Abstract group 96.183: D8:C6D_8:C_6

• Label: 96.183
• Order: $2^{5} \cdot 3$
• Exponent: $2^{3} \cdot 3$
• Nilpotent: yes
• Solvable: yes
• #Gab\card{G^{\mathrm{ab}}}: $2^{3} \cdot 3$
• #Z(G)\card{Z(G)}: $2 \cdot 3$
• #Aut⁡(G)\card{\Aut(G)}: $2^{7}$
• #Out(G)\card{\mathrm{Out}(G)}: $2^{3}$
• Perm deg.: $11$
• Trans deg.: $24$
• Rank: $3$

Group information

Description:	$D_8:C_6$
Order:	$96$$\medspace = 2^{5} \cdot 3$
Exponent:	$24$$\medspace = 2^{3} \cdot 3$
Automorphism group:	$C_2\times D_4^2$, of order $128$$\medspace = 2^{7}$
Composition factors:	$C_2$ x 5, $C_3$
Nilpotency class:	$3$
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	8	12	24
Elements	1	15	2	8	30	8	16	16	96
Conjugacy classes	1	5	2	3	10	2	6	4	33
Divisions	1	5	1	3	5	2	3	2	22
Autjugacy classes	1	4	1	3	4	1	3	1	18

Dimension	1	2	4	8
Irr. complex chars.	24	6	3	0	33
Irr. rational chars.	8	10	3	1	22

Minimal presentations

Permutation degree:	$11$
Transitive degree:	$24$
Rank:	$3$
Inequivalent generating triples:	$2184$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	4	8	8
Arbitrary	4	6	6

Constructions

Presentation:	$\langle a, b, c \mid a^{2}=b^{2}=c^{24}=[a,b]=1, c^{a}=c^{13}, c^{b}=c^{19} \rangle$
Permutation group:	Degree $11$ $\langle(1,5)(2,6)(3,7)(4,8), (3,4)(5,8)(6,7), (1,2)(3,4), (9,11,10), (1,4,2,3)(5,7,6,8), (1,2)(3,4)(5,6)(7,8)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{rrrrrr} 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 1 & 0 & 0 \\ 0 & 1 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & -1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{array}\right), \left(\begin{array}{rrrrrr} 1 & -1 & -1 & -1 & 0 & 0 \\ 1 & -1 & 0 & -1 & 0 & 0 \\ 1 & -1 & 0 & 0 & 0 & 0 \\ 0 & 1 & -1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{array}\right), \left(\begin{array}{rrrrrr} 0 & -1 & 0 & -1 & 0 & 0 \\ 0 & -1 & 0 & 0 & 0 & 0 \\ 0 & -1 & 1 & 0 & 0 & 0 \\ -1 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & -1 & 1 \end{array}\right) \right\rangle \subseteq \GL_{6}(\Z)$
	$\left\langle \left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 23 & 9 \\ 0 & 5 \end{array}\right), \left(\begin{array}{rr} 1 & 6 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 11 & 0 \\ 0 & 5 \end{array}\right), \left(\begin{array}{rr} 1 & 0 \\ 0 & 19 \end{array}\right), \left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/24\Z)$
Transitive group:	24T114				more information
Direct product:	$C_3$ $\, \times\,$ $(D_8:C_2)$
Semidirect product:	$D_8$ $\,\rtimes\,$ $C_6$ (2)	$\SD_{16}$ $\,\rtimes\,$ $C_6$ (2)	$\OD_{16}$ $\,\rtimes\,$ $C_6$	$C_{24}$ $\,\rtimes\,$ $C_2^2$ (2)	all 16
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_{12}$ . $D_4$	$C_{12}$ . $C_2^3$	$(C_2\times C_6)$ . $D_4$	$D_4$ . $(C_2\times C_6)$ (2)	all 12

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

Homology

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

Subgroups

There are 116 subgroups in 68 conjugacy classes, 40 normal (24 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_6$	$G/Z \simeq$ $C_2\times D_4$
Commutator:	$G' \simeq$ $C_4$	$G/G' \simeq$ $C_2^2\times C_6$
Frattini:	$\Phi \simeq$ $C_4$	$G/\Phi \simeq$ $C_2^2\times C_6$
Fitting:	$\operatorname{Fit} \simeq$ $D_8:C_6$	$G/\operatorname{Fit} \simeq$ $C_1$
Radical:	$R \simeq$ $D_8:C_6$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_6$	$G/\operatorname{soc} \simeq$ $C_2\times D_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $D_8:C_2$
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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Subgroup information

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

Classes of subgroups up to conjugation

Order 96: $D_8:C_6$

Order 48: $C_3\times \SD_{16}$ x 2, $C_3\times D_8$ x 2, $D_4:C_6$, $C_6\times D_4$, $C_3\times \OD_{16}$

Order 32: $D_8:C_2$

Order 24: $C_3\times D_4$ x 5, $C_2\times C_{12}$ x 2, $C_{24}$ x 2, $C_2^2\times C_6$, $C_3\times Q_8$

