Abstract group 576.8609: $C_6^2:C_2^4$

• Label: 576.8609
• Order: \( 2^{6} \cdot 3^{2} \)
• Exponent: \( 2^{2} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{5} \cdot 3 \)
• $\card{Z(G)}$: \( 2^{3} \cdot 3 \)
• $\card{\Aut(G)}$: \( 2^{15} \cdot 3^{2} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{12} \cdot 3 \)
• Perm deg.: $14$
• Trans deg.: $96$
• Rank: $5$

Group information

Description:	$C_6^2:C_2^4$
Order:	\(576\)\(\medspace = 2^{6} \cdot 3^{2} \)
Exponent:	\(12\)\(\medspace = 2^{2} \cdot 3 \)
Automorphism group:	$C_2^7.C_2^4.D_6^2$, of order \(294912\)\(\medspace = 2^{15} \cdot 3^{2} \)
Composition factors:	$C_2$ x 6, $C_3$ x 2
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	95	8	32	328	112	576
Conjugacy classes	1	31	5	8	107	28	180
Divisions	1	31	3	8	61	16	120
Autjugacy classes	1	5	3	2	11	4	26

Dimension	1	2	4	8
Irr. complex chars.	96	72	12	0	180
Irr. rational chars.	32	56	28	4	120

Minimal presentations

Permutation degree:	$14$
Transitive degree:	$96$
Rank:	$5$
Inequivalent generating 5-tuples:	$63016800$

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:	$\langle a, b, c, d, e \mid a^{2}=b^{2}=c^{2}=d^{6}=e^{12}=[a,b]=[a,c]=[a,d]=[b,c]=[b,d]=[b,e]=[c,d]=[c,e]=1, e^{a}=e^{7}, e^{d}=e^{11} \rangle$
Permutation group:	Degree $14$ $\langle(2,3)(4,5)(6,7), (4,5)(6,7), (4,6)(5,7)(8,9)(10,11), (4,7)(5,6)(10,11), (8,10)(9,11), (12,13,14), (8,9)(10,11), (1,2,3)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{rrrrrr} -1 & 0 & 0 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{array}\right), \left(\begin{array}{rrrrrr} 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{array}\right), \left(\begin{array}{rrrrrr} 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & -1 \end{array}\right), \left(\begin{array}{rrrrrr} 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & -1 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{array}\right), \left(\begin{array}{rrrrrr} 0 & -1 & 0 & 0 & 0 & 0 \\ 1 & -1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1 & 0 & 0 \\ 0 & 0 & 1 & -1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 1 & 0 \end{array}\right) \right\rangle \subseteq \GL_{6}(\Z)$
	$\left\langle \left(\begin{array}{rr} 0 & 9 \\ 25 & 0 \end{array}\right), \left(\begin{array}{rr} 25 & 0 \\ 0 & 25 \end{array}\right), \left(\begin{array}{rr} 15 & 0 \\ 14 & 15 \end{array}\right), \left(\begin{array}{rr} 20 & 21 \\ 21 & 20 \end{array}\right), \left(\begin{array}{rr} 1 & 14 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 25 & 0 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 14 \\ 14 & 1 \end{array}\right), \left(\begin{array}{rr} 13 & 0 \\ 0 & 13 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/28\Z)$
Direct product:	$C_2$ ${}^2$ $\, \times\, $ $C_3$ $\, \times\, $ $S_3$ $\, \times\, $ $D_4$
Semidirect product:	$C_6^2$ $\,\rtimes\,$ $C_2^4$	$(C_6\times S_3)$ $\,\rtimes\,$ $C_2^4$	$C_2^4$ $\,\rtimes\,$ $(C_6\times S_3)$	$(C_6^2:D_4)$ $\,\rtimes\,$ $C_2$	all 68
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_6^2$ . $C_2^4$	$(C_6\times S_3)$ . $C_2^4$	$(C_6\times D_6)$ . $C_2^3$	$D_6$ . $(C_2^3\times C_6)$	all 18
Aut. group:	$\Aut(C_{168}:C_2)$	$\Aut(C_6:C_{56})$	$\Aut(C_{12}.C_{28})$	$\Aut(C_{21}:D_8)$	all 23

Elements of the group are displayed as matrices in $\GL_{2}(\Z/{28}\Z)$.

Homology

Abelianization:	$C_{2}^{4} \times C_{6} \simeq C_{2}^{5} \times C_{3}$
Schur multiplier:	$C_{2}^{10}$
Commutator length:	$1$

Subgroups

There are 7180 subgroups in 2842 conjugacy classes, 1006 normal (30 characteristic).

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

Special subgroups

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

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 576: $C_6^2:C_2^4$

Order 288: $C_6^2:C_2^3$ x 24, $C_6^2:C_2^3$ x 2, $C_6^2:D_4$ x 2, $C_6^2:D_4$, $C_6^2.C_2^3$, $D_4\times C_6^2$

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

Order 144: $C_{12}:D_6$ x 64, $C_6^2:C_2^2$ x 43, $C_6^2:C_2^2$ x 24, $C_4:C_6^2$ 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 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: $C_6\times D_6$ x 162, $C_6\wr C_2$ 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 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: $C_6\times S_3$ x 124, $C_6^2$ x 39, $C_3\times C_{12}$ 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 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: $C_3\times S_3$ 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: $C_3^2$

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 576: $C_6^2:C_2^4$

Order 288: $C_6^2:C_2^3$, $C_6^2:D_4$, $C_6^2.C_2^3$, $C_6^2:C_2^3$, $D_4\times C_6^2$, $C_6^2:D_4$

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

Order 144: $C_6^2:C_2^2$ 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 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: $C_6\times D_6$ x 6, $C_2\times C_6^2$ 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 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: $C_6\times S_3$ x 5, $C_6^2$ x 4, $C_3\times C_{12}$, $C_3:C_{12}$

