Abstract group 576.2207: $C_4^2.S_3^2$

• Label: 576.2207
• Order: \( 2^{6} \cdot 3^{2} \)
• Exponent: \( 2^{3} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{5} \)
• $\card{Z(G)}$: \( 2^{3} \)
• $\card{\Aut(G)}$: \( 2^{10} \cdot 3^{2} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{7} \)
• Perm deg.: $18$
• Trans deg.: $192$
• Rank: $3$

Group information

Description:	$C_4^2.S_3^2$
Order:	\(576\)\(\medspace = 2^{6} \cdot 3^{2} \)
Exponent:	\(24\)\(\medspace = 2^{3} \cdot 3 \)
Automorphism group:	$C_2^2\times C_6^2.C_2^6$, of order \(9216\)\(\medspace = 2^{10} \cdot 3^{2} \)
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	8	12	24
Elements	1	27	8	36	72	192	144	96	576
Conjugacy classes	1	7	3	12	13	20	36	16	108
Divisions	1	7	3	7	13	5	17	3	56
Autjugacy classes	1	4	3	5	10	3	13	2	41

Dimension	1	2	4	8	16
Irr. complex chars.	32	56	20	0	0	108
Irr. rational chars.	8	18	19	9	2	56

Minimal presentations

Permutation degree:	$18$
Transitive degree:	$192$
Rank:	$3$
Inequivalent generating triples:	$5376$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	4	6	8

Constructions

Presentation:	$\langle a, b, c \mid a^{8}=b^{6}=c^{12}=[a,c]=1, b^{a}=b^{5}, c^{b}=c^{11} \rangle$
Permutation group:	Degree $18$ $\langle(1,2)(3,4)(6,7)(8,9)(10,12)(11,13)(14,15), (8,10,11,14,9,12,13,15)(17,18) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 7 & 0 \\ 0 & 7 \end{array}\right), \left(\begin{array}{rr} 4 & 15 \\ 25 & 14 \end{array}\right), \left(\begin{array}{rr} 16 & 15 \\ 15 & 1 \end{array}\right), \left(\begin{array}{rr} 11 & 0 \\ 0 & 11 \end{array}\right), \left(\begin{array}{rr} 22 & 9 \\ 1 & 8 \end{array}\right), \left(\begin{array}{rr} 13 & 10 \\ 10 & 23 \end{array}\right), \left(\begin{array}{rr} 19 & 0 \\ 0 & 19 \end{array}\right), \left(\begin{array}{rr} 13 & 12 \\ 6 & 1 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/30\Z)$
Direct product:	$(C_3:C_8)$ $\, \times\, $ $D_{12}$
Semidirect product:	$D_6$ $\,\rtimes\,$ $(C_6:C_8)$ (4)	$(C_{12}:C_8)$ $\,\rtimes\,$ $S_3$	$C_{12}$ $\,\rtimes\,$ $(S_3\times C_8)$ (2)	$(C_3:C_{24})$ $\,\rtimes\,$ $D_4$ (2)	all 16
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_4^2$ . $S_3^2$	$C_{12}^2$ . $C_2^2$	$(C_4\times D_6)$ . $D_6$ (2)	$(C_6:C_8)$ . $D_6$ (2)	all 51

Elements of the group are displayed as words in the presentation generators from the presentation above.

Homology

Abelianization:	$C_{2}^{2} \times C_{8} $
Schur multiplier:	$C_{2}^{3}$
Commutator length:	$1$

Subgroups

There are 884 subgroups in 287 conjugacy classes, 108 normal (58 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2\times C_4$	$G/Z \simeq$ $S_3\times D_6$
Commutator:	$G' \simeq$ $C_3\times C_6$	$G/G' \simeq$ $C_2^2\times C_8$
Frattini:	$\Phi \simeq$ $C_2\times C_4$	$G/\Phi \simeq$ $S_3\times D_6$
Fitting:	$\operatorname{Fit} \simeq$ $C_{12}^2$	$G/\operatorname{Fit} \simeq$ $C_2^2$
Radical:	$R \simeq$ $C_4^2.S_3^2$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_6^2$	$G/\operatorname{soc} \simeq$ $C_2^2\times C_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_8\times D_4$
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: $C_4^2.S_3^2$

Order 288: $S_3\times C_6:C_8$ x 2, $C_{12}.D_{12}$ x 2, $C_{12}\times D_{12}$, $C_{12}.D_{12}$, $C_{12}:C_{24}$

Order 192: $C_4^2.D_6$, $C_8\times D_{12}$

Order 144: $C_{12}.D_6$ x 4, $C_{12}.D_6$ x 2, $D_6:C_{12}$ x 2, $C_6:C_{24}$ x 2, $C_{12}\times D_6$ x 2, $C_{12}:C_{12}$, $C_6\times D_{12}$, $C_{12}^2$

Order 96: $C_{12}:C_8$ x 3, $C_{12}.D_4$ x 2, $D_6:C_8$ x 2, $C_{12}.C_2^3$ x 2, $C_8\times D_6$ x 2, $C_{12}:C_8$, $C_4\times D_{12}$, $C_4\times C_{24}$, $D_4\times C_{12}$

Order 72: $C_3\times D_{12}$ x 4, $S_3\times C_{12}$ x 4, $C_6\times C_{12}$ x 3, $C_3:C_{24}$ x 3, $C_6\times D_6$ x 2, $C_6:C_{12}$ x 2, $C_3^2:C_8$ x 2

Order 64: $C_8\times D_4$

Order 48: $C_6:C_8$ x 12, $S_3\times C_8$ x 4, $C_4\times C_{12}$ x 3, $C_2^2\times C_{12}$ x 2, $C_4\times D_6$ x 2, $C_2\times C_{24}$ x 2, $C_2^2:C_{12}$ x 2, $D_6:C_4$ x 2, $C_6\times D_4$, $C_2\times D_{12}$, $C_4:C_{12}$, $C_{12}:C_4$

Order 36: $C_6\times S_3$ x 8, $C_3\times C_{12}$ x 5, $C_3:C_{12}$ x 2, $C_6^2$

Order 32: $C_2^2:C_8$ x 2, $C_2^2\times C_8$ x 2, $C_4\times C_8$, $C_4\times D_4$, $C_4:C_8$

Order 24: $C_2\times C_{12}$ x 15, $C_3:C_8$ x 9, $D_{12}$ x 4, $C_4\times S_3$ x 4, $C_3\times D_4$ x 4, $C_{24}$ x 3, $C_6:C_4$ x 2, $C_2^2\times C_6$ x 2, $C_2\times D_6$ x 2

Order 18: $C_3\times S_3$ x 4, $C_3\times C_6$ x 3

Order 16: $C_2\times C_8$ x 8, $C_2^2:C_4$ x 2, $C_2^2\times C_4$ x 2, $C_4:C_4$, $C_4^2$, $C_2\times D_4$

Order 12: $C_{12}$ x 17, $C_2\times C_6$ x 11, $D_6$ x 8, $C_3:C_4$ x 2

Order 9: $C_3^2$

Order 8: $C_2\times C_4$ x 9, $C_8$ x 5, $D_4$ x 4, $C_2^3$ x 2

Order 6: $C_6$ x 13, $S_3$ x 4

Order 4: $C_2^2$ x 9, $C_4$ x 7

Order 3: $C_3$ x 3

Order 2: $C_2$ x 7

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 576: $C_4^2.S_3^2$

Order 288: $C_{12}\times D_{12}$, $S_3\times C_6:C_8$, $C_{12}.D_{12}$, $C_{12}:C_{24}$, $C_{12}.D_{12}$

Order 192: $C_4^2.D_6$, $C_8\times D_{12}$

Order 144: $C_6:C_{24}$ x 2, $C_{12}.D_6$, $D_6:C_{12}$, $C_{12}:C_{12}$, $C_{12}.D_6$, $C_6\times D_{12}$, $C_{12}\times D_6$, $C_{12}^2$

Order 96: $C_{12}:C_8$ x 3, $C_{12}:C_8$, $C_4\times D_{12}$, $C_4\times C_{24}$, $C_{12}.D_4$, $D_6:C_8$, $D_4\times C_{12}$, $C_{12}.C_2^3$, $C_8\times D_6$

Order 72: $C_6\times C_{12}$ x 3, $S_3\times C_{12}$ x 2, $C_3:C_{24}$ x 2, $C_6\times D_6$, $C_6:C_{12}$, $C_3\times D_{12}$, $C_3^2:C_8$

Order 64: $C_8\times D_4$

Order 48: $C_6:C_8$ x 6, $C_4\times C_{12}$ x 3, $C_2\times C_{24}$ x 2, $C_6\times D_4$, $C_2^2\times C_{12}$, $S_3\times C_8$, $C_2\times D_{12}$, $C_4\times D_6$, $C_4:C_{12}$, $C_2^2:C_{12}$, $D_6:C_4$, $C_{12}:C_4$

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

Order 32: $C_2^2:C_8$, $C_2^2\times C_8$, $C_4\times C_8$, $C_4\times D_4$, $C_4:C_8$

Order 24: $C_2\times C_{12}$ x 12, $C_3:C_8$ x 5, $C_4\times S_3$ x 2, $C_{24}$ x 2, $C_6:C_4$, $D_{12}$, $C_2^2\times C_6$, $C_2\times D_6$, $C_3\times D_4$

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

Order 16: $C_2\times C_8$ x 4, $C_4:C_4$, $C_2^2:C_4$, $C_4^2$, $C_2\times D_4$, $C_2^2\times C_4$

Order 12: $C_{12}$ x 13, $C_2\times C_6$ x 6, $D_6$ x 3, $C_3:C_4$

Order 9: $C_3^2$

Order 8: $C_2\times C_4$ x 6, $C_8$ x 3, $C_2^3$, $D_4$

Order 6: $C_6$ x 10, $S_3$

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

Order 3: $C_3$ x 3

Order 2: $C_2$ x 4

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 576: $C_4^2.S_3^2$ ($C_1$)

Order 288: $S_3\times C_6:C_8$ ($C_2$) x 2, $C_{12}.D_{12}$ ($C_2$) x 2, $C_{12}\times D_{12}$ ($C_2$), $C_{12}.D_{12}$ ($C_2$), $C_{12}:C_{24}$ ($C_2$)

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

Order 96: $C_{12}:C_8$ ($S_3$), $C_4\times D_{12}$ ($S_3$)

Order 72: $C_3\times D_{12}$ ($C_8$) x 4, $C_6\times D_6$ ($C_2\times C_4$) x 2, $C_6\times C_{12}$ ($C_2\times C_4$) x 2, $C_6:C_{12}$ ($C_2\times C_4$) x 2, $C_3:C_{24}$ ($D_4$) x 2, $C_6\times C_{12}$ ($C_2^3$)

Order 48: $C_6:C_8$ ($D_6$) x 2, $C_4\times D_6$ ($D_6$) x 2, $C_4\times C_{12}$ ($D_6$) x 2, $D_6:C_4$ ($C_3:C_4$) x 2, $C_2\times D_{12}$ ($C_3:C_4$), $C_{12}:C_4$ ($C_3:C_4$)

Order 36: $C_6\times S_3$ ($C_2\times C_8$) x 4, $C_3\times C_{12}$ ($C_2\times C_8$) x 2, $C_3\times C_{12}$ ($D_4:C_2$), $C_3\times C_{12}$ ($C_2\times D_4$), $C_6^2$ ($C_2^2\times C_4$)

Order 24: $D_{12}$ ($C_3:C_8$) x 4, $C_2\times C_{12}$ ($C_6:C_4$) x 2, $C_2\times C_{12}$ ($C_4\times S_3$) x 2, $C_2\times C_{12}$ ($C_2\times D_6$) x 2, $C_6:C_4$ ($C_6:C_4$) x 2, $C_2\times D_6$ ($C_6:C_4$) x 2, $C_3:C_8$ ($D_{12}$) x 2

Order 18: $C_3\times C_6$ ($\OD_{16}:C_2$), $C_3\times C_6$ ($C_2^2\times C_8$), $C_3\times C_6$ ($C_4\times D_4$)

Order 16: $C_4^2$ ($S_3^2$)

Order 12: $D_6$ ($C_6:C_8$) x 4, $C_{12}$ ($C_6:C_8$) x 2, $C_{12}$ ($S_3\times C_8$) x 2, $C_2\times C_6$ ($C_6.C_2^3$), $C_2\times C_6$ ($C_4\times D_6$), $C_{12}$ ($D_4:S_3$), $C_{12}$ ($S_3\times D_4$), $C_{12}$ ($D_{12}:C_2$), $C_{12}$ ($C_2\times D_{12}$)

Order 9: $C_3^2$ ($C_8\times D_4$)

Order 8: $C_2\times C_4$ ($C_6.D_6$) x 2, $C_2\times C_4$ ($S_3\times D_6$)

Order 6: $C_6$ ($C_4\times D_{12}$), $C_6$ ($C_{12}.C_2^3$), $C_6$ ($C_2^3.D_6$), $C_6$ ($C_{12}.C_2^3$), $C_6$ ($C_8.D_6$), $C_6$ ($C_8\times D_6$)

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

Order 3: $C_3$ ($C_4^2.D_6$), $C_3$ ($C_8\times D_{12}$)

Order 2: $C_2$ ($C_6^2.C_2^3$), $C_2$ ($C_{12}.(C_4\times S_3)$), $C_2$ ($S_3\times C_6:C_8$)

Order 1: $C_1$ ($C_4^2.S_3^2$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 576: $C_4^2.S_3^2$ ($C_1$)

Order 288: $C_{12}\times D_{12}$ ($C_2$), $S_3\times C_6:C_8$ ($C_2$), $C_{12}.D_{12}$ ($C_2$), $C_{12}:C_{24}$ ($C_2$), $C_{12}.D_{12}$ ($C_2$)

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

Order 96: $C_{12}:C_8$ ($S_3$), $C_4\times D_{12}$ ($S_3$)

Order 72: $C_6\times C_{12}$ ($C_2\times C_4$) x 2, $C_6\times D_6$ ($C_2\times C_4$), $C_6\times C_{12}$ ($C_2^3$), $C_6:C_{12}$ ($C_2\times C_4$), $C_3\times D_{12}$ ($C_8$), $C_3:C_{24}$ ($D_4$)

Order 48: $C_6:C_8$ ($D_6$) x 2, $C_4\times C_{12}$ ($D_6$) x 2, $C_2\times D_{12}$ ($C_3:C_4$), $C_4\times D_6$ ($D_6$), $D_6:C_4$ ($C_3:C_4$), $C_{12}:C_4$ ($C_3:C_4$)

Order 36: $C_3\times C_{12}$ ($C_2\times C_8$), $C_3\times C_{12}$ ($D_4:C_2$), $C_3\times C_{12}$ ($C_2\times D_4$), $C_6^2$ ($C_2^2\times C_4$), $C_6\times S_3$ ($C_2\times C_8$)

Order 24: $C_2\times C_{12}$ ($C_6:C_4$) x 2, $C_2\times C_{12}$ ($C_4\times S_3$) x 2, $C_2\times C_{12}$ ($C_2\times D_6$) x 2, $C_6:C_4$ ($C_6:C_4$), $D_{12}$ ($C_3:C_8$), $C_2\times D_6$ ($C_6:C_4$), $C_3:C_8$ ($D_{12}$)

Order 18: $C_3\times C_6$ ($\OD_{16}:C_2$), $C_3\times C_6$ ($C_2^2\times C_8$), $C_3\times C_6$ ($C_4\times D_4$)

Order 16: $C_4^2$ ($S_3^2$)

Order 12: $C_2\times C_6$ ($C_6.C_2^3$), $C_2\times C_6$ ($C_4\times D_6$), $D_6$ ($C_6:C_8$), $C_{12}$ ($C_6:C_8$), $C_{12}$ ($S_3\times C_8$), $C_{12}$ ($D_4:S_3$), $C_{12}$ ($S_3\times D_4$), $C_{12}$ ($D_{12}:C_2$), $C_{12}$ ($C_2\times D_{12}$)

Order 9: $C_3^2$ ($C_8\times D_4$)

Order 8: $C_2\times C_4$ ($C_6.D_6$) x 2, $C_2\times C_4$ ($S_3\times D_6$)

Order 6: $C_6$ ($C_4\times D_{12}$), $C_6$ ($C_{12}.C_2^3$), $C_6$ ($C_2^3.D_6$), $C_6$ ($C_{12}.C_2^3$), $C_6$ ($C_8.D_6$), $C_6$ ($C_8\times D_6$)

Order 4: $C_2^2$ ($D_6.D_6$), $C_4$ ($C_{12}.D_6$), $C_4$ ($S_3\times D_{12}$), $C_4$ ($D_{12}:S_3$)

Order 3: $C_3$ ($C_4^2.D_6$), $C_3$ ($C_8\times D_{12}$)

Order 2: $C_2$ ($C_6^2.C_2^3$), $C_2$ ($C_{12}.(C_4\times S_3)$), $C_2$ ($S_3\times C_6:C_8$)

Order 1: $C_1$ ($C_4^2.S_3^2$)

Series

Derived series

$C_4^2.S_3^2$

$\rhd$

$C_3\times C_6$

$\rhd$

$C_1$

Chief series

$C_4^2.S_3^2$

$\rhd$

$C_{12}:C_{24}$

$\rhd$

$C_{12}^2$

$\rhd$

$C_6\times C_{12}$

$\rhd$

$C_3\times C_{12}$

$\rhd$

$C_{12}$

$\rhd$

$C_6$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$C_4^2.S_3^2$

$\rhd$

$C_3\times C_6$

$\rhd$

$C_3^2$

Upper central series

$C_1$

$\lhd$

$C_2\times C_4$

$\lhd$

$C_4^2$

Supergroups

This group is a maximal subgroup of 10 larger groups in the database.

This group is a maximal quotient of 9 larger groups in the database.

Character theory

Complex character table

See the $108 \times 108$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $56 \times 56$ rational character table.