Abstract group 96.103: $D_6:Q_8$

• Label: 96.103
• Order: \( 2^{5} \cdot 3 \)
• Exponent: \( 2^{2} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: \( 2^{2} \)
• $\card{\Aut(G)}$: \( 2^{7} \cdot 3 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{4} \)
• Perm deg.: $15$
• Trans deg.: $48$
• Rank: $3$

Group information

Description:	$D_6:Q_8$
Order:	\(96\)\(\medspace = 2^{5} \cdot 3 \)
Exponent:	\(12\)\(\medspace = 2^{2} \cdot 3 \)
Automorphism group:	$D_6\times C_2^5$, of order \(384\)\(\medspace = 2^{7} \cdot 3 \)
Composition factors:	$C_2$ x 5, $C_3$
Derived length:	$2$

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

Group statistics

Order	1	2	3	4	6	12
Elements	1	15	2	48	6	24	96
Conjugacy classes	1	5	1	8	3	6	24
Divisions	1	5	1	7	3	3	20
Autjugacy classes	1	4	1	6	3	3	18

Dimension	1	2	4	8
Irr. complex chars.	8	14	2	0	24
Irr. rational chars.	8	8	3	1	20

Minimal presentations

Permutation degree:	$15$
Transitive degree:	$48$
Rank:	$3$
Inequivalent generating triples:	$672$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	4	6	8

Constructions

Presentation:	$\langle a, b, c \mid a^{2}=b^{4}=c^{12}=[a,b]=1, c^{a}=b^{2}c^{11}, c^{b}=c^{7} \rangle$
Permutation group:	Degree $15$ $\langle(9,10)(11,12)(14,15), (1,2,4,6)(3,7,8,5)(9,11)(10,12), (1,3,4,8)(2,5,6,7) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 11 & 1 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 13 & 0 \\ 0 & 13 \end{array}\right), \left(\begin{array}{rr} 11 & 4 \\ 12 & 7 \end{array}\right), \left(\begin{array}{rr} 13 & 21 \\ 0 & 7 \end{array}\right), \left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/24\Z)$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$D_6$ $\,\rtimes\,$ $Q_8$ (2)	$(C_4:C_4)$ $\,\rtimes\,$ $S_3$	$(C_6:Q_8)$ $\,\rtimes\,$ $C_2$	$(C_6.D_4)$ $\,\rtimes\,$ $C_2$ (2)	all 6
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_2$ . $(S_3\times Q_8)$	$C_2$ . $(S_3\times D_4)$	$C_6$ . $(C_2\times Q_8)$	$(C_4\times D_6)$ . $C_2$	all 15

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

Homology

Abelianization:	$C_{2}^{3} $
Schur multiplier:	$C_{2}^{2}$
Commutator length:	$1$

Subgroups

There are 170 subgroups in 74 conjugacy classes, 33 normal (29 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2^2$	$G/Z \simeq$ $C_2\times D_6$
Commutator:	$G' \simeq$ $C_2\times C_6$	$G/G' \simeq$ $C_2^3$
Frattini:	$\Phi \simeq$ $C_2^2$	$G/\Phi \simeq$ $C_2\times D_6$
Fitting:	$\operatorname{Fit} \simeq$ $C_4:C_{12}$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $D_6:Q_8$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2\times C_6$	$G/\operatorname{soc} \simeq$ $C_2^3$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^2:Q_8$
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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Subgroup information

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

Classes of subgroups up to conjugation

Order 96: $D_6:Q_8$

Order 48: $D_6:C_4$ x 2, $C_6.D_4$ x 2, $C_4\times D_6$, $C_6:Q_8$, $C_4:C_{12}$

Order 32: $C_2^2:Q_8$

Order 24: $C_2\times C_{12}$ x 3, $C_6:C_4$ x 3, $C_4\times S_3$ x 2, $C_3:Q_8$ x 2, $C_2\times D_6$

