Abstract group 1728.44111: $C_6^3.C_2^3$

• Label: 1728.44111
• Order: \( 2^{6} \cdot 3^{3} \)
• Exponent: \( 2^{2} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{5} \)
• $\card{Z(G)}$: \( 2^{3} \)
• $\card{\Aut(G)}$: \( 2^{16} \cdot 3^{4} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{13} \cdot 3 \)
• Perm deg.: $19$
• Trans deg.: $288$
• Rank: $4$

Group information

Description:	$C_6^3.C_2^3$
Order:	\(1728\)\(\medspace = 2^{6} \cdot 3^{3} \)
Exponent:	\(12\)\(\medspace = 2^{2} \cdot 3 \)
Automorphism group:	$C_6^4:D_6$, of order \(5308416\)\(\medspace = 2^{16} \cdot 3^{4} \)
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	12
Elements	1	231	26	408	390	672	1728
Conjugacy classes	1	15	9	24	119	48	216
Divisions	1	15	9	14	99	22	160
Autjugacy classes	1	6	3	5	15	4	34

Dimension	1	2	4	8
Irr. complex chars.	32	104	80	0	216
Irr. rational chars.	16	52	74	18	160

Minimal presentations

Permutation degree:	$19$
Transitive degree:	$288$
Rank:	$4$
Inequivalent generating quadruples:	$425880$

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:	${\langle a, b, c, d, e \mid a^{4}=b^{2}=c^{6}=d^{6}=e^{6}=[a,b]=[a,d]=[a,e]= \!\cdots\! \rangle}$
Permutation group:	Degree $19$ $\langle(2,3)(5,6)(7,8,9,10), (7,8,9,10)(12,13)(17,19), (7,9)(8,10)(14,15), (16,17)(18,19), (16,18)(17,19), (7,9)(8,10), (1,2,3)(4,5,6), (1,3,2), (11,12,13)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 13 & 12 \\ 36 & 1 \end{array}\right), \left(\begin{array}{rr} 37 & 8 \\ 9 & 23 \end{array}\right), \left(\begin{array}{rr} 1 & 20 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 49 & 0 \\ 0 & 49 \end{array}\right), \left(\begin{array}{rr} 13 & 15 \\ 0 & 43 \end{array}\right), \left(\begin{array}{rr} 23 & 15 \\ 30 & 53 \end{array}\right), \left(\begin{array}{rr} 16 & 15 \\ 45 & 31 \end{array}\right), \left(\begin{array}{rr} 41 & 0 \\ 0 & 41 \end{array}\right), \left(\begin{array}{rr} 19 & 0 \\ 0 & 19 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/60\Z)$
Direct product:	$C_2$ $\, \times\, $ $(C_6^3.C_2^2)$
Semidirect product:	$(C_6^2.D_4)$ $\,\rtimes\,$ $S_3$	$(C_6.D_6^2)$ $\,\rtimes\,$ $C_2$	$(C_6^2.C_6)$ $\,\rtimes\,$ $D_4$	$(C_6^2:C_4)$ $\,\rtimes\,$ $D_6$	all 47
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_6^3$ . $C_2^3$ (2)	$C_6^2$ . $(D_4:S_3)$	$C_6^2$ . $(S_3\times D_4)$	$(C_6^2:C_4)$ . $D_6$ (2)	all 66

Elements of the group are displayed as matrices in $\GL_{2}(\Z/{60}\Z)$.

Homology

Abelianization:	$C_{2}^{3} \times C_{4} $
Schur multiplier:	$C_{2}^{5} \times C_{6}$
Commutator length:	$1$

Subgroups

There are 26176 subgroups in 3500 conjugacy classes, 508 normal (48 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2^3$	$G/Z \simeq$ $C_6:S_3^2$
Commutator:	$G' \simeq$ $C_3^2\times C_6$	$G/G' \simeq$ $C_2^3\times C_4$
Frattini:	$\Phi \simeq$ $C_2^2$	$G/\Phi \simeq$ $C_3:D_6^2$
Fitting:	$\operatorname{Fit} \simeq$ $C_2\times C_6^3$	$G/\operatorname{Fit} \simeq$ $C_2^2$
Radical:	$R \simeq$ $C_6^3.C_2^3$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_6^3$	$G/\operatorname{soc} \simeq$ $C_2^3$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_4^2:C_2^2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^3$

