Abstract group 96.210: $D_4:D_6$

• Label: 96.210
• Order: \( 2^{5} \cdot 3 \)
• Exponent: \( 2^{2} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \)
• $\card{Z(G)}$: \( 2^{2} \)
• $\card{\Aut(G)}$: \( 2^{9} \cdot 3 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{6} \)
• Perm deg.: $13$
• Trans deg.: $48$
• Rank: $4$

Group information

Description:	$D_4:D_6$
Order:	\(96\)\(\medspace = 2^{5} \cdot 3 \)
Exponent:	\(12\)\(\medspace = 2^{2} \cdot 3 \)
Automorphism group:	$S_3\times C_2^5:D_4$, of order \(1536\)\(\medspace = 2^{9} \cdot 3 \)
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	23	2	40	22	8	96
Conjugacy classes	1	9	1	10	7	2	30
Divisions	1	9	1	8	7	2	28
Autjugacy classes	1	4	1	3	3	1	13

Dimension	1	2	4
Irr. complex chars.	16	12	2	30
Irr. rational chars.	16	8	4	28

Minimal presentations

Permutation degree:	$13$
Transitive degree:	$48$
Rank:	$4$
Inequivalent generating quadruples:	$16380$

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

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

Homology

Abelianization:	$C_{2}^{4} $
Schur multiplier:	$C_{2}^{5}$
Commutator length:	$1$

Subgroups

There are 306 subgroups in 164 conjugacy classes, 89 normal (15 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:	$G' \simeq$ $C_6$	$G/G' \simeq$ $C_2^4$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $C_2^2\times D_6$
Fitting:	$\operatorname{Fit} \simeq$ $C_6\times D_4$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $D_4:D_6$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2\times C_6$	$G/\operatorname{soc} \simeq$ $C_2^3$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $D_4:C_2^2$
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.

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

Subgroup information

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

Classes of subgroups up to conjugation

Order 96: $D_4:D_6$

Order 48: $D_4:S_3$ x 8, $C_6:D_4$ x 2, $C_6.C_2^3$ x 2, $C_6\times D_4$, $C_4\times D_6$, $C_6:Q_8$

Order 32: $D_4:C_2^2$

Order 24: $C_6:C_4$ x 11, $C_3:D_4$ x 8, $C_4\times S_3$ x 4, $C_3:Q_8$ x 4, $C_3\times D_4$ x 4, $C_2^2\times C_6$ x 2, $C_2\times C_{12}$, $C_2\times D_6$

Order 16: $D_4:C_2$ x 8, $C_2\times D_4$ x 3, $C_2^2\times C_4$ x 3, $C_2\times Q_8$

Order 12: $C_2\times C_6$ x 9, $C_3:C_4$ x 6, $D_6$ x 4, $C_{12}$ x 2

Order 8: $C_2\times C_4$ x 16, $D_4$ x 12, $Q_8$ x 4, $C_2^3$ x 3

Order 6: $C_6$ x 7, $S_3$ x 2

Order 4: $C_2^2$ x 13, $C_4$ x 8

Order 3: $C_3$

Order 2: $C_2$ x 9

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 96: $D_4:D_6$

Order 48: $C_6\times D_4$, $C_6:D_4$, $C_6.C_2^3$, $D_4:S_3$, $C_4\times D_6$, $C_6:Q_8$

Order 32: $D_4:C_2^2$

Order 24: $C_6:C_4$ x 3, $C_2\times C_{12}$, $C_3:D_4$, $C_4\times S_3$, $C_3:Q_8$, $C_2^2\times C_6$, $C_2\times D_6$, $C_3\times D_4$

Order 16: $C_2\times D_4$ x 2, $C_2^2\times C_4$ x 2, $D_4:C_2$, $C_2\times Q_8$

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

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

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

Order 4: $C_2^2$ x 5, $C_4$ x 3

Order 3: $C_3$

Order 2: $C_2$ x 4

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 96: $D_4:D_6$ ($C_1$)

