Abstract group 288.305: $C_6^2:C_8$

• Label: 288.305
• Order: \( 2^{5} \cdot 3^{2} \)
• Exponent: \( 2^{3} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \)
• $\card{Z(G)}$: \( 2^{3} \)
• $\card{\Aut(G)}$: \( 2^{10} \cdot 3^{3} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{8} \cdot 3 \)
• Perm deg.: $18$
• Trans deg.: $144$
• Rank: $3$

Group information

Description:	$C_6^2:C_8$
Order:	\(288\)\(\medspace = 2^{5} \cdot 3^{2} \)
Exponent:	\(24\)\(\medspace = 2^{3} \cdot 3 \)
Automorphism group:	$\ASL(2,3).C_2^6.C_2$, of order \(27648\)\(\medspace = 2^{10} \cdot 3^{3} \)
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	8	12
Elements	1	7	8	8	56	144	64	288
Conjugacy classes	1	5	4	6	28	8	32	84
Divisions	1	5	4	3	20	2	12	47
Autjugacy classes	1	4	1	2	4	1	2	15

Dimension	1	2	4	8
Irr. complex chars.	16	68	0	0	84
Irr. rational chars.	4	20	19	4	47

Minimal presentations

Permutation degree:	$18$
Transitive degree:	$144$
Rank:	$3$
Inequivalent generating triples:	$336$

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:	$\langle a, b, c \mid a^{8}=b^{6}=c^{6}=[b,c]=1, b^{a}=b^{5}c^{3}, c^{a}=c^{5} \rangle$
Permutation group:	Degree $18$ $\langle(2,4)(5,6,8,10,7,9,11,12)(14,15)(17,18), (1,2)(3,4)(5,7)(6,9)(8,11)(10,12) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 17 & 0 \\ 0 & 17 \end{array}\right), \left(\begin{array}{rr} 16 & 15 \\ 5 & 11 \end{array}\right), \left(\begin{array}{rr} 9 & 0 \\ 0 & 9 \end{array}\right), \left(\begin{array}{rr} 8 & 11 \\ 17 & 11 \end{array}\right), \left(\begin{array}{rr} 19 & 10 \\ 10 & 9 \end{array}\right), \left(\begin{array}{rr} 9 & 14 \\ 19 & 11 \end{array}\right), \left(\begin{array}{rr} 3 & 0 \\ 0 & 3 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/20\Z)$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$C_6^2$ $\,\rtimes\,$ $C_8$	$(C_{12}.D_6)$ $\,\rtimes\,$ $C_2$	$C_3$ $\,\rtimes\,$ $(C_{12}.D_4)$	$C_2^2$ $\,\rtimes\,$ $(C_3^2:C_8)$	all 5
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_6$ . $(C_6.D_4)$	$C_6$ . $(C_6:C_8)$	$(C_3\times C_{12})$ . $D_4$	$(C_2\times C_{12})$ . $D_6$	all 28

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

Homology

Abelianization:	$C_{2} \times C_{8} $
Schur multiplier:	$C_{2} \times C_{6}$
Commutator length:	$1$

Subgroups

There are 316 subgroups in 150 conjugacy classes, 81 normal (19 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2\times C_4$	$G/Z \simeq$ $C_6:S_3$
Commutator:	$G' \simeq$ $C_3\times C_6$	$G/G' \simeq$ $C_2\times C_8$
Frattini:	$\Phi \simeq$ $C_2\times C_4$	$G/\Phi \simeq$ $C_6:S_3$
Fitting:	$\operatorname{Fit} \simeq$ $C_2\times C_6\times C_{12}$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $C_6^2:C_8$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_6^2$	$G/\operatorname{soc} \simeq$ $C_2\times C_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^2:C_8$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^2$