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

No diagram available

Normal subgroups

No diagram available: subgroups only stored up to automorphism

Normal subgroups up to automorphism

No diagram available

Classes of subgroups up to conjugation

Order 1728: $C_6^3.C_2^3$

Order 864: $C_6^3.C_2^2$ x 8, $C_6^3:C_2^2$, $C_6^3:C_4$, $C_6.D_6^2$, $C_6^3:C_4$, $C_6^3.C_2^2$, $C_6^2.D_{12}$, $C_6^3.C_2^2$

Order 576: $C_2^2.D_6^2$ x 4, $C_6^2.C_2^4$

Order 432: $C_6^3:C_2$ x 12, $C_6^2:C_{12}$ x 10, $C_6^2.D_6$ x 8, $C_6^2:C_{12}$ x 4, $C_6^2.D_6$ x 4, $C_6^2.D_6$ x 4, $C_6^2.D_6$ x 4, $C_6^2:C_{12}$ x 2, $C_2\times C_6^3$, $C_6^3:C_2$, $C_6^3.C_2$

Order 288: $C_2^3.S_3^2$ x 32, $C_6^2:D_4$ x 9, $C_6^2.C_2^3$ x 8, $C_2.D_6^2$ x 4, $C_6^2.D_4$ x 4, $C_6^2.D_4$ x 4, $C_6^2.D_4$ x 4, $C_2^3.S_3^2$ x 4, $C_6^2.C_2^3$ x 4, $C_6^2.D_4$, $C_6^2.D_4$, $C_6^2.D_4$, $C_6^2.C_2^3$, $C_6^2.C_2^3$, $C_6^2.C_2^3$

Order 216: $C_6^2.C_6$ x 18, $C_6^2:S_3$ x 16, $C_6^3$ x 11, $C_6^2:S_3$ x 10, $C_6.C_6^2$ x 8, $C_6.S_3^2$ x 8, $C_6^2.S_3$ x 6

Order 192: $C_2^4.D_6$ x 4, $C_2^4.D_6$

Order 144: $C_6^2:C_2^2$ x 108, $C_6^2:C_4$ x 48, $C_2^2.S_3^2$ x 32, $C_6^2:C_4$ x 19, $C_6^2:C_4$ x 16, $C_2^2.S_3^2$ x 16, $C_6.D_{12}$ x 16, $C_3^2:C_4^2$ x 16, $C_2^2\times C_6^2$ x 9, $C_6^2:C_2^2$ x 9, $C_{12}:D_6$ x 8, $C_6.D_{12}$ x 4, $C_6^2.C_2^2$ x 4, $C_3^2:C_4^2$ x 4, $C_6^2:C_4$ x 4, $C_2\times C_6\times C_{12}$ x 2

Order 108: $C_3\times C_6^2$ x 23, $C_3^2:D_6$ x 16, $C_3^2:C_{12}$ x 6, $C_3^3:C_4$ x 4, $C_3^2:C_{12}$ x 4

Order 96: $C_2^3.D_6$ x 32, $C_2^3:D_6$ x 9, $C_{12}:D_4$ x 8, $C_{12}:C_2^3$ x 5, $C_2^2.D_{12}$ x 5, $C_2^3.D_6$ x 5, $C_6:C_4^2$ x 5, $C_6.C_2^4$ x 4, $C_2^3:C_{12}$ x 4, $C_2^3\times C_{12}$, $C_2^3.D_6$

Order 72: $C_6^2:C_2$ x 144, $C_6:C_{12}$ x 104, $C_2\times C_6^2$ x 99, $C_6:D_6$ x 90, $C_6.D_6$ x 72, $C_6.D_6$ x 32, $C_6\times C_{12}$ x 8, $C_{12}:S_3$ x 8

