Abstract group 192.605: $D_{12}.D_4$

• Label: 192.605
• Order: \( 2^{6} \cdot 3 \)
• Exponent: \( 2^{3} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: \( 2^{2} \)
• $\card{\Aut(G)}$: \( 2^{10} \cdot 3 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{6} \)
• Perm deg.: $15$
• Trans deg.: $48$
• Rank: $3$

Group information

Description:	$D_{12}.D_4$
Order:	\(192\)\(\medspace = 2^{6} \cdot 3 \)
Exponent:	\(24\)\(\medspace = 2^{3} \cdot 3 \)
Automorphism group:	$C_2^7:D_4$, of order \(3072\)\(\medspace = 2^{10} \cdot 3 \)
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
Elements	1	55	2	24	14	48	48	192
Conjugacy classes	1	9	1	5	5	4	8	33
Divisions	1	9	1	5	5	2	5	28
Autjugacy classes	1	5	1	4	4	1	4	20

Dimension	1	2	4	8
Irr. complex chars.	8	18	7	0	33
Irr. rational chars.	8	10	9	1	28

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	not computed	not computed	not computed

Constructions

Presentation:	$\langle a, b, c, d \mid b^{8}=c^{2}=d^{6}=[a,c]=[b,c]=[c,d]=1, a^{2}=b^{4}, b^{a}=b^{3}, d^{a}=cd, d^{b}=cd^{5} \rangle$
Permutation group:	Degree $15$ $\langle(1,2)(3,4)(6,7)(9,12)(10,14)(13,15), (1,3)(2,4), (3,4)(8,9,11,13)(10,15,14,12) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rrrrrr} 0 & 0 & 1 & 1 & 0 & 0 \\ 1 & 0 & 0 & 1 & 0 & 0 \\ -1 & 1 & 1 & 1 & 0 & 0 \\ 0 & -1 & -1 & -1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & -1 \end{array}\right), \left(\begin{array}{rrrrrr} 1 & -1 & 0 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 1 & 0 & 0 \\ -1 & 1 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{array}\right), \left(\begin{array}{rrrrrr} -1 & 1