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

Order 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: $C_3\times C_6$ 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: $C_3^2$

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 576: $C_6^2:C_2^4$ ($C_1$)

Order 288: $C_6^2:C_2^3$ ($C_2$) x 24, $C_6^2:C_2^3$ ($C_2$) x 2, $C_6^2:D_4$ ($C_2$) x 2, $C_6^2:D_4$ ($C_2$), $C_6^2.C_2^3$ ($C_2$), $D_4\times C_6^2$ ($C_2$)

Order 192: $C_{12}:C_2^4$ ($C_3$)

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

Order 96: $D_4\times D_6$ ($C_6$) x 24, $C_2^3\times D_6$ ($C_6$) x 2, $C_2^3:D_6$ ($C_6$) x 2, $C_{12}:C_2^3$ ($C_6$), $C_{12}:C_2^3$ ($S_3$), $C_2^2\times D_{12}$ ($C_6$), $C_{12}:C_2^3$ ($C_6$)

Order 72: $C_6\times D_6$ ($C_2^3$) x 50, $C_6\wr C_2$ ($C_2^3$) x 32, $D_4\times C_3^2$ ($C_2^3$) x 16, $C_3\times D_{12}$ ($C_2^3$) x 16, $S_3\times C_{12}$ ($C_2^3$) x 16, $C_2\times C_6^2$ ($C_2^3$) x 13, $C_6\times D_6$ ($D_4$) x 8, $C_6\times C_{12}$ ($C_2^3$) x 6, $C_6:C_{12}$ ($C_2^3$) x 6

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

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

Order 32: $C_2^2\times D_4$ ($C_3\times S_3$)

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

Order 18: $C_3\times S_3$ ($C_2^2\times D_4$) x 8, $C_3\times C_6$ ($C_2^2\times D_4$) x 6, $C_3\times C_6$ ($C_2^5$)

Order 16: $C_2\times D_4$ ($C_6\times S_3$) x 12, $C_2^4$ ($C_6\times S_3$) x 2, $C_2^2\times C_4$ ($C_6\times S_3$)

Order 12: $D_6$ ($C_6\times D_4$) x 24, $D_6$ ($C_2^3\times C_6$) x 12, $C_2\times C_6$ ($C_2^3\times C_6$) x 11, $C_2\times C_6$ ($C_2^2\times D_6$) x 11, $C_2\times C_6$ ($C_6\times D_4$) x 4, $C_2\times C_6$ ($S_3\times D_4$) x 4, $C_{12}$ ($C_2^3\times C_6$) x 4, $C_{12}$ ($C_2^2\times D_6$) x 4, $C_3:C_4$ ($C_2^3\times C_6$) x 4

Order 9: $C_3^2$ ($D_4\times C_2^3$)

Order 8: $D_4$ ($C_6\times D_6$) x 16, $C_2^3$ ($C_6\times D_6$) x 13, $C_2\times C_4$ ($C_6\times D_6$) x 6

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

Order 4: $C_2^2$ ($C_6^2:C_2^2$) x 11, $C_2^2$ ($C_{12}:D_6$) x 4, $C_4$ ($C_6^2:C_2^2$) x 4

Order 3: $C_3$ ($C_{12}:C_2^4$), $C_3$ ($C_{12}:C_2^4$)

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

Order 1: $C_1$ ($C_6^2:C_2^4$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 576: $C_6^2:C_2^4$ ($C_1$)

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

Order 192: $C_{12}:C_2^4$ ($C_3$)

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

Order 96: $C_2^3\times D_6$ ($C_6$), $C_{12}:C_2^3$ ($C_6$), $C_{12}:C_2^3$ ($S_3$), $C_2^3:D_6$ ($C_6$), $D_4\times D_6$ ($C_6$), $C_2^2\times D_{12}$ ($C_6$), $C_{12}:C_2^3$ ($C_6$)

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

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

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

Order 32: $C_2^2\times D_4$ ($C_3\times S_3$)

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

Order 18: $C_3\times C_6$ ($C_2^5$), $C_3\times C_6$ ($C_2^2\times D_4$), $C_3\times S_3$ ($C_2^2\times D_4$)

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

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

Order 9: $C_3^2$ ($D_4\times C_2^3$)

Order 8: $C_2^3$ ($C_6\times D_6$) x 2, $D_4$ ($C_6\times D_6$), $C_2\times C_4$ ($C_6\times D_6$)

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

Order 4: $C_2^2$ ($C_6^2:C_2^2$) x 2, $C_2^2$ ($C_{12}:D_6$), $C_4$ ($C_6^2:C_2^2$)

Order 3: $C_3$ ($C_{12}:C_2^4$), $C_3$ ($C_{12}:C_2^4$)

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

Order 1: $C_1$ ($C_6^2:C_2^4$)

Series

Derived series

$C_6^2:C_2^4$

$\rhd$

$C_6$

$\rhd$

$C_1$

Chief series

$C_6^2:C_2^4$

$\rhd$

$D_4\times C_6^2$

$\rhd$

$C_4:C_6^2$

$\rhd$

$D_4\times C_3^2$

$\rhd$

$C_3\times C_{12}$

$\rhd$

$C_{12}$

$\rhd$

$C_6$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$C_6^2:C_2^4$

$\rhd$

$C_6$

$\rhd$

$C_3$

Upper central series

$C_1$

$\lhd$

$C_2^2\times C_6$

$\lhd$

$C_{12}:C_2^3$

Supergroups

This group is a maximal subgroup of 21 larger groups in the database.

This group is a maximal quotient of 14 larger groups in the database.

Character theory

Complex character table

See the $180 \times 180$ 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 $120 \times 120$ rational character table.