Order 64: $C_4^2:C_2^2$

Order 54: $C_3^2\times C_6$ x 11, $C_3^2:S_3$ x 4

Order 48: $C_6:D_4$ x 108, $C_6.C_2^3$ x 51, $C_4\times D_6$ x 40, $D_6:C_4$ x 20, $C_6.D_4$ x 20, $C_{12}:C_4$ x 20, $C_2^2\times C_{12}$ x 18, $C_2^2:C_{12}$ x 16, $C_2^3\times C_6$ x 9, $C_2^2\times D_6$ x 9, $C_6.D_4$ x 4

Order 36: $C_6^2$ x 207, $C_6:S_3$ x 144, $C_3^2:C_4$ x 42, $C_3:C_{12}$ x 40, $C_3\times C_{12}$ x 4

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

Order 27: $C_3^3$

Order 24: $C_3:D_4$ x 144, $C_6:C_4$ x 134, $C_2^2\times C_6$ x 99, $C_2\times D_6$ x 90, $C_2\times C_{12}$ x 50, $C_4\times S_3$ x 40

Order 18: $C_3\times C_6$ x 99, $C_3:S_3$ x 36

Order 16: $C_2^2\times C_4$ x 21, $C_2\times D_4$ x 12, $C_2^2:C_4$ x 8, $C_4:C_4$ x 4, $C_4^2$ x 4, $C_2^4$ x 2

Order 12: $C_2\times C_6$ x 207, $D_6$ x 144, $C_3:C_4$ x 64, $C_{12}$ x 22

Order 9: $C_3^2$ x 9

Order 8: $C_2\times C_4$ x 40, $C_2^3$ x 21, $D_4$ x 16

Order 6: $C_6$ x 99, $S_3$ x 36

Order 4: $C_2^2$ x 39, $C_4$ x 14

Order 3: $C_3$ x 9

Order 2: $C_2$ x 15

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 1728: $C_6^3.C_2^3$

Order 864: $C_6^3:C_2^2$, $C_6^3:C_4$, $C_6.D_6^2$, $C_6^3:C_4$, $C_6^3.C_2^2$, $C_6^3.C_2^2$, $C_6^2.D_{12}$, $C_6^3.C_2^2$

Order 576: $C_6^2.C_2^4$, $C_2^2.D_6^2$

Order 432: $C_6^2:C_{12}$ x 4, $C_6^3:C_2$ x 2, $C_6^2:C_{12}$ x 2, $C_6^2.D_6$ x 2, $C_2\times C_6^3$, $C_6^3:C_2$, $C_6^3.C_2$, $C_6^2:C_{12}$, $C_6^2.D_6$, $C_6^2.D_6$, $C_6^2.D_6$

Order 288: $C_6^2:D_4$ x 3, $C_2.D_6^2$, $C_6^2.D_4$, $C_6^2.D_4$, $C_6^2.D_4$, $C_6^2.C_2^3$, $C_6^2.C_2^3$, $C_6^2.D_4$, $C_6^2.D_4$, $C_6^2.D_4$, $C_2^3.S_3^2$, $C_2^3.S_3^2$, $C_6^2.C_2^3$, $C_6^2.C_2^3$, $C_6^2.C_2^3$

Order 216: $C_6^3$ x 5, $C_6^2.C_6$ x 5, $C_6^2:S_3$ x 4, $C_6.C_6^2$ x 4, $C_6^2.S_3$ x 2, $C_6^2:S_3$, $C_6.S_3^2$

Order 192: $C_2^4.D_6$, $C_2^4.D_6$

Order 144: $C_6^2:C_4$ x 7, $C_6^2:C_2^2$ x 6, $C_6^2:C_4$ x 6, $C_2^2\times C_6^2$ x 3, $C_6^2:C_2^2$ x 3, $C_2\times C_6\times C_{12}$ x 2, $C_{12}:D_6$ x 2, $C_2^2.S_3^2$ x 2, $C_6.D_{12}$, $C_6^2.C_2^2$, $C_3^2:C_4^2$, $C_6^2:C_4$, $C_2^2.S_3^2$, $C_6.D_{12}$, $C_3^2:C_4^2$, $C_6^2:C_4$

Order 108: $C_3\times C_6^2$ x 8, $C_3^2:D_6$ x 4, $C_3^2:C_{12}$ x 2, $C_3^2:C_{12}$ x 2, $C_3^3:C_4$

