Abstract group 192.328: $D_6.D_8$

• Label: 192.328
• Order: \( 2^{6} \cdot 3 \)
• Exponent: \( 2^{3} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \)
• $\card{Z(G)}$: \( 2^{2} \)
• $\card{\Aut(G)}$: \( 2^{9} \cdot 3 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{5} \)
• Perm deg.: $15$
• Trans deg.: $48$
• Rank: $3$

Group information

Description:	$D_6.D_8$
Order:	\(192\)\(\medspace = 2^{6} \cdot 3 \)
Exponent:	\(24\)\(\medspace = 2^{3} \cdot 3 \)
Automorphism group:	$C_2^5\times S_3\times D_4$, of order \(1536\)\(\medspace = 2^{9} \cdot 3 \)
Composition factors:	$C_2$ x 6, $C_3$
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	8	12	24
Elements	1	47	2	48	22	32	24	16	192
Conjugacy classes	1	11	1	8	5	8	4	4	42
Divisions	1	11	1	6	5	2	3	1	30
Autjugacy classes	1	6	1	5	4	2	3	1	23

Dimension	1	2	4	8
Irr. complex chars.	16	20	6	0	42
Irr. rational chars.	8	12	8	2	30

Minimal presentations

Permutation degree:	$15$
Transitive degree:	$48$
Rank:	$3$
Inequivalent generating triples:	$1344$

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:	$\langle a, b, c, d \mid a^{2}=b^{2}=c^{4}=d^{12}=[a,b]=[b,c]=1, c^{a}=cd^{3}, d^{a}=d^{7}, d^{b}=d^{5}, d^{c}=d^{7} \rangle$
Permutation group:	Degree $15$ $\langle(2,3)(4,5)(6,7), (9,12)(11,14)(13,15), (4,6,5,7)(8,9,10,13)(11,15,14,12) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 8 & 3 \\ 9 & 2 \end{array}\right), \left(\begin{array}{rr} 4 & 3 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 14 & 0 \\ 10 & 4 \end{array}\right), \left(\begin{array}{rr} 8 & 5 \\ 5 & 3 \end{array}\right), \left(\begin{array}{rr} 13 & 3 \\ 9 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 5 \\ 5 & 11 \end{array}\right), \left(\begin{array}{rr} 4 & 0 \\ 0 & 4 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/15\Z)$
Direct product:	$S_3$ $\, \times\, $ $(D_4:C_4)$
Semidirect product:	$(S_3\times D_4)$ $\,\rtimes\,$ $C_4$ (4)	$D_4$ $\,\rtimes\,$ $(C_4\times S_3)$ (2)	$(D_6.D_4)$ $\,\rtimes\,$ $C_2$	$(C_8\times D_6)$ $\,\rtimes\,$ $C_2$	all 17
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$D_6$ . $D_8$ (2)	$D_6$ . $\SD_{16}$ (2)	$(C_4\times S_3)$ . $D_4$ (2)	$C_4$ . $(S_3\times D_4)$	all 27

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

Homology

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

Subgroups

There are 712 subgroups in 202 conjugacy classes, 63 normal (37 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2^2$	$G/Z \simeq$ $S_3\times D_4$
Commutator:	$G' \simeq$ $C_{12}$	$G/G' \simeq$ $C_2^2\times C_4$
Frattini:	$\Phi \simeq$ $C_2\times C_4$	$G/\Phi \simeq$ $C_2\times D_6$
Fitting:	$\operatorname{Fit} \simeq$ $D_4:C_{12}$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $D_6.D_8$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2\times C_6$	$G/\operatorname{soc} \simeq$ $C_2\times D_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^2.D_8$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

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Classes of subgroups up to automorphism

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Normal subgroups

Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 192: $D_6.D_8$

Order 96: $D_6.D_4$, $D_4:C_{12}$, $C_6.D_8$, $D_{12}:C_4$, $D_4\times D_6$, $D_{12}:C_4$, $C_8\times D_6$

Order 64: $C_2^2.D_8$

Order 48: $S_3\times D_4$ x 6, $S_3\times C_8$ x 2, $C_4\times D_6$ x 2, $C_6:C_8$, $C_2^2\times D_6$, $C_6\times D_4$, $C_6:D_4$, $C_2\times D_{12}$, $C_2\times C_{24}$, $C_4:C_{12}$, $C_{12}:C_4$, $C_6.D_4$