Order 48: $D_4:S_3$ ($C_2$) x 8, $C_6:D_4$ ($C_2$) x 2, $C_6.C_2^3$ ($C_2$) x 2, $C_6\times D_4$ ($C_2$), $C_4\times D_6$ ($C_2$), $C_6:Q_8$ ($C_2$)

Order 24: $C_6:C_4$ ($C_2^2$) x 11, $C_3:D_4$ ($C_2^2$) x 8, $C_4\times S_3$ ($C_2^2$) x 4, $C_3:Q_8$ ($C_2^2$) x 4, $C_3\times D_4$ ($C_2^2$) x 4, $C_2^2\times C_6$ ($C_2^2$) x 2, $C_2\times C_{12}$ ($C_2^2$), $C_2\times D_6$ ($C_2^2$)

Order 16: $C_2\times D_4$ ($S_3$)

Order 12: $C_3:C_4$ ($C_2^3$) x 6, $C_2\times C_6$ ($C_2^3$) x 5, $D_6$ ($C_2^3$) x 2, $C_{12}$ ($C_2^3$) x 2

Order 8: $D_4$ ($D_6$) x 4, $C_2^3$ ($D_6$) x 2, $C_2\times C_4$ ($D_6$)

Order 6: $C_6$ ($D_4:C_2$) x 2, $C_6$ ($C_2^4$)

Order 4: $C_2^2$ ($C_2\times D_6$) x 5, $C_4$ ($C_2\times D_6$) x 2

Order 3: $C_3$ ($D_4:C_2^2$)

Order 2: $C_2$ ($D_4:S_3$) x 2, $C_2$ ($C_2^2\times D_6$)

Order 1: $C_1$ ($D_4:D_6$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 96: $D_4:D_6$ ($C_1$)

Order 48: $C_6\times D_4$ ($C_2$), $C_6:D_4$ ($C_2$), $C_6.C_2^3$ ($C_2$), $D_4:S_3$ ($C_2$), $C_4\times D_6$ ($C_2$), $C_6:Q_8$ ($C_2$)

Order 24: $C_6:C_4$ ($C_2^2$) x 3, $C_2\times C_{12}$ ($C_2^2$), $C_3:D_4$ ($C_2^2$), $C_4\times S_3$ ($C_2^2$), $C_3:Q_8$ ($C_2^2$), $C_2^2\times C_6$ ($C_2^2$), $C_2\times D_6$ ($C_2^2$), $C_3\times D_4$ ($C_2^2$)

Order 16: $C_2\times D_4$ ($S_3$)

Order 12: $C_2\times C_6$ ($C_2^3$) x 2, $C_3:C_4$ ($C_2^3$) x 2, $D_6$ ($C_2^3$), $C_{12}$ ($C_2^3$)

Order 8: $C_2^3$ ($D_6$), $D_4$ ($D_6$), $C_2\times C_4$ ($D_6$)

Order 6: $C_6$ ($C_2^4$), $C_6$ ($D_4:C_2$)

Order 4: $C_2^2$ ($C_2\times D_6$) x 2, $C_4$ ($C_2\times D_6$)

Order 3: $C_3$ ($D_4:C_2^2$)

Order 2: $C_2$ ($C_2^2\times D_6$), $C_2$ ($D_4:S_3$)

Order 1: $C_1$ ($D_4:D_6$)

Series

Derived series

$D_4:D_6$

$\rhd$

$C_6$

$\rhd$

$C_1$

Chief series

$D_4:D_6$

$\rhd$

$C_4\times D_6$

$\rhd$

$C_2\times C_{12}$

$\rhd$

$C_2\times C_4$

$\rhd$

$C_2^2$

$\rhd$

$C_2$

$\rhd$

$C_1$

Lower central series

$D_4:D_6$

$\rhd$

$C_6$

$\rhd$

$C_3$

Upper central series

$C_1$

$\lhd$

$C_2^2$

$\lhd$

$C_2\times D_4$

Supergroups

This group is a maximal subgroup of 100 larger groups in the database.

This group is a maximal quotient of 124 larger groups in the database.

Character theory

Complex character table

See the $30 \times 30$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $28 \times 28$ rational character table.