Abstract group 288.630: $C_6^2:Q_8$

• Label: 288.630
• Order: \( 2^{5} \cdot 3^{2} \)
• Exponent: \( 2^{2} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: \( 2^{2} \)
• $\card{\Aut(G)}$: \( 2^{9} \cdot 3^{2} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{6} \)
• Perm deg.: $18$
• Trans deg.: $48$
• Rank: $3$

Group information

Description:	$C_6^2:Q_8$
Order:	\(288\)\(\medspace = 2^{5} \cdot 3^{2} \)
Exponent:	\(12\)\(\medspace = 2^{2} \cdot 3 \)
Automorphism group:	$D_6^2.C_2^5$, 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	7	8	120	56	96	288
Conjugacy classes	1	5	3	8	17	8	42
Divisions	1	5	3	7	17	4	37
Autjugacy classes	1	4	2	4	8	2	21

Dimension	1	2	4
Irr. complex chars.	8	22	12	42
Irr. rational chars.	8	12	17	37

Minimal presentations

Permutation degree:	$18$
Transitive degree:	$48$
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	6	8	8

Constructions

Presentation:	$\langle a, b, c, d \mid a^{2}=b^{4}=c^{12}=d^{3}=[a,d]=[b,d]=1, b^{a}=b^{3}c^{6}, c^{a}=b^{2}c^{7}, c^{b}=c^{11}, d^{c}=d^{2} \rangle$
Permutation group:	Degree $18$ $\langle(1,2)(3,5)(4,6)(7,8,10,12)(9,13,14,11)(15,16), (1,2)(3,6)(4,5)(7,9,10,14) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 29 & 23 \\ 8 & 25 \end{array}\right), \left(\begin{array}{rr} 17 & 0 \\ 0 & 17 \end{array}\right), \left(\begin{array}{rr} 13 & 12 \\ 12 & 25 \end{array}\right), \left(\begin{array}{rr} 1 & 18 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 21 & 35 \\ 28 & 15 \end{array}\right), \left(\begin{array}{rr} 19 & 32 \\ 32 & 35 \end{array}\right), \left(\begin{array}{rr} 1 & 12 \\ 24 & 1 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/36\Z)$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$C_6^2$ $\,\rtimes\,$ $Q_8$ (2)	$(C_6.D_{12})$ $\,\rtimes\,$ $C_2$ (2)	$C_3$ $\,\rtimes\,$ $(C_2^3.D_6)$ (2)	$(C_2^2.S_3^2)$ $\,\rtimes\,$ $C_2$	all 8
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_2^3$ . $S_3^2$	$C_6^2$ . $C_2^3$	$(C_6.D_4)$ . $S_3$ (2)	$C_6$ . $(D_4:S_3)$ (2)	all 22

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

Homology

Abelianization:	$C_{2}^{3} $
Schur multiplier:	$C_{2}^{2}$
Commutator length:	$2$

Subgroups

There are 594 subgroups in 175 conjugacy classes, 54 normal (18 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:	$G' \simeq$ $C_6^2$	$G/G' \simeq$ $C_2^3$
Frattini:	$\Phi \simeq$ $C_2^2$	$G/\Phi \simeq$ $S_3\times D_6$
Fitting:	$\operatorname{Fit} \simeq$ $C_2\times C_6^2$	$G/\operatorname{Fit} \simeq$ $C_2^2$
Radical:	$R \simeq$ $C_6^2:Q_8$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_6^2$	$G/\operatorname{soc} \simeq$ $C_2^3$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^2:Q_8$
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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Classes of subgroups up to conjugation

Order 288: $C_6^2:Q_8$

Order 144: $C_6^2:C_4$ x 2, $C_6.D_{12}$ x 2, $C_2^2.S_3^2$, $C_6^2:C_4$, $C_2^2.S_3^2$

Order 96: $C_2^3.D_6$ x 2

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

Order 48: $C_6.D_4$ x 4, $C_6.C_2^3$ x 3, $C_6:Q_8$ x 2, $C_2^2:C_{12}$ x 2, $C_6.D_4$ x 2, $C_{12}:C_4$ x 2

Order 36: $C_6^2$ x 5, $C_3:C_{12}$ x 4, $C_3^2:C_4$ x 3

Order 32: $C_2^2:Q_8$

Order 24: $C_6:C_4$ x 18, $C_2\times C_{12}$ x 4, $C_3:Q_8$ x 4, $C_2^2\times C_6$ x 3