Order 96: $C_2^3:D_6$ x 3, $C_{12}:C_2^3$ x 2, $C_2^2.D_{12}$ x 2, $C_2^3.D_6$ x 2, $C_6:C_4^2$ x 2, $C_2^3.D_6$, $C_2^3\times C_{12}$, $C_6.C_2^4$, $C_2^3:C_{12}$, $C_2^3.D_6$, $C_{12}:D_4$

Order 72: $C_2\times C_6^2$ x 15, $C_6:D_6$ x 12, $C_6.D_6$ x 11, $C_6:C_{12}$ x 9, $C_6\times C_{12}$ x 4, $C_6^2:C_2$ x 3, $C_{12}:S_3$, $C_6.D_6$

Order 64: $C_4^2:C_2^2$

Order 54: $C_3^2\times C_6$ x 5, $C_3^2:S_3$

Order 48: $C_6.C_2^3$ x 9, $C_2^2\times C_{12}$ x 6, $C_6:D_4$ x 6, $C_4\times D_6$ x 4, $C_2^3\times C_6$ x 3, $C_2^2\times D_6$ x 3, $D_6:C_4$ x 2, $C_6.D_4$ x 2, $C_{12}:C_4$ x 2, $C_2^2:C_{12}$, $C_6.D_4$

Order 36: $C_6^2$ x 24, $C_6:S_3$ x 12, $C_3^2:C_4$ x 5, $C_3:C_{12}$ x 4, $C_3\times C_{12}$ x 2

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

Order 27: $C_3^3$

Order 24: $C_6:C_4$ x 15, $C_2^2\times C_6$ x 15, $C_2\times D_6$ x 12, $C_2\times C_{12}$ x 9, $C_3:D_4$ x 3, $C_4\times S_3$ x 2

Order 18: $C_3\times C_6$ x 15, $C_3:S_3$ x 3

Order 16: $C_2^2\times C_4$ x 9, $C_2^2:C_4$ x 2, $C_2^4$ x 2, $C_2\times D_4$ x 2, $C_4:C_4$, $C_4^2$

Order 12: $C_2\times C_6$ x 24, $D_6$ x 12, $C_3:C_4$ x 7, $C_{12}$ x 4

Order 9: $C_3^2$ x 3

Order 8: $C_2\times C_4$ x 12, $C_2^3$ x 9, $D_4$

Order 6: $C_6$ x 15, $S_3$ x 3

Order 4: $C_2^2$ x 12, $C_4$ x 5

Order 3: $C_3$ x 3

Order 2: $C_2$ x 6

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1728: $C_6^3.C_2^3$ ($C_1$)

Order 864: $C_6^3.C_2^2$ ($C_2$) x 8, $C_6^3:C_2^2$ ($C_2$), $C_6^3:C_4$ ($C_2$), $C_6.D_6^2$ ($C_2$), $C_6^3:C_4$ ($C_2$), $C_6^3.C_2^2$ ($C_2$), $C_6^2.D_{12}$ ($C_2$), $C_6^3.C_2^2$ ($C_2$)

Order 432: $C_6^3:C_2$ ($C_4$) x 8, $C_6^2:C_{12}$ ($C_2^2$) x 6, $C_6^3:C_2$ ($C_2^2$) x 4, $C_6^2.D_6$ ($C_2^2$) x 4, $C_6^2:C_{12}$ ($C_2^2$) x 4, $C_6^2.D_6$ ($C_2^2$) x 4, $C_6^2.D_6$ ($C_2^2$) x 4, $C_6^2.D_6$ ($C_2^2$) x 4, $C_6^2:C_{12}$ ($C_2^2$) x 2, $C_2\times C_6^3$ ($C_2^2$), $C_6^3:C_2$ ($C_2^2$), $C_6^3.C_2$ ($C_2^2$)

Order 288: $C_6^2.D_4$ ($S_3$) x 4, $C_6^2.C_2^3$ ($S_3$)

Order 216: $C_6^2:S_3$ ($C_2\times C_4$) x 16, $C_6^3$ ($C_2\times C_4$) x 4, $C_6^2:S_3$ ($C_2\times C_4$) x 4, $C_6^2.S_3$ ($C_2\times C_4$) x 4, $C_6^2.C_6$ ($C_2^3$) x 4, $C_6^2.C_6$ ($D_4$) x 4, $C_6.C_6^2$ ($C_2^3$) x 4, $C_6^3$ ($C_2^3$) x 3, $C_6^2:S_3$ ($C_2^3$) x 2, $C_6^2.S_3$ ($C_2^3$) x 2

