Abstract group 96.222: $C_6.C_2^4$

• Label: 96.222
• Order: \( 2^{5} \cdot 3 \)
• Exponent: \( 2^{2} \cdot 3 \)
• Nilpotent: yes
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \cdot 3 \)
• $\card{Z(G)}$: \( 2^{3} \cdot 3 \)
• $\card{\Aut(G)}$: \( 2^{11} \cdot 3^{2} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{9} \cdot 3^{2} \)
• Perm deg.: $15$
• Trans deg.: $96$
• Rank: $4$

Group information

Description:	$C_6.C_2^4$
Order:	\(96\)\(\medspace = 2^{5} \cdot 3 \)
Exponent:	\(12\)\(\medspace = 2^{2} \cdot 3 \)
Automorphism group:	$C_2^6.(D_6\times S_4)$, of order \(18432\)\(\medspace = 2^{11} \cdot 3^{2} \)
Composition factors:	$C_2$ x 5, $C_3$
Nilpotency class:	$2$
Derived length:	$2$

This group is nonabelian, elementary for $p = 2$ (hence nilpotent, solvable, supersolvable, monomial, and hyperelementary), and metabelian.

Group statistics

Order	1	2	3	4	6	12
Elements	1	7	2	24	14	48	96
Conjugacy classes	1	7	2	12	14	24	60
Divisions	1	7	1	12	7	12	40
Autjugacy classes	1	2	1	1	2	1	8

Dimension	1	2	4
Irr. complex chars.	48	12	0	60
Irr. rational chars.	16	20	4	40

Minimal presentations

Permutation degree:	$15$
Transitive degree:	$96$
Rank:	$4$
Inequivalent generating quadruples:	$1400$

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:	$\langle a, b, c, d \mid a^{2}=c^{2}=d^{12}=[a,b]=[a,c]=[a,d]=[b,c]=[c,d]=1, b^{2}=d^{6}, d^{b}=d^{7} \rangle$
Permutation group:	Degree $15$ $\langle(1,2,4,6)(3,7,8,5), (1,3,4,8)(2,5,6,7)(9,10)(11,12), (1,4)(2,6)(3,8)(5,7)(9,11)(10,12), (9,10)(11,12), (13,15,14), (1,4)(2,6)(3,8)(5,7)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{rrrrrr} 1 & -1 & -1 & 0 & 0 & 0 \\ 1 & -1 & -1 & -1 & 0 & 0 \\ 1 & 0 & 0 & 1 & 0 & 0 \\ -1 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{array}\right), \left(\begin{array}{rrrrrr} 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & 0 & 0 & -1 \end{array}\right), \left(\begin{array}{rrrrrr} -1 & 0 & 1 & 0 & 0 & 0 \\ 0 & -1 & 0 & -1 & 0 & 0 \\ -1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & -1 \end{array}\right), \left(\begin{array}{rrrrrr} 0 & 1 & 0 & 1 & 0 & 0 \\ 0 & 0 & -1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ -1 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{array}\right) \right\rangle \subseteq \GL_{6}(\Z)$
	$\left\langle \left(\begin{array}{llll}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ \alpha & 0 & 1 & \alpha \\ 0 & 0 & 0 & 1 \\ \end{array}\right), \left(\begin{array}{llll}\alpha & 0 & 0 & 0 \\ 0 & \alpha & 0 & 0 \\ 0 & 0 & \alpha & 0 \\ 0 & 0 & 0 & \alpha \\ \end{array}\right), \left(\begin{array}{llll}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ \alpha^{2} & 0 & 1 & \alpha^{2} \\ 0 & 0 & 0 & 1 \\ \end{array}\right), \left(\begin{array}{llll}\alpha & 0 & 0 & 0 \\ \alpha & \alpha & 0 & \alpha \\ 1 & \alpha & \alpha & \alpha \\ 0 & 0 & 0 & \alpha \\ \end{array}\right), \left(\begin{array}{llll}0 & 0 & 0 & \alpha \\ 0 & \alpha & 0 & 0 \\ 0 & 1 & \alpha & \alpha \\ \alpha & 0 & 0 & 0 \\ \end{array}\right), \left(\begin{array}{llll}\alpha & 0 & 0 & \alpha^{2} \\ 1 & 1 & 0 & 1 \\ \alpha^{2} & 0 & 1 & \alpha^{2} \\ \alpha^{2} & 0 & 0 & \alpha \\ \end{array}\right) \right\rangle \subseteq \GL_{4}(\F_{4}) = \GL_{4}(\F_{2}[\alpha]/(\alpha^{2} + \alpha + 1))$
Direct product:	$C_2$ ${}^2$ $\, \times\, $ $C_3$ $\, \times\, $ $Q_8$
Semidirect product:	not computed
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_6$ . $C_2^4$	$C_{12}$ . $C_2^3$	$(C_2^2\times C_4)$ . $C_6$	$(C_2\times C_6)$ . $C_2^3$	all 12

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

Homology

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

Subgroups

There are 156 subgroups, all normal (8 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2^2\times C_6$	$G/Z \simeq$ $C_2^2$
Commutator:	$G' \simeq$ $C_2$	$G/G' \simeq$ $C_2^3\times C_6$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $C_2^3\times C_6$
Fitting:	$\operatorname{Fit} \simeq$ $C_6.C_2^4$	$G/\operatorname{Fit} \simeq$ $C_1$
Radical:	$R \simeq$ $C_6.C_2^4$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^2\times C_6$	$G/\operatorname{soc} \simeq$ $C_2^2$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^2\times Q_8$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3$

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

For the default diagram, subgroups are sorted vertically by the number of prime divisors (counted with multiplicity) in their orders.
To see subgroups sorted vertically by order instead, check this box.

