Abstract group 96.9: $C_{12}:C_8$

• Label: 96.9
• Order: \( 2^{5} \cdot 3 \)
• Exponent: \( 2^{3} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{5} \)
• $\card{Z(G)}$: \( 2^{4} \)
• $\card{\Aut(G)}$: \( 2^{8} \cdot 3 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{7} \)
• Perm deg.: $15$
• Trans deg.: $96$
• Rank: $2$

Group information

Description:	$C_{12}:C_8$
Order:	\(96\)\(\medspace = 2^{5} \cdot 3 \)
Exponent:	\(24\)\(\medspace = 2^{3} \cdot 3 \)
Automorphism group:	$S_3\times C_2^4:D_4$, of order \(768\)\(\medspace = 2^{8} \cdot 3 \)
Composition factors:	$C_2$ x 5, $C_3$
Derived length:	$2$

This group is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), hyperelementary for $p = 2$, and an A-group.

Group statistics

Order	1	2	3	4	6	8	12
Elements	1	3	2	12	6	48	24	96
Conjugacy classes	1	3	1	12	3	16	12	48
Divisions	1	3	1	6	3	4	6	24
Autjugacy classes	1	2	1	2	2	1	2	11

Dimension	1	2	4
Irr. complex chars.	32	16	0	48
Irr. rational chars.	4	10	10	24

Minimal presentations

Permutation degree:	$15$
Transitive degree:	$96$
Rank:	$2$
Inequivalent generating pairs:	$3$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	3	6	6

Constructions

Presentation:	$\langle a, b \mid a^{8}=b^{12}=1, b^{a}=b^{5} \rangle$
Permutation group:	Degree $15$ $\langle(1,2,3,5,4,6,7,8)(10,11)(12,13,15,14), (12,14,15,13), (1,3,4,7)(2,5,6,8)(12,15)(13,14), (12,15)(13,14), (1,4)(2,6)(3,7)(5,8), (9,10,11)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{rrrrrr} 1 & -1 & 0 & -1 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & -1 & -1 & 0 & 0 \\ -1 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{array}\right), \left(\begin{array}{rrrrrr} -1 & 0 & 1 & 0 & 0 & 0 \\ 0 & -1 & 0 & -1 & 0 & 0 \\ -1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & -1 \\ 0 & 0 & 0 & 0 & 1 & 0 \end{array}\right) \right\rangle \subseteq \GL_{6}(\Z)$
	$\left\langle \left(\begin{array}{rrr} 3 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 3 \end{array}\right), \left(\begin{array}{rrr} 2 & 3 & 0 \\ 0 & 1 & 4 \\ 1 & 1 & 0 \end{array}\right), \left(\begin{array}{rrr} 4 & 1 & 3 \\ 1 & 3 & 1 \\ 3 & 1 & 4 \end{array}\right), \left(\begin{array}{rrr} 1 & 1 & 3 \\ 1 & 0 & 1 \\ 3 & 1 & 1 \end{array}\right), \left(\begin{array}{rrr} 2 & 1 & 2 \\ 1 & 4 & 3 \\ 2 & 3 & 3 \end{array}\right), \left(\begin{array}{rrr} 4 & 0 & 0 \\ 0 & 4 & 0 \\ 0 & 0 & 4 \end{array}\right) \right\rangle \subseteq \GL_{3}(\F_{5})$
Direct product:	$C_4$ $\, \times\, $ $(C_3:C_8)$
Semidirect product:	$C_{12}$ $\,\rtimes\,$ $C_8$ (4)	$C_3$ $\,\rtimes\,$ $(C_4\times C_8)$			more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_4^2$ . $S_3$	$C_6$ . $C_4^2$	$C_4$ . $(C_4\times S_3)$ (2)	$(C_2\times C_4)$ . $D_6$	all 15

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

Homology

Abelianization:	$C_{4} \times C_{8} $
Schur multiplier:	$C_{4}$
Commutator length:	$1$

