Abstract group 288.491: $C_6^2.C_2^3$

• Label: 288.491
• Order: \( 2^{5} \cdot 3^{2} \)
• Exponent: \( 2^{2} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \)
• $\card{Z(G)}$: \( 2^{2} \)
• $\card{\Aut(G)}$: \( 2^{9} \cdot 3^{2} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{6} \)
• Perm deg.: $18$
• Trans deg.: $96$
• Rank: $3$

Group information

Description:	$C_6^2.C_2^3$
Order:	\(288\)\(\medspace = 2^{5} \cdot 3^{2} \)
Exponent:	\(12\)\(\medspace = 2^{2} \cdot 3 \)
Automorphism group:	$C_6^2.C_2^6.C_2$, of order \(4608\)\(\medspace = 2^{9} \cdot 3^{2} \)
Composition factors:	$C_2$ x 5, $C_3$ 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	3	8	124	24	128	288
Conjugacy classes	1	3	3	16	9	16	48
Divisions	1	3	3	11	9	12	39
Autjugacy classes	1	3	3	5	9	5	26

Dimension	1	2	4	8
Irr. complex chars.	16	20	12	0	48
Irr. rational chars.	8	18	11	2	39

Minimal presentations

Permutation degree:	$18$
Transitive degree:	$96$
Rank:	$3$
Inequivalent generating triples:	$1344$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	5	8	10

Constructions

Presentation:	$\langle a, b, c \mid b^{12}=c^{12}=1, a^{2}=c^{6}, b^{a}=b^{5}, c^{a}=c^{7}, c^{b}=c^{11} \rangle$
Permutation group:	Degree $18$ $\langle(1,2)(3,5)(4,6)(7,8,11,9)(10,13,14,12), (1,2)(3,6)(4,5)(7,9,11,8)(10,12,14,13) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 1 & 20 \\ 20 & 11 \end{array}\right), \left(\begin{array}{rr} 7 & 21 \\ 0 & 13 \end{array}\right), \left(\begin{array}{rr} 11 & 5 \\ 20 & 1 \end{array}\right), \left(\begin{array}{rr} 16 & 15 \\ 15 & 1 \end{array}\right), \left(\begin{array}{rr} 11 & 0 \\ 0 & 11 \end{array}\right), \left(\begin{array}{rr} 19 & 0 \\ 0 & 19 \end{array}\right), \left(\begin{array}{rr} 13 & 12 \\ 6 & 1 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/30\Z)$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(C_3^2:Q_8)$ $\,\rtimes\,$ $C_4$ (4)	$(C_3^2:C_4)$ $\,\rtimes\,$ $Q_8$ (2)	$C_3^2$ $\,\rtimes\,$ $(C_4\times Q_8)$	$C_3$ $\,\rtimes\,$ $(C_3:C_4\times Q_8)$	all 6
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_6^2$ . $C_2^3$	$(C_6:Q_8)$ . $S_3$	$C_6$ . $(S_3\times Q_8)$ (2)	$C_6$ . $(D_4:S_3)$	all 33

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

Homology

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

Subgroups

There are 426 subgroups in 155 conjugacy classes, 70 normal (32 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2^2$	$G/Z \simeq$ $S_3\times D_6$
Commutator:	$G' \simeq$ $C_3\times C_6$	$G/G' \simeq$ $C_2^2\times C_4$
Frattini:	$\Phi \simeq$ $C_2^2$	$G/\Phi \simeq$ $S_3\times D_6$
Fitting:	$\operatorname{Fit} \simeq$ $C_6\times C_{12}$	$G/\operatorname{Fit} \simeq$ $C_2^2$
Radical:	$R \simeq$ $C_6^2.C_2^3$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_6^2$	$G/\operatorname{soc} \simeq$ $C_2^3$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_4\times Q_8$
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

Normal subgroups up to automorphism

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Subgroup information

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Classes of subgroups up to conjugation

Order 288: $C_6^2.C_2^3$

Order 144: $C_6.D_{12}$ x 2, $C_3^2:C_4^2$ x 2, $C_3^2:C_4^2$, $C_{12}:C_{12}$, $C_{12}.D_6$