Sorry, your browser does not support the subgroup diagram.

Subgroup information Click on a subgroup in the diagram to see information about it.

See a full page version of the diagram

See a full page version of the diagram

See a full page version of the diagram

See a full page version of the diagram

Classes of subgroups up to conjugation

Order 96: $C_6.C_2^4$

Order 48: $C_6\times Q_8$ x 12, $C_2^2\times C_{12}$ x 3

Order 32: $C_2^2\times Q_8$

Order 24: $C_2\times C_{12}$ x 18, $C_3\times Q_8$ x 16, $C_2^2\times C_6$

Order 16: $C_2\times Q_8$ x 12, $C_2^2\times C_4$ x 3

Order 12: $C_{12}$ x 12, $C_2\times C_6$ x 7

Order 8: $C_2\times C_4$ x 18, $Q_8$ x 16, $C_2^3$

Order 6: $C_6$ x 7

Order 4: $C_4$ x 12, $C_2^2$ x 7

Order 3: $C_3$

Order 2: $C_2$ x 7

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 96: $C_6.C_2^4$

Order 48: $C_6\times Q_8$, $C_2^2\times C_{12}$

Order 32: $C_2^2\times Q_8$

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

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

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

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

Order 6: $C_6$ x 2

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

Order 3: $C_3$

Order 2: $C_2$ x 2

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 96: $C_6.C_2^4$ ($C_1$)

Order 48: $C_6\times Q_8$ ($C_2$) x 12, $C_2^2\times C_{12}$ ($C_2$) x 3

Order 32: $C_2^2\times Q_8$ ($C_3$)

Order 24: $C_2\times C_{12}$ ($C_2^2$) x 18, $C_3\times Q_8$ ($C_2^2$) x 16, $C_2^2\times C_6$ ($C_2^2$)

Order 16: $C_2\times Q_8$ ($C_6$) x 12, $C_2^2\times C_4$ ($C_6$) x 3

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

Order 8: $C_2\times C_4$ ($C_2\times C_6$) x 18, $Q_8$ ($C_2\times C_6$) x 16, $C_2^3$ ($C_2\times C_6$)

Order 6: $C_6$ ($C_2\times Q_8$) x 6, $C_6$ ($C_2^4$)

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

Order 3: $C_3$ ($C_2^2\times Q_8$)

Order 2: $C_2$ ($C_6\times Q_8$) x 6, $C_2$ ($C_2^3\times C_6$)

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

Normal subgroups up to automorphism (quotient in parentheses)

Order 96: $C_6.C_2^4$ ($C_1$)

Order 48: $C_6\times Q_8$ ($C_2$), $C_2^2\times C_{12}$ ($C_2$)

Order 32: $C_2^2\times Q_8$ ($C_3$)

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

Order 16: $C_2\times Q_8$ ($C_6$), $C_2^2\times C_4$ ($C_6$)

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

Order 8: $C_2^3$ ($C_2\times C_6$), $Q_8$ ($C_2\times C_6$), $C_2\times C_4$ ($C_2\times C_6$)

Order 6: $C_6$ ($C_2^4$), $C_6$ ($C_2\times Q_8$)

Order 4: $C_2^2$ ($C_2^2\times C_6$), $C_2^2$ ($C_3\times Q_8$), $C_4$ ($C_2^2\times C_6$)

Order 3: $C_3$ ($C_2^2\times Q_8$)

Order 2: $C_2$ ($C_2^3\times C_6$), $C_2$ ($C_6\times Q_8$)

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

Series

Derived series

$C_6.C_2^4$

$\rhd$

$C_2$

$\rhd$

$C_1$

Chief series

$C_6.C_2^4$

$\rhd$

$C_2^2\times C_{12}$

$\rhd$

$C_2\times C_{12}$

$\rhd$

$C_2\times C_6$

$\rhd$

$C_6$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$C_6.C_2^4$

$\rhd$

$C_2$

$\rhd$

$C_1$

Upper central series

$C_1$

$\lhd$

$C_2^2\times C_6$

$\lhd$

$C_6.C_2^4$

Supergroups

This group is a maximal subgroup of 86 larger groups in the database.

This group is a maximal quotient of 46 larger groups in the database.

Character theory

Complex character table

See the $60 \times 60$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $40 \times 40$ rational character table.