Order 16: $\SD_{16}$ x 2, $D_8$ x 2, $\OD_{16}$, $D_4:C_2$, $C_2\times D_4$

Order 12: $C_2\times C_6$ x 6, $C_{12}$ x 3

Order 8: $D_4$ x 5, $C_2\times C_4$ x 2, $C_8$ x 2, $C_2^3$, $Q_8$

Order 6: $C_6$ x 5

Order 4: $C_2^2$ x 6, $C_4$ x 3

Order 3: $C_3$

Order 2: $C_2$ x 5

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 96: $D_8:C_6$

Order 48: $D_4:C_6$, $C_6\times D_4$, $C_3\times \SD_{16}$, $C_3\times D_8$, $C_3\times \OD_{16}$

Order 32: $D_8:C_2$

Order 24: $C_3\times D_4$ x 4, $C_2\times C_{12}$ x 2, $C_{24}$, $C_2^2\times C_6$, $C_3\times Q_8$

Order 16: $\SD_{16}$, $D_8$, $\OD_{16}$, $D_4:C_2$, $C_2\times D_4$

Order 12: $C_2\times C_6$ x 4, $C_{12}$ x 3

Order 8: $D_4$ x 4, $C_2\times C_4$ x 2, $C_2^3$, $Q_8$, $C_8$

Order 6: $C_6$ x 4

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

Order 3: $C_3$

Order 2: $C_2$ x 4

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 96: $D_8:C_6$ ($C_1$)

Order 48: $C_3\times \SD_{16}$ ($C_2$) x 2, $C_3\times D_8$ ($C_2$) x 2, $D_4:C_6$ ($C_2$), $C_6\times D_4$ ($C_2$), $C_3\times \OD_{16}$ ($C_2$)

Order 32: $D_8:C_2$ ($C_3$)

Order 24: $C_3\times D_4$ ($C_2^2$) x 3, $C_{24}$ ($C_2^2$) x 2, $C_2\times C_{12}$ ($C_2^2$), $C_3\times Q_8$ ($C_2^2$)

Order 16: $\SD_{16}$ ($C_6$) x 2, $D_8$ ($C_6$) x 2, $\OD_{16}$ ($C_6$), $D_4:C_2$ ($C_6$), $C_2\times D_4$ ($C_6$)

Order 12: $C_2\times C_6$ ($D_4$), $C_{12}$ ($C_2^3$), $C_{12}$ ($D_4$)

Order 8: $D_4$ ($C_2\times C_6$) x 3, $C_8$ ($C_2\times C_6$) x 2, $Q_8$ ($C_2\times C_6$), $C_2\times C_4$ ($C_2\times C_6$)

Order 6: $C_6$ ($C_2\times D_4$)

Order 4: $C_2^2$ ($C_3\times D_4$), $C_4$ ($C_2^2\times C_6$), $C_4$ ($C_3\times D_4$)

Order 3: $C_3$ ($D_8:C_2$)

Order 2: $C_2$ ($C_6\times D_4$)

Order 1: $C_1$ ($D_8:C_6$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 96: $D_8:C_6$ ($C_1$)

Order 48: $D_4:C_6$ ($C_2$), $C_6\times D_4$ ($C_2$), $C_3\times \SD_{16}$ ($C_2$), $C_3\times D_8$ ($C_2$), $C_3\times \OD_{16}$ ($C_2$)

Order 32: $D_8:C_2$ ($C_3$)

Order 24: $C_3\times D_4$ ($C_2^2$) x 2, $C_2\times C_{12}$ ($C_2^2$), $C_{24}$ ($C_2^2$), $C_3\times Q_8$ ($C_2^2$)

Order 16: $\SD_{16}$ ($C_6$), $D_8$ ($C_6$), $\OD_{16}$ ($C_6$), $D_4:C_2$ ($C_6$), $C_2\times D_4$ ($C_6$)

Order 12: $C_2\times C_6$ ($D_4$), $C_{12}$ ($C_2^3$), $C_{12}$ ($D_4$)

Order 8: $D_4$ ($C_2\times C_6$) x 2, $Q_8$ ($C_2\times C_6$), $C_2\times C_4$ ($C_2\times C_6$), $C_8$ ($C_2\times C_6$)

Order 6: $C_6$ ($C_2\times D_4$)

Order 4: $C_2^2$ ($C_3\times D_4$), $C_4$ ($C_2^2\times C_6$), $C_4$ ($C_3\times D_4$)

Order 3: $C_3$ ($D_8:C_2$)

Order 2: $C_2$ ($C_6\times D_4$)

Order 1: $C_1$ ($D_8:C_6$)

Series

Derived series

$D_8:C_6$

$\rhd$

$C_4$

$\rhd$

$C_1$

Chief series

$D_8:C_6$

$\rhd$

$C_6\times D_4$

$\rhd$

$C_2\times C_{12}$

$\rhd$

$C_{12}$

$\rhd$

$C_6$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$D_8:C_6$

$\rhd$

$C_4$

$\rhd$

$C_2$

$\rhd$

$C_1$

Upper central series

$C_1$

$\lhd$

$C_6$

$\lhd$

$C_2\times C_{12}$

$\lhd$

$D_8:C_6$

Supergroups

This group is a maximal subgroup of 85 larger groups in the database.

This group is a maximal quotient of 90 larger groups in the database.

Character theory

Complex character table

See the $33 \times 33$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $22 \times 22$ rational character table.