Order 144: $C_6^2:C_4$ ($D_6$) x 16, $C_6^2:C_4$ ($D_6$) x 8, $C_6^2:C_4$ ($D_6$) x 6, $C_2^2\times C_6^2$ ($D_6$) x 5

Order 108: $C_3\times C_6^2$ ($C_2^2\times C_4$) x 6, $C_3^2:D_6$ ($C_2^2\times C_4$) x 4, $C_3^3:C_4$ ($C_2^2\times C_4$) x 4, $C_3^2:C_{12}$ ($C_2\times D_4$) x 4, $C_3\times C_6^2$ ($D_4:C_2$) x 2, $C_3\times C_6^2$ ($C_2\times D_4$) x 2, $C_3\times C_6^2$ ($C_2^4$)

Order 96: $C_2^3.D_6$ ($C_3:S_3$)

Order 72: $C_2\times C_6^2$ ($C_4\times S_3$) x 20, $C_6:C_{12}$ ($C_2\times D_6$) x 16, $C_2\times C_6^2$ ($C_2\times D_6$) x 15, $C_6.D_6$ ($C_3:D_4$) x 4, $C_6.D_6$ ($C_2\times D_6$) x 4

Order 54: $C_3^2\times C_6$ ($C_4\times D_4$) x 4, $C_3^2\times C_6$ ($D_4:C_2^2$), $C_3^2\times C_6$ ($C_2^2\times D_4$), $C_3^2\times C_6$ ($C_2^3\times C_4$)

Order 48: $C_2^3\times C_6$ ($S_3^2$) x 4, $C_6.D_4$ ($C_6:S_3$) x 4, $C_6.C_2^3$ ($C_6:S_3$) x 2, $C_2^3\times C_6$ ($C_6:S_3$)

Order 36: $C_6^2$ ($C_4\times D_6$) x 30, $C_6^2$ ($D_4:S_3$) x 8, $C_6^2$ ($S_3\times D_4$) x 8, $C_6^2$ ($C_2^2\times D_6$) x 5, $C_3^2:C_4$ ($C_6:D_4$) x 4, $C_6^2$ ($C_6:D_4$) x 2, $C_6^2$ ($D_{12}:C_2$) x 2

Order 27: $C_3^3$ ($C_4^2:C_2^2$)

Order 24: $C_2^2\times C_6$ ($C_6.D_6$) x 16, $C_2^2\times C_6$ ($S_3\times D_6$) x 12, $C_6:C_4$ ($C_6:D_6$) x 4, $C_2^2\times C_6$ ($C_{12}:S_3$) x 4, $C_2^2\times C_6$ ($C_6:D_6$) x 3

Order 18: $C_3\times C_6$ ($C_2^3.D_6$) x 16, $C_3\times C_6$ ($C_{12}:C_2^3$) x 5, $C_3\times C_6$ ($D_4:D_6$) x 4, $C_3\times C_6$ ($D_4\times D_6$) x 4, $C_3\times C_6$ ($C_{12}:D_4$) x 4, $C_3\times C_6$ ($C_2^3:D_6$), $C_3\times C_6$ ($D_{12}:C_2^2$)

Order 16: $C_2^4$ ($C_3:S_3^2$)

Order 12: $C_2\times C_6$ ($C_2^2.S_3^2$) x 24, $C_2\times C_6$ ($D_6:D_6$) x 8, $C_2\times C_6$ ($D_6.D_6$) x 8, $C_2\times C_6$ ($C_{12}:D_6$) x 6, $C_2\times C_6$ ($D_6^2$) x 4, $C_2\times C_6$ ($C_{12}.D_6$) x 2, $C_2\times C_6$ ($C_{12}:D_6$) x 2, $C_2\times C_6$ ($C_6^2:C_2^2$)

Order 9: $C_3^2$ ($C_2^4.D_6$) x 4, $C_3^2$ ($C_2^4.D_6$)

