^2$) x 7, $C_6^2.D_6$ ($C_2^2$) x 6, $D_{18}:C_{12}$ ($C_2^2$) x 6, $C_{12}.C_6^2$ ($C_2^2$) x 3, $D_{36}:C_6$ ($C_2^2$) x 3, $D_{18}:C_{12}$ ($C_2^2$) x 3, $C_6^2.C_{12}$ ($C_2^2$) x 3, $C_6^2.D_6$ ($C_2^2$) x 3, $C_2^2\times C_{18}:C_6$ ($C_2^2$)

Order 288: $C_2^2:D_{36}$ ($C_6$) x 3, $D_{18}.D_4$ ($C_6$) x 3, $D_4\times D_{18}$ ($C_6$) x 3, $D_{18}:D_4$ ($C_6$) x 3, $C_2^3\times D_{18}$ ($C_6$), $C_6^2:D_4$ ($S_3$), $C_2^4:C_{18}$ ($C_6$), $C_2^4:D_9$ ($C_6$)

Order 216: $D_{18}:C_6$ ($D_4$) x 12, $D_{18}:C_6$ ($C_2^3$) x 5, $C_6.C_6^2$ ($C_2^3$) x 4, $C_{18}:C_{12}$ ($C_2^3$) x 3, $C_{18}:C_{12}$ ($C_2^3$) x 3

Order 144: $C_2^2\times D_{18}$ ($C_2\times C_6$) x 7, $C_{18}:D_4$ ($C_2\times C_6$) x 6, $D_{18}:C_4$ ($C_2\times C_6$) x 6, $D_4\times C_{18}$ ($C_2\times C_6$) x 3, $C_2\times D_{36}$ ($C_2\times C_6$) x 3, $C_4\times D_{18}$ ($C_2\times C_6$) x 3, $C_2^2:C_{36}$ ($C_2\times C_6$) x 3, $C_{18}.D_4$ ($C_2\times C_6$) x 3, $C_4:C_6^2$ ($D_6$) x 3, $C_6^2:C_4$ ($D_6$) x 3, $C_2^2\times C_6^2$ ($D_6$), $C_2^3\times C_{18}$ ($C_2\times C_6$)

Order 108: $C_{18}:C_6$ ($C_2\times D_4$) x 12, $C_{18}:C_6$ ($C_2\times D_4$) x 6, $C_{18}:C_6$ ($C_2^4$)

Order 96: $C_2^4:C_6$ ($C_3\times S_3$)

Order 72: $C_2\times D_{18}$ ($C_3\times D_4$) x 12, $C_2\times D_{18}$ ($C_2^2\times C_6$) x 5, $C_2\times C_6^2$ ($C_2\times D_6$) x 4, $C_2^2\times C_{18}$ ($C_2^2\times C_6$) x 4, $C_2\times C_{36}$ ($C_2^2\times C_6$) x 3, $C_{18}:C_4$ ($C_2^2\times C_6$) x 3, $C_6\times C_{12}$ ($C_2\times D_6$) x 3

Order 54: $C_9:C_6$ ($C_2^2\wr C_2$) x 4, $C_9:C_6$ ($C_2^2\times D_4$) x 3

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

Order 36: $D_{18}$ ($C_6\times D_4$) x 12, $C_2\times C_{18}$ ($C_6\times D_4$) x 6, $C_6^2$ ($S_3\times D_4$) x 6, $C_2\times C_{18}$ ($C_2^3\times C_6$), $C_6^2$ ($C_2^2\times D_6$)

Order 32: $C_2^2\wr C_2$ ($C_9:C_6$)

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

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

Order 18: $D_9$ ($C_2^4:C_6$) x 4, $C_3\times C_6$ ($D_4\times D_6$) x 3, $C_{18}$ ($C_{12}:C_2^3$) x 3

Order 16: $C_2^2:C_4$ ($C_{18}:C_6$) x 3, $C_2\times D_4$ ($C_{18}:C_6$) x 3, $C_2^4$ ($C_{18}:C_6$)

Order 12: $C_2\times C_6$ ($C_{12}:D_6$) x 6, $C_2\times C_6$ ($C_6^2:C_2^2$)

Order 9: $C_3^2$ ($C_2^4:D_6$), $C_9$ ($C_2^5:C_6$)

Order 8: $C_2^3$ ($D_{18}:C_6$) x 4, $C_2\times C_4$ ($D_{18}:C_6$) x 3

Order 6: $C_6$ ($C_6^2:C_2^3$) x 3

Order 4: $C_2^2$ ($D_{36}:C_6$) x 6, $C_2^2$ ($C_6^2.D_6$)

Order 3: $C_3$ ($C_6^2.C_2^4$)