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

Normal subgroups

No diagram available: subgroups only stored up to automorphism

Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 288: $C_6^2:C_8$

Order 144: $C_{12}.D_6$ x 2, $C_2\times C_6\times C_{12}$

Order 96: $C_{12}.D_4$ x 4

Order 72: $C_6\times C_{12}$ x 4, $C_3^2:C_8$ x 2, $C_2\times C_6^2$

Order 48: $C_6:C_8$ x 8, $C_2^2\times C_{12}$ x 4

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

Order 32: $C_2^2:C_8$

Order 24: $C_2\times C_{12}$ x 16, $C_3:C_8$ x 8, $C_2^2\times C_6$ x 4

Order 18: $C_3\times C_6$ x 5

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

Order 12: $C_2\times C_6$ x 20, $C_{12}$ x 12

Order 9: $C_3^2$

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

Order 6: $C_6$ x 20

Order 4: $C_2^2$ x 5, $C_4$ x 3

Order 3: $C_3$ x 4

Order 2: $C_2$ x 5

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 288: $C_6^2:C_8$

Order 144: $C_{12}.D_6$, $C_2\times C_6\times C_{12}$

Order 96: $C_{12}.D_4$

Order 72: $C_6\times C_{12}$ x 3, $C_2\times C_6^2$, $C_3^2:C_8$

Order 48: $C_6:C_8$, $C_2^2\times C_{12}$

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

Order 32: $C_2^2:C_8$

Order 24: $C_2\times C_{12}$ x 3, $C_2^2\times C_6$, $C_3:C_8$

Order 18: $C_3\times C_6$ x 4

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

Order 12: $C_2\times C_6$ x 4, $C_{12}$ x 2

Order 9: $C_3^2$

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

Order 6: $C_6$ x 4

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

Order 3: $C_3$

Order 2: $C_2$ x 4

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 288: $C_6^2:C_8$ ($C_1$)

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

Order 72: $C_2\times C_6^2$ ($C_4$), $C_6\times C_{12}$ ($C_2^2$), $C_6\times C_{12}$ ($C_4$)

Order 48: $C_2^2\times C_{12}$ ($S_3$) x 4

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

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

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

Order 16: $C_2^2\times C_4$ ($C_3:S_3$)

Order 12: $C_2\times C_6$ ($C_3:C_8$) x 8, $C_{12}$ ($C_3:D_4$) x 8, $C_2\times C_6$ ($C_6:C_4$) x 4

Order 9: $C_3^2$ ($C_2^2:C_8$)

Order 8: $C_2^3$ ($C_3^2:C_4$), $C_2\times C_4$ ($C_3^2:C_4$), $C_2\times C_4$ ($C_6:S_3$)

Order 6: $C_6$ ($C_6:C_8$) x 4, $C_6$ ($C_6.D_4$) x 4, $C_6$ ($C_3:\OD_{16}$) x 4

Order 4: $C_2^2$ ($C_3^2:C_8$) x 2, $C_4$ ($C_6^2:C_2$) x 2, $C_2^2$ ($C_6.D_6$)

Order 3: $C_3$ ($C_{12}.D_4$) x 4

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

Order 1: $C_1$ ($C_6^2:C_8$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 288: $C_6^2:C_8$ ($C_1$)

Order 144: $C_{12}.D_6$ ($C_2$), $C_2\times C_6\times C_{12}$ ($C_2$)

Order 72: $C_2\times C_6^2$ ($C_4$), $C_6\times C_{12}$ ($C_2^2$), $C_6\times C_{12}$ ($C_4$)

Order 48: $C_2^2\times C_{12}$ ($S_3$)

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

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

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

Order 16: $C_2^2\times C_4$ ($C_3:S_3$)

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

Order 9: $C_3^2$ ($C_2^2:C_8$)

Order 8: $C_2^3$ ($C_3^2:C_4$), $C_2\times C_4$ ($C_3^2:C_4$), $C_2\times C_4$ ($C_6:S_3$)

Order 6: $C_6$ ($C_6:C_8$), $C_6$ ($C_6.D_4$), $C_6$ ($C_3:\OD_{16}$)

Order 4: $C_2^2$ ($C_6.D_6$), $C_2^2$ ($C_3^2:C_8$), $C_4$ ($C_6^2:C_2$)

Order 3: $C_3$ ($C_{12}.D_4$)

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

Order 1: $C_1$ ($C_6^2:C_8$)

Series

Derived series

$C_6^2:C_8$

$\rhd$

$C_3\times C_6$

$\rhd$

$C_1$

Chief series

$C_6^2:C_8$

$\rhd$

$C_{12}.D_6$

$\rhd$

$C_6\times C_{12}$

$\rhd$

$C_6^2$

$\rhd$

$C_3\times C_6$

$\rhd$

$C_6$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$C_6^2:C_8$

$\rhd$

$C_3\times C_6$

$\rhd$

$C_3^2$

Upper central series

$C_1$

$\lhd$

$C_2\times C_4$

$\lhd$

$C_2^2\times C_4$

Supergroups

This group is a maximal subgroup of 91 larger groups in the database.

This group is a maximal quotient of 20 larger groups in the database.

Character theory

Complex character table

See the $84 \times 84$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $47 \times 47$ rational character table.