Order 8: $C_2^3$ ($C_6.S_3^2$) x 4, $C_2^3$ ($C_6:S_3^2$) x 3

Order 6: $C_6$ ($C_2^3.S_3^2$) x 16, $C_6$ ($D_6^2:C_2$) x 4, $C_6$ ($C_2.D_6^2$) x 4, $C_6$ ($C_2.D_6^2$) x 4, $C_6$ ($C_6^2.C_2^3$) x 4, $C_6$ ($C_6^2.C_2^3$), $C_6$ ($C_6^2:C_2^3$), $C_6$ ($C_6^2.C_2^3$)

Order 4: $C_2^2$ ($C_6^2.D_6$) x 6, $C_2^2$ ($C_6^2:D_6$) x 2, $C_2^2$ ($C_6^2.D_6$) x 2, $C_2^2$ ($C_3:D_6^2$)

Order 3: $C_3$ ($C_2^2.D_6^2$) x 4, $C_3$ ($C_6^2.C_2^4$)

Order 2: $C_2$ ($C_6^3.C_2^2$) x 4, $C_2$ ($C_6^3:C_2^2$), $C_2$ ($C_6.D_6^2$), $C_2$ ($C_6.D_6^2$)

Order 1: $C_1$ ($C_6^3.C_2^3$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1728: $C_6^3.C_2^3$ ($C_1$)

Order 864: $C_6^3:C_2^2$ ($C_2$), $C_6^3:C_4$ ($C_2$), $C_6.D_6^2$ ($C_2$), $C_6^3:C_4$ ($C_2$), $C_6^3.C_2^2$ ($C_2$), $C_6^3.C_2^2$ ($C_2$), $C_6^2.D_{12}$ ($C_2$), $C_6^3.C_2^2$ ($C_2$)

Order 432: $C_6^2:C_{12}$ ($C_2^2$) x 3, $C_6^2:C_{12}$ ($C_2^2$) x 2, $C_2\times C_6^3$ ($C_2^2$), $C_6^3:C_2$ ($C_2^2$), $C_6^3:C_2$ ($C_2^2$), $C_6^3:C_2$ ($C_4$), $C_6^3.C_2$ ($C_2^2$), $C_6^2.D_6$ ($C_2^2$), $C_6^2:C_{12}$ ($C_2^2$), $C_6^2.D_6$ ($C_2^2$), $C_6^2.D_6$ ($C_2^2$), $C_6^2.D_6$ ($C_2^2$)

Order 288: $C_6^2.D_4$ ($S_3$), $C_6^2.C_2^3$ ($S_3$)

Order 216: $C_6^3$ ($C_2^3$) x 2, $C_6^2.C_6$ ($C_2^3$) x 2, $C_6.C_6^2$ ($C_2^3$) x 2, $C_6^3$ ($C_2\times C_4$), $C_6^2:S_3$ ($C_2^3$), $C_6^2:S_3$ ($C_2\times C_4$), $C_6^2:S_3$ ($C_2\times C_4$), $C_6^2.S_3$ ($C_2^3$), $C_6^2.S_3$ ($C_2\times C_4$), $C_6^2.C_6$ ($D_4$)

Order 144: $C_6^2:C_4$ ($D_6$) x 3, $C_2^2\times C_6^2$ ($D_6$) x 2, $C_6^2:C_4$ ($D_6$) x 2, $C_6^2:C_4$ ($D_6$)

Order 108: $C_3\times C_6^2$ ($C_2^2\times C_4$) x 2, $C_3\times C_6^2$ ($C_2^4$), $C_3\times C_6^2$ ($D_4:C_2$), $C_3\times C_6^2$ ($C_2\times D_4$), $C_3^2:D_6$ ($C_2^2\times C_4$), $C_3^3:C_4$ ($C_2^2\times C_4$), $C_3^2:C_{12}$ ($C_2\times D_4$)

Order 96: $C_2^3.D_6$ ($C_3:S_3$)

Order 72: $C_2\times C_6^2$ ($C_2\times D_6$) x 4, $C_2\times C_6^2$ ($C_4\times S_3$) x 2, $C_6.D_6$ ($C_2\times D_6$) x 2, $C_6:C_{12}$ ($C_2\times D_6$) x 2, $C_6.D_6$ ($C_3:D_4$)