Order 32: $D_4:C_4$ x 4, $C_2^2\times D_4$, $C_2^2\times C_8$, $C_2^2.D_4$

Order 24: $C_2\times D_6$ x 10, $C_4\times S_3$ x 6, $C_3:D_4$ x 4, $D_{12}$ x 3, $C_3\times D_4$ x 3, $C_2\times C_{12}$ x 2, $C_6:C_4$ x 2, $C_{24}$, $C_2^2\times C_6$, $C_3:C_8$

Order 16: $C_2\times D_4$ x 9, $C_2\times C_8$ x 4, $C_4:C_4$ x 3, $C_2^2\times C_4$ x 2, $C_2^4$

Order 12: $D_6$ x 18, $C_2\times C_6$ x 5, $C_{12}$ x 3, $C_3:C_4$ x 3

Order 8: $C_2^3$ x 11, $D_4$ x 10, $C_2\times C_4$ x 10, $C_8$ x 2

Order 6: $S_3$ x 6, $C_6$ x 5

Order 4: $C_2^2$ x 23, $C_4$ x 6

Order 3: $C_3$

Order 2: $C_2$ x 11

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 192: $D_6.D_8$

Order 96: $D_6.D_4$, $D_4:C_{12}$, $C_6.D_8$, $D_{12}:C_4$, $D_4\times D_6$, $D_{12}:C_4$, $C_8\times D_6$

Order 64: $C_2^2.D_8$

Order 48: $S_3\times D_4$ x 2, $C_4\times D_6$ x 2, $C_6:C_8$, $C_2^2\times D_6$, $C_6\times D_4$, $C_6:D_4$, $S_3\times C_8$, $C_2\times D_{12}$, $C_2\times C_{24}$, $C_4:C_{12}$, $C_{12}:C_4$, $C_6.D_4$

Order 32: $D_4:C_4$ x 4, $C_2^2\times D_4$, $C_2^2\times C_8$, $C_2^2.D_4$

Order 24: $C_2\times D_6$ x 5, $C_4\times S_3$ x 3, $C_2\times C_{12}$ x 2, $C_6:C_4$ x 2, $D_{12}$ x 2, $C_3\times D_4$ x 2, $C_3:D_4$, $C_{24}$, $C_2^2\times C_6$, $C_3:C_8$

Order 16: $C_2\times D_4$ x 5, $C_2\times C_8$ x 3, $C_4:C_4$ x 3, $C_2^2\times C_4$ x 2, $C_2^4$

Order 12: $D_6$ x 7, $C_2\times C_6$ x 4, $C_{12}$ x 3, $C_3:C_4$ x 2

Order 8: $C_2\times C_4$ x 7, $C_2^3$ x 6, $D_4$ x 5, $C_8$ x 2

Order 6: $C_6$ x 4, $S_3$ x 2

Order 4: $C_2^2$ x 11, $C_4$ x 5

Order 3: $C_3$

Order 2: $C_2$ x 6

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 192: $D_6.D_8$ ($C_1$)

Order 96: $D_6.D_4$ ($C_2$), $D_4:C_{12}$ ($C_2$), $C_6.D_8$ ($C_2$), $D_{12}:C_4$ ($C_2$), $D_4\times D_6$ ($C_2$), $D_{12}:C_4$ ($C_2$), $C_8\times D_6$ ($C_2$)

Order 48: $S_3\times D_4$ ($C_4$) x 4, $C_6:C_8$ ($C_2^2$), $C_6\times D_4$ ($C_2^2$), $C_2\times D_{12}$ ($C_2^2$), $C_4\times D_6$ ($C_2^2$), $C_2\times C_{24}$ ($C_2^2$), $C_4:C_{12}$ ($C_2^2$), $C_{12}:C_4$ ($C_2^2$)

Order 32: $D_4:C_4$ ($S_3$)

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

Order 16: $C_2\times C_8$ ($D_6$), $C_4:C_4$ ($D_6$), $C_2\times D_4$ ($D_6$)