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

Order 12: $D_6$ x 4, $C_3:C_4$ x 4, $C_{12}$ x 3, $C_2\times C_6$

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

Order 6: $C_6$ x 3, $S_3$ x 2

Order 4: $C_4$ x 7, $C_2^2$ x 5

Order 3: $C_3$

Order 2: $C_2$ x 5

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 96: $D_6:Q_8$

Order 48: $D_6:C_4$ x 2, $C_6.D_4$ x 2, $C_4\times D_6$, $C_6:Q_8$, $C_4:C_{12}$

Order 32: $C_2^2:Q_8$

Order 24: $C_2\times C_{12}$ x 3, $C_6:C_4$ x 3, $C_4\times S_3$, $C_3:Q_8$, $C_2\times D_6$

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

Order 12: $D_6$ x 3, $C_{12}$ x 3, $C_3:C_4$ x 3, $C_2\times C_6$

Order 8: $C_2\times C_4$ x 7, $C_2^3$, $Q_8$

Order 6: $C_6$ x 3, $S_3$

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

Order 3: $C_3$

Order 2: $C_2$ x 4

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 96: $D_6:Q_8$ ($C_1$)

Order 48: $D_6:C_4$ ($C_2$) x 2, $C_6.D_4$ ($C_2$) x 2, $C_4\times D_6$ ($C_2$), $C_6:Q_8$ ($C_2$), $C_4:C_{12}$ ($C_2$)

Order 24: $C_2\times C_{12}$ ($C_2^2$) x 3, $C_6:C_4$ ($C_2^2$) x 3, $C_2\times D_6$ ($C_2^2$)

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

Order 12: $D_6$ ($Q_8$) x 2, $C_3:C_4$ ($D_4$) x 2, $C_2\times C_6$ ($C_2^3$)

Order 8: $C_2\times C_4$ ($D_6$) x 3

Order 6: $C_6$ ($D_4:C_2$), $C_6$ ($C_2\times Q_8$), $C_6$ ($C_2\times D_4$)

Order 4: $C_2^2$ ($C_2\times D_6$)

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

Order 2: $C_2$ ($S_3\times Q_8$), $C_2$ ($S_3\times D_4$), $C_2$ ($D_{12}:C_2$)

Order 1: $C_1$ ($D_6:Q_8$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 96: $D_6:Q_8$ ($C_1$)

Order 48: $D_6:C_4$ ($C_2$) x 2, $C_6.D_4$ ($C_2$) x 2, $C_4\times D_6$ ($C_2$), $C_6:Q_8$ ($C_2$), $C_4:C_{12}$ ($C_2$)

Order 24: $C_2\times C_{12}$ ($C_2^2$) x 3, $C_6:C_4$ ($C_2^2$) x 3, $C_2\times D_6$ ($C_2^2$)

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

Order 12: $C_2\times C_6$ ($C_2^3$), $D_6$ ($Q_8$), $C_3:C_4$ ($D_4$)

Order 8: $C_2\times C_4$ ($D_6$) x 3

Order 6: $C_6$ ($D_4:C_2$), $C_6$ ($C_2\times Q_8$), $C_6$ ($C_2\times D_4$)

Order 4: $C_2^2$ ($C_2\times D_6$)

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

Order 2: $C_2$ ($S_3\times Q_8$), $C_2$ ($S_3\times D_4$), $C_2$ ($D_{12}:C_2$)

Order 1: $C_1$ ($D_6:Q_8$)

Series

Derived series

$D_6:Q_8$

$\rhd$

$C_2\times C_6$

$\rhd$

$C_1$

Chief series

$D_6:Q_8$

$\rhd$

$C_4\times D_6$

$\rhd$

$C_2\times D_6$

$\rhd$

$C_2\times C_6$

$\rhd$

$C_6$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$D_6:Q_8$

$\rhd$

$C_2\times C_6$

$\rhd$

$C_3$

Upper central series

$C_1$

$\lhd$

$C_2^2$

$\lhd$

$C_4:C_4$

Supergroups

This group is a maximal subgroup of 121 larger groups in the database.

This group is a maximal quotient of 84 larger groups in the database.

Character theory

Complex character table

See the $24 \times 24$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $20 \times 20$ rational character table.