Abstract group 96.47: $C_8:C_{12}$

• Label: 96.47
• Order: \( 2^{5} \cdot 3 \)
• Exponent: \( 2^{3} \cdot 3 \)
• Nilpotent: yes
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \cdot 3 \)
• $\card{Z(G)}$: \( 2^{3} \cdot 3 \)
• $\card{\Aut(G)}$: \( 2^{8} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{6} \)
• Perm deg.: $15$
• Trans deg.: $96$
• Rank: $2$

Group information

Description:	$C_8:C_{12}$
Order:	\(96\)\(\medspace = 2^{5} \cdot 3 \)
Exponent:	\(24\)\(\medspace = 2^{3} \cdot 3 \)
Automorphism group:	$C_2^5:D_4$, of order \(256\)\(\medspace = 2^{8} \)
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 metacyclic (hence metabelian).

Group statistics

Order	1	2	3	4	6	8	12	24
Elements	1	3	2	12	6	16	24	32	96
Conjugacy classes	1	3	2	8	6	8	16	16	60
Divisions	1	3	1	4	3	4	4	4	24
Autjugacy classes	1	2	1	2	2	1	2	1	12

Dimension	1	2	4	8
Irr. complex chars.	48	12	0	0	60
Irr. rational chars.	4	10	8	2	24

Minimal presentations

Permutation degree:	$15$
Transitive degree:	$96$
Rank:	$2$
Inequivalent generating pairs:	$12$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	3	6	8

Constructions

Presentation:	$\langle a, b \mid a^{4}=b^{24}=1, b^{a}=b^{13} \rangle$
Permutation group:	Degree $15$ $\langle(1,2,4,6,5,7,3,8)(9,10)(11,12), (1,3,5,4)(2,6,7,8)(9,11,10,12), (13,15,14) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rrr} 9 & 11 & 3 \\ 2 & 3 & 2 \\ 8 & 7 & 0 \end{array}\right), \left(\begin{array}{rrr} 4 & 7 & 7 \\ 2 & 9 & 11 \\ 3 & 10 & 8 \end{array}\right), \left(\begin{array}{rrr} 1 & 3 & 6 \\ 11 & 8 & 2 \\ 11 & 0 & 9 \end{array}\right), \left(\begin{array}{rrr} 0 & 9 & 9 \\ 10 & 12 & 3 \\ 2 & 11 & 7 \end{array}\right), \left(\begin{array}{rrr} 12 & 0 & 0 \\ 0 & 12 & 0 \\ 0 & 0 & 12 \end{array}\right), \left(\begin{array}{rrr} 8 & 2 & 2 \\ 8 & 2 & 5 \\ 12 & 1 & 11 \end{array}\right) \right\rangle \subseteq \GL_{3}(\F_{13})$
Direct product:	$C_3$ $\, \times\, $ $(C_8:C_4)$
Semidirect product:	$C_{24}$ $\,\rtimes\,$ $C_4$ (4)	$C_8$ $\,\rtimes\,$ $C_{12}$ (4)			more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_4^2$ . $C_6$	$C_6$ . $C_4^2$	$C_6$ . $\OD_{16}$ (2)	$(C_2\times C_8)$ . $C_6$ (2)	all 16

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

Homology

Abelianization:	$C_{4} \times C_{12} \simeq C_{4}^{2} \times C_{3}$
Schur multiplier:	$C_{2}$
Commutator length:	$1$

Subgroups

There are 44 subgroups in 40 conjugacy classes, 36 normal (12 characteristic).

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

Special subgroups

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

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: $C_8:C_{12}$

Order 48: $C_2\times C_{24}$ x 2, $C_4\times C_{12}$

Order 32: $C_8:C_4$

Order 24: $C_{24}$ 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 4, $C_2\times C_6$

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

Order 6: $C_6$ x 3

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

Order 3: $C_3$

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 96: $C_8:C_{12}$

Order 48: $C_2\times C_{24}$, $C_4\times C_{12}$

Order 32: $C_8:C_4$

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

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

Order 48: $C_2\times C_{24}$ ($C_2$) x 2, $C_4\times C_{12}$ ($C_2$)

Order 32: $C_8:C_4$ ($C_3$)

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

Order 16: $C_2\times C_8$ ($C_6$) x 2, $C_4^2$ ($C_6$)

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

Order 8: $C_8$ ($C_{12}$) x 4, $C_2\times C_4$ ($C_{12}$) x 2, $C_2\times C_4$ ($C_2\times C_6$)

Order 6: $C_6$ ($\OD_{16}$) x 2, $C_6$ ($C_4^2$)

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

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

Order 2: $C_2$ ($C_3\times \OD_{16}$) x 2, $C_2$ ($C_4\times C_{12}$)

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

Normal subgroups up to automorphism (quotient in parentheses)

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

Order 48: $C_2\times C_{24}$ ($C_2$), $C_4\times C_{12}$ ($C_2$)

Order 32: $C_8:C_4$ ($C_3$)

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

Order 16: $C_2\times C_8$ ($C_6$), $C_4^2$ ($C_6$)

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

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

Order 6: $C_6$ ($\OD_{16}$), $C_6$ ($C_4^2$)

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

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

Order 2: $C_2$ ($C_3\times \OD_{16}$), $C_2$ ($C_4\times C_{12}$)

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

Series

Derived series

$C_8:C_{12}$

$\rhd$

$C_2$

$\rhd$

$C_1$

Chief series

$C_8:C_{12}$

$\rhd$

$C_4\times C_{12}$

$\rhd$

$C_2\times C_{12}$

$\rhd$

$C_2\times C_6$

$\rhd$

$C_6$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$C_8:C_{12}$

$\rhd$

$C_2$

$\rhd$

$C_1$

Upper central series

$C_1$

$\lhd$

$C_2\times C_{12}$

$\lhd$

$C_8:C_{12}$

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 $24 \times 24$ rational character table.