Order 2: $C_2$ ($C_2\times D_{36}:C_6$) x 3

Order 1: $C_1$ ($D_{18}:(C_6\times D_4)$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1728: $D_{18}:(C_6\times D_4)$ ($C_1$)

Order 864: $C_2^3\times C_{18}:C_6$ ($C_2$), $C_2\times D_{36}:C_6$ ($C_2$), $(D_4\times C_{18}):C_6$ ($C_2$), $(C_2^3\times C_{18}):C_6$ ($C_2$), $(D_4\times C_{18}):C_6$ ($C_2$), $C_6^2.D_{12}$ ($C_2$), $(C_2\times D_{18}):C_{12}$ ($C_2$)

Order 576: $C_2^4:D_{18}$ ($C_3$)

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

Order 288: $C_2^2:D_{36}$ ($C_6$), $D_{18}.D_4$ ($C_6$), $C_2^3\times D_{18}$ ($C_6$), $C_6^2:D_4$ ($S_3$), $D_4\times D_{18}$ ($C_6$), $C_2^4:C_{18}$ ($C_6$), $C_2^4:D_9$ ($C_6$), $D_{18}:D_4$ ($C_6$)

Order 216: $D_{18}:C_6$ ($C_2^3$) x 3, $C_6.C_6^2$ ($C_2^3$) x 2, $C_{18}:C_{12}$ ($C_2^3$), $C_{18}:C_{12}$ ($C_2^3$), $D_{18}:C_6$ ($D_4$)

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

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

Order 96: $C_2^4:C_6$ ($C_3\times S_3$)

Order 72: $C_2\times D_{18}$ ($C_2^2\times C_6$) x 3, $C_2\times C_6^2$ ($C_2\times D_6$) x 2, $C_2^2\times C_{18}$ ($C_2^2\times C_6$) x 2, $C_2\times C_{36}$ ($C_2^2\times C_6$), $C_{18}:C_4$ ($C_2^2\times C_6$), $C_6\times C_{12}$ ($C_2\times D_6$), $C_2\times D_{18}$ ($C_3\times D_4$)

Order 54: $C_9:C_6$ ($C_2^2\wr C_2$), $C_9:C_6$ ($C_2^2\times D_4$)

Order 48: $C_2^3\times C_6$ ($C_6\times S_3$), $C_6\times D_4$ ($C_6\times S_3$), $C_2^2:C_{12}$ ($C_6\times S_3$)

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

Order 32: $C_2^2\wr C_2$ ($C_9:C_6$)

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

Order 24: $C_2^2\times C_6$ ($C_6\times D_6$) x 2, $C_2\times C_{12}$ ($C_6\times D_6$)

Order 18: $C_3\times C_6$ ($D_4\times D_6$), $C_{18}$ ($C_{12}:C_2^3$), $D_9$ ($C_2^4:C_6$)

Order 16: $C_2^2:C_4$ ($C_{18}:C_6$), $C_2^4$ ($C_{18}:C_6$), $C_2\times D_4$ ($C_{18}:C_6$)

Order 12: $C_2\times C_6$ ($C_6^2:C_2^2$), $C_2\times C_6$ ($C_{12}:D_6$)

Order 9: $C_3^2$ ($C_2^4:D_6$), $C_9$ ($C_2^5:C_6$)

Order 8: $C_2^3$ ($D_{18}:C_6$) x 2, $C_2\times C_4$ ($D_{18}:C_6$)

Order 6: $C_6$ ($C_6^2:C_2^3$)

Order 4: $C_2^2$ ($C_6^2.D_6$), $C_2^2$ ($D_{36}:C_6$)

Order 3: $C_3$ ($C_6^2.C_2^4$)

Order 2: $C_2$ ($C_2\times D_{36}:C_6$)

Order 1: $C_1$ ($D_{18}:(C_6\times D_4)$)

Series

Derived series

$D_{18}:(C_6\times D_4)$

$\rhd$

$C_2\times C_{18}$

$\rhd$

$C_1$

Chief series

$D_{18}:(C_6\times D_4)$

$\rhd$

$(D_4\times C_{18}):C_6$

$\rhd$

$C_{12}.C_6^2$

$\rhd$

$C_6.C_6^2$

$\rhd$

$C_{18}:C_6$

$\rhd$

$C_9:C_6$

$\rhd$

$C_{18}$

$\rhd$

$C_9$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$D_{18}:(C_6\times D_4)$

$\rhd$

$C_2\times C_{18}$

$\rhd$

$C_9$

Upper central series

$C_1$

$\lhd$

$C_2^2$

$\lhd$

$C_2^2\wr C_2$

Supergroups

This group is a maximal subgroup of 1 larger groups in the database.

This group is a maximal quotient of 1 larger groups in the database.

Character theory

Complex character table

See the $140 \times 140$ character table (warning: may be slow to load). Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $98 \times 98$ rational character table.