Order 18: $C_3\times C_6$ x 5

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

Order 12: $C_2\times C_6$ x 17, $C_3:C_4$ x 14, $C_{12}$ x 4

Order 9: $C_3^2$

Order 8: $C_2\times C_4$ x 8, $Q_8$ x 2, $C_2^3$

Order 6: $C_6$ x 17

Order 4: $C_4$ x 7, $C_2^2$ x 5

Order 3: $C_3$ x 3

Order 2: $C_2$ x 5

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 288: $C_6^2:Q_8$

Order 144: $C_6^2:C_4$, $C_2^2.S_3^2$, $C_6.D_{12}$, $C_6^2:C_4$, $C_2^2.S_3^2$

Order 96: $C_2^3.D_6$

Order 72: $C_6.D_6$ x 3, $C_6:C_{12}$ x 2, $C_2\times C_6^2$, $C_3^2:Q_8$

Order 48: $C_6.C_2^3$ x 2, $C_6.D_4$ x 2, $C_6:Q_8$, $C_2^2:C_{12}$, $C_6.D_4$, $C_{12}:C_4$

Order 36: $C_6^2$ x 4, $C_3^2:C_4$ x 2, $C_3:C_{12}$ x 2

Order 32: $C_2^2:Q_8$

Order 24: $C_6:C_4$ x 8, $C_2\times C_{12}$ x 2, $C_2^2\times C_6$ x 2, $C_3:Q_8$

Order 18: $C_3\times C_6$ x 4

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

Order 12: $C_2\times C_6$ x 8, $C_3:C_4$ x 6, $C_{12}$ x 2

Order 9: $C_3^2$

Order 8: $C_2\times C_4$ x 5, $C_2^3$, $Q_8$

Order 6: $C_6$ x 8

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

Order 3: $C_3$ x 2

Order 2: $C_2$ x 4

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 288: $C_6^2:Q_8$ ($C_1$)

Order 144: $C_6^2:C_4$ ($C_2$) x 2, $C_6.D_{12}$ ($C_2$) x 2, $C_2^2.S_3^2$ ($C_2$), $C_6^2:C_4$ ($C_2$), $C_2^2.S_3^2$ ($C_2$)

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

Order 48: $C_6.D_4$ ($S_3$) x 2

Order 36: $C_3^2:C_4$ ($D_4$) x 2, $C_6^2$ ($Q_8$) x 2, $C_6^2$ ($C_2^3$)

Order 24: $C_6:C_4$ ($D_6$) x 4, $C_2^2\times C_6$ ($D_6$) x 2

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

Order 12: $C_2\times C_6$ ($C_3:Q_8$) x 4, $C_2\times C_6$ ($C_2\times D_6$) x 2

Order 9: $C_3^2$ ($C_2^2:Q_8$)

Order 8: $C_2^3$ ($S_3^2$)

Order 6: $C_6$ ($D_4:S_3$) x 2, $C_6$ ($S_3\times D_4$) x 2, $C_6$ ($C_6:Q_8$) x 2

Order 4: $C_2^2$ ($C_3^2:Q_8$) x 2, $C_2^2$ ($S_3\times D_6$)

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

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

Order 1: $C_1$ ($C_6^2:Q_8$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 288: $C_6^2:Q_8$ ($C_1$)

Order 144: $C_6^2:C_4$ ($C_2$), $C_2^2.S_3^2$ ($C_2$), $C_6.D_{12}$ ($C_2$), $C_6^2:C_4$ ($C_2$), $C_2^2.S_3^2$ ($C_2$)

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

Order 48: $C_6.D_4$ ($S_3$)

Order 36: $C_3^2:C_4$ ($D_4$), $C_6^2$ ($C_2^3$), $C_6^2$ ($Q_8$)

Order 24: $C_6:C_4$ ($D_6$) x 2, $C_2^2\times C_6$ ($D_6$)

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

Order 12: $C_2\times C_6$ ($C_3:Q_8$), $C_2\times C_6$ ($C_2\times D_6$)

Order 9: $C_3^2$ ($C_2^2:Q_8$)

Order 8: $C_2^3$ ($S_3^2$)

Order 6: $C_6$ ($D_4:S_3$), $C_6$ ($S_3\times D_4$), $C_6$ ($C_6:Q_8$)

Order 4: $C_2^2$ ($S_3\times D_6$), $C_2^2$ ($C_3^2:Q_8$)

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

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

Order 1: $C_1$ ($C_6^2:Q_8$)

Series

Derived series

$C_6^2:Q_8$

$\rhd$

$C_6^2$

$\rhd$

$C_1$

Chief series

$C_6^2:Q_8$

$\rhd$

$C_6^2:C_4$

$\rhd$

$C_2\times C_6^2$

$\rhd$

$C_6^2$

$\rhd$

$C_3\times C_6$

$\rhd$

$C_3^2$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$C_6^2:Q_8$

$\rhd$

$C_6^2$

$\rhd$

$C_3^2$

Upper central series

$C_1$

$\lhd$

$C_2^2$

$\lhd$

$C_2^3$

Supergroups

This group is a maximal subgroup of 108 larger groups in the database.

This group is a maximal quotient of 32 larger groups in the database.

Character theory

Complex character table

See the $42 \times 42$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $37 \times 37$ rational character table.