Order 54: $C_3^2\times C_6$ ($D_4:C_2^2$), $C_3^2\times C_6$ ($C_2^2\times D_4$), $C_3^2\times C_6$ ($C_2^3\times C_4$), $C_3^2\times C_6$ ($C_4\times D_4$)

Order 48: $C_6.C_2^3$ ($C_6:S_3$) x 2, $C_2^3\times C_6$ ($C_6:S_3$), $C_2^3\times C_6$ ($S_3^2$), $C_6.D_4$ ($C_6:S_3$)

Order 36: $C_6^2$ ($C_4\times D_6$) x 4, $C_6^2$ ($C_2^2\times D_6$) x 2, $C_3^2:C_4$ ($C_6:D_4$), $C_6^2$ ($C_6:D_4$), $C_6^2$ ($D_4:S_3$), $C_6^2$ ($S_3\times D_4$), $C_6^2$ ($D_{12}:C_2$)

Order 27: $C_3^3$ ($C_4^2:C_2^2$)

Order 24: $C_6:C_4$ ($C_6:D_6$) x 2, $C_2^2\times C_6$ ($C_6:D_6$) x 2, $C_2^2\times C_6$ ($S_3\times D_6$) x 2, $C_2^2\times C_6$ ($C_{12}:S_3$), $C_2^2\times C_6$ ($C_6.D_6$)

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

Order 16: $C_2^4$ ($C_3:S_3^2$)

Order 12: $C_2\times C_6$ ($C_{12}:D_6$) x 2, $C_2\times C_6$ ($C_2^2.S_3^2$) x 2, $C_2\times C_6$ ($C_6^2:C_2^2$), $C_2\times C_6$ ($D_6^2$), $C_2\times C_6$ ($C_{12}.D_6$), $C_2\times C_6$ ($C_{12}:D_6$), $C_2\times C_6$ ($D_6:D_6$), $C_2\times C_6$ ($D_6.D_6$)

Order 9: $C_3^2$ ($C_2^4.D_6$), $C_3^2$ ($C_2^4.D_6$)

Order 8: $C_2^3$ ($C_6:S_3^2$) x 2, $C_2^3$ ($C_6.S_3^2$)

Order 6: $C_6$ ($D_6^2:C_2$), $C_6$ ($C_2.D_6^2$), $C_6$ ($C_2.D_6^2$), $C_6$ ($C_6^2.C_2^3$), $C_6$ ($C_2^3.S_3^2$), $C_6$ ($C_6^2.C_2^3$), $C_6$ ($C_6^2:C_2^3$), $C_6$ ($C_6^2.C_2^3$)

Order 4: $C_2^2$ ($C_6^2.D_6$) x 2, $C_2^2$ ($C_3:D_6^2$), $C_2^2$ ($C_6^2:D_6$), $C_2^2$ ($C_6^2.D_6$)

Order 3: $C_3$ ($C_6^2.C_2^4$), $C_3$ ($C_2^2.D_6^2$)

Order 2: $C_2$ ($C_6^3:C_2^2$), $C_2$ ($C_6.D_6^2$), $C_2$ ($C_6.D_6^2$), $C_2$ ($C_6^3.C_2^2$)

Order 1: $C_1$ ($C_6^3.C_2^3$)

Series

Derived series

$C_6^3.C_2^3$

$\rhd$

$C_3^2\times C_6$

$\rhd$

$C_1$

Chief series

$C_6^3.C_2^3$

$\rhd$

$C_6.D_6^2$

$\rhd$

$C_6^2:C_{12}$

$\rhd$

$C_6^3$

$\rhd$

$C_3\times C_6^2$

$\rhd$

$C_6^2$

$\rhd$

$C_3\times C_6$

$\rhd$

$C_6$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$C_6^3.C_2^3$

$\rhd$

$C_3^2\times C_6$

$\rhd$

$C_3^3$

Upper central series

$C_1$

$\lhd$

$C_2^3$

$\lhd$

$C_2^4$

Character theory

Complex character table

See the $216 \times 216$ 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 $160 \times 160$ rational character table (warning: may be slow to load).