Subgroups

There are 58 subgroups in 44 conjugacy classes, 37 normal (11 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_4^2$	$G/Z \simeq$ $S_3$
Commutator:	$G' \simeq$ $C_3$	$G/G' \simeq$ $C_4\times C_8$
Frattini:	$\Phi \simeq$ $C_2\times C_4$	$G/\Phi \simeq$ $D_6$
Fitting:	$\operatorname{Fit} \simeq$ $C_4\times C_{12}$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $C_{12}:C_8$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2\times C_6$	$G/\operatorname{soc} \simeq$ $C_2\times C_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_4\times C_8$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

Classes of subgroups up to automorphism

Normal subgroups

Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 96: $C_{12}:C_8$

Order 48: $C_6:C_8$ x 2, $C_4\times C_{12}$

Order 32: $C_4\times C_8$

Order 24: $C_3:C_8$ x 4, $C_2\times C_{12}$ x 3

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

Order 12: $C_{12}$ x 6, $C_2\times C_6$

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

Order 6: $C_6$ x 3

Order 4: $C_4$ x 6, $C_2^2$

Order 3: $C_3$

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 96: $C_{12}:C_8$

Order 48: $C_6:C_8$, $C_4\times C_{12}$

Order 32: $C_4\times C_8$

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

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

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

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

Order 6: $C_6$ x 2

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

Order 3: $C_3$

Order 2: $C_2$ x 2

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 96: $C_{12}:C_8$ ($C_1$)

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

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

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

Order 12: $C_{12}$ ($C_8$) x 4, $C_{12}$ ($C_2\times C_4$) x 2, $C_2\times C_6$ ($C_2\times C_4$)

Order 8: $C_2\times C_4$ ($C_3:C_4$) x 2, $C_2\times C_4$ ($D_6$)

Order 6: $C_6$ ($C_2\times C_8$) x 2, $C_6$ ($C_4^2$)

Order 4: $C_4$ ($C_3:C_8$) x 4, $C_4$ ($C_4\times S_3$) x 2, $C_2^2$ ($C_6:C_4$)

Order 3: $C_3$ ($C_4\times C_8$)

Order 2: $C_2$ ($C_6:C_8$) x 2, $C_2$ ($C_{12}:C_4$)

Order 1: $C_1$ ($C_{12}:C_8$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 96: $C_{12}:C_8$ ($C_1$)

Order 48: $C_6:C_8$ ($C_2$), $C_4\times C_{12}$ ($C_2$)

Order 24: $C_2\times C_{12}$ ($C_2^2$), $C_2\times C_{12}$ ($C_4$), $C_3:C_8$ ($C_4$)

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

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

Order 8: $C_2\times C_4$ ($D_6$), $C_2\times C_4$ ($C_3:C_4$)

Order 6: $C_6$ ($C_2\times C_8$), $C_6$ ($C_4^2$)

Order 4: $C_2^2$ ($C_6:C_4$), $C_4$ ($C_4\times S_3$), $C_4$ ($C_3:C_8$)

Order 3: $C_3$ ($C_4\times C_8$)

Order 2: $C_2$ ($C_6:C_8$), $C_2$ ($C_{12}:C_4$)

Order 1: $C_1$ ($C_{12}:C_8$)

Series

Derived series

$C_{12}:C_8$

$\rhd$

$C_3$

$\rhd$

$C_1$

Chief series

$C_{12}:C_8$

$\rhd$

$C_6:C_8$

$\rhd$

$C_2\times C_{12}$

$\rhd$

$C_{12}$

$\rhd$

$C_6$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$C_{12}:C_8$

$\rhd$

$C_3$

Upper central series

$C_1$

$\lhd$

$C_4^2$

Supergroups

This group is a maximal subgroup of 83 larger groups in the database.

This group is a maximal quotient of 40 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 $24 \times 24$ rational character table.