Order 12: $D_6$ ($\SD_{16}$) x 2, $D_6$ ($D_8$) x 2, $D_6$ ($C_2^2:C_4$) x 2, $C_3:C_4$ ($C_2^2:C_4$) x 2, $C_2\times C_6$ ($C_2\times D_4$), $C_{12}$ ($C_2\times D_4$), $C_{12}$ ($C_2^2\times C_4$)

Order 8: $D_4$ ($C_4\times S_3$) x 2, $C_2\times C_4$ ($C_2\times D_6$)

Order 6: $S_3$ ($D_4:C_4$) x 4, $C_6$ ($C_2\times \SD_{16}$), $C_6$ ($C_2\times D_8$), $C_6$ ($C_2^3:C_4$)

Order 4: $C_2^2$ ($S_3\times D_4$), $C_4$ ($S_3\times D_4$), $C_4$ ($C_4\times D_6$)

Order 3: $C_3$ ($C_2^2.D_8$)

Order 2: $C_2$ ($D_6.D_4$), $C_2$ ($S_3\times \SD_{16}$), $C_2$ ($S_3\times D_8$)

Order 1: $C_1$ ($D_6.D_8$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 192: $D_6.D_8$ ($C_1$)

Order 96: $D_6.D_4$ ($C_2$), $D_4:C_{12}$ ($C_2$), $C_6.D_8$ ($C_2$), $D_{12}:C_4$ ($C_2$), $D_4\times D_6$ ($C_2$), $D_{12}:C_4$ ($C_2$), $C_8\times D_6$ ($C_2$)

Order 48: $C_6:C_8$ ($C_2^2$), $C_6\times D_4$ ($C_2^2$), $S_3\times D_4$ ($C_4$), $C_2\times D_{12}$ ($C_2^2$), $C_4\times D_6$ ($C_2^2$), $C_2\times C_{24}$ ($C_2^2$), $C_4:C_{12}$ ($C_2^2$), $C_{12}:C_4$ ($C_2^2$)

Order 32: $D_4:C_4$ ($S_3$)

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

Order 16: $C_2\times C_8$ ($D_6$), $C_4:C_4$ ($D_6$), $C_2\times D_4$ ($D_6$)

Order 12: $C_2\times C_6$ ($C_2\times D_4$), $D_6$ ($\SD_{16}$), $D_6$ ($D_8$), $D_6$ ($C_2^2:C_4$), $C_{12}$ ($C_2\times D_4$), $C_{12}$ ($C_2^2\times C_4$), $C_3:C_4$ ($C_2^2:C_4$)

Order 8: $D_4$ ($C_4\times S_3$), $C_2\times C_4$ ($C_2\times D_6$)

Order 6: $C_6$ ($C_2\times \SD_{16}$), $C_6$ ($C_2\times D_8$), $C_6$ ($C_2^3:C_4$), $S_3$ ($D_4:C_4$)

Order 4: $C_2^2$ ($S_3\times D_4$), $C_4$ ($S_3\times D_4$), $C_4$ ($C_4\times D_6$)

Order 3: $C_3$ ($C_2^2.D_8$)

Order 2: $C_2$ ($D_6.D_4$), $C_2$ ($S_3\times \SD_{16}$), $C_2$ ($S_3\times D_8$)

Order 1: $C_1$ ($D_6.D_8$)

Series

Derived series

$D_6.D_8$

$\rhd$

$C_{12}$

$\rhd$

$C_1$

Chief series

$D_6.D_8$

$\rhd$

$D_4\times D_6$

$\rhd$

$C_6\times D_4$

$\rhd$

$C_2\times C_{12}$

$\rhd$

$C_2\times C_6$

$\rhd$

$C_6$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$D_6.D_8$

$\rhd$

$C_{12}$

$\rhd$

$C_6$

$\rhd$

$C_3$

Upper central series

$C_1$

$\lhd$

$C_2^2$

$\lhd$

$C_2\times C_4$

$\lhd$

$D_4:C_4$

Supergroups

This group is a maximal subgroup of 158 larger groups in the database.

This group is a maximal quotient of 79 larger groups in the database.

Character theory

Complex character table

See the $42 \times 42$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $30 \times 30$ rational character table.