Order 96: $C_3:(C_4\times Q_8)$, $C_3:C_4\times Q_8$

Order 72: $C_6:C_{12}$ x 4, $C_3^2:Q_8$ x 4, $C_6.D_6$ x 2, $C_6\times C_{12}$

Order 48: $C_{12}:C_4$ x 7, $C_{12}:C_4$ x 3, $C_6.D_4$ x 2, $C_6\times Q_8$, $C_6:Q_8$, $C_4:C_{12}$

Order 36: $C_3:C_{12}$ x 6, $C_3^2:C_4$ x 3, $C_3\times C_{12}$ x 2, $C_6^2$

Order 32: $C_4\times Q_8$

Order 24: $C_6:C_4$ x 10, $C_2\times C_{12}$ x 7, $C_3:Q_8$ x 4, $C_3\times Q_8$ x 4

Order 18: $C_3\times C_6$ x 3

Order 16: $C_4:C_4$ x 3, $C_4^2$ x 3, $C_2\times Q_8$

Order 12: $C_3:C_4$ x 16, $C_{12}$ x 12, $C_2\times C_6$ x 3

Order 9: $C_3^2$

Order 8: $C_2\times C_4$ x 7, $Q_8$ x 4

Order 6: $C_6$ x 9

Order 4: $C_4$ x 11, $C_2^2$

Order 3: $C_3$ x 3

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 288: $C_6^2.C_2^3$

Order 144: $C_3^2:C_4^2$, $C_{12}:C_{12}$, $C_6.D_{12}$, $C_3^2:C_4^2$, $C_{12}.D_6$

Order 96: $C_3:(C_4\times Q_8)$, $C_3:C_4\times Q_8$

Order 72: $C_6.D_6$ x 2, $C_6:C_{12}$ x 2, $C_6\times C_{12}$, $C_3^2:Q_8$

Order 48: $C_{12}:C_4$ x 5, $C_{12}:C_4$ x 2, $C_6\times Q_8$, $C_6:Q_8$, $C_4:C_{12}$, $C_6.D_4$

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

Order 32: $C_4\times Q_8$

Order 24: $C_6:C_4$ x 8, $C_2\times C_{12}$ x 5, $C_3:Q_8$, $C_3\times Q_8$

Order 18: $C_3\times C_6$ x 3

Order 16: $C_4:C_4$ x 2, $C_4^2$ x 2, $C_2\times Q_8$

Order 12: $C_3:C_4$ x 8, $C_{12}$ x 5, $C_2\times C_6$ x 3

Order 9: $C_3^2$

Order 8: $C_2\times C_4$ x 5, $Q_8$

Order 6: $C_6$ x 9

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

Order 3: $C_3$ x 3

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 288: $C_6^2.C_2^3$ ($C_1$)

Order 144: $C_6.D_{12}$ ($C_2$) x 2, $C_3^2:C_4^2$ ($C_2$) x 2, $C_3^2:C_4^2$ ($C_2$), $C_{12}:C_{12}$ ($C_2$), $C_{12}.D_6$ ($C_2$)

Order 72: $C_6:C_{12}$ ($C_2^2$) x 4, $C_3^2:Q_8$ ($C_4$) x 4, $C_6.D_6$ ($C_2^2$) x 2, $C_6\times C_{12}$ ($C_2^2$)

Order 48: $C_6:Q_8$ ($S_3$), $C_{12}:C_4$ ($S_3$)

Order 36: $C_3:C_{12}$ ($C_2\times C_4$) x 4, $C_3\times C_{12}$ ($C_2\times C_4$) x 2, $C_3^2:C_4$ ($Q_8$) x 2, $C_6^2$ ($C_2^3$)

Order 24: $C_6:C_4$ ($D_6$) x 4, $C_3:Q_8$ ($C_3:C_4$) x 4, $C_2\times C_{12}$ ($D_6$) x 2

Order 18: $C_3\times C_6$ ($D_4:C_2$), $C_3\times C_6$ ($C_2\times Q_8$), $C_3\times C_6$ ($C_2^2\times C_4$)

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

Order 9: $C_3^2$ ($C_4\times Q_8$)

Order 8: $C_2\times C_4$ ($S_3^2$)

Order 6: $C_6$ ($S_3\times Q_8$) x 2, $C_6$ ($C_6.C_2^3$), $C_6$ ($D_{12}:C_2$), $C_6$ ($D_4:S_3$), $C_6$ ($C_4\times D_6$)

Order 4: $C_4$ ($C_6.D_6$) x 2, $C_2^2$ ($S_3\times D_6$)

Order 3: $C_3$ ($C_3:(C_4\times Q_8)$), $C_3$ ($C_3:C_4\times Q_8$)

Order 2: $C_2$ ($D_6.D_6$), $C_2$ ($C_{12}.D_6$), $C_2$ ($D_{12}:S_3$)

Order 1: $C_1$ ($C_6^2.C_2^3$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 288: $C_6^2.C_2^3$ ($C_1$)

Order 144: $C_3^2:C_4^2$ ($C_2$), $C_{12}:C_{12}$ ($C_2$), $C_6.D_{12}$ ($C_2$), $C_3^2:C_4^2$ ($C_2$), $C_{12}.D_6$ ($C_2$)

Order 72: $C_6.D_6$ ($C_2^2$) x 2, $C_6:C_{12}$ ($C_2^2$) x 2, $C_6\times C_{12}$ ($C_2^2$), $C_3^2:Q_8$ ($C_4$)

Order 48: $C_6:Q_8$ ($S_3$), $C_{12}:C_4$ ($S_3$)

Order 36: $C_3\times C_{12}$ ($C_2\times C_4$), $C_3^2:C_4$ ($Q_8$), $C_3:C_{12}$ ($C_2\times C_4$), $C_6^2$ ($C_2^3$)

Order 24: $C_2\times C_{12}$ ($D_6$) x 2, $C_6:C_4$ ($D_6$) x 2, $C_3:Q_8$ ($C_3:C_4$)

Order 18: $C_3\times C_6$ ($D_4:C_2$), $C_3\times C_6$ ($C_2\times Q_8$), $C_3\times C_6$ ($C_2^2\times C_4$)

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

Order 9: $C_3^2$ ($C_4\times Q_8$)

Order 8: $C_2\times C_4$ ($S_3^2$)

Order 6: $C_6$ ($S_3\times Q_8$) x 2, $C_6$ ($C_6.C_2^3$), $C_6$ ($D_{12}:C_2$), $C_6$ ($D_4:S_3$), $C_6$ ($C_4\times D_6$)

Order 4: $C_2^2$ ($S_3\times D_6$), $C_4$ ($C_6.D_6$)

Order 3: $C_3$ ($C_3:(C_4\times Q_8)$), $C_3$ ($C_3:C_4\times Q_8$)

Order 2: $C_2$ ($D_6.D_6$), $C_2$ ($C_{12}.D_6$), $C_2$ ($D_{12}:S_3$)

Order 1: $C_1$ ($C_6^2.C_2^3$)

Series

Derived series

$C_6^2.C_2^3$

$\rhd$

$C_3\times C_6$

$\rhd$

$C_1$

Chief series

$C_6^2.C_2^3$

$\rhd$

$C_{12}.D_6$

$\rhd$

$C_6\times C_{12}$

$\rhd$

$C_3\times C_{12}$

$\rhd$

$C_{12}$

$\rhd$

$C_6$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$C_6^2.C_2^3$

$\rhd$

$C_3\times C_6$

$\rhd$

$C_3^2$

Upper central series

$C_1$

$\lhd$

$C_2^2$

$\lhd$

$C_2\times C_4$

Supergroups

This group is a maximal subgroup of 143 larger groups in the database.

This group is a maximal quotient of 29 larger groups in the database.

Character theory

Complex character table

See the $48 \times 48$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $39 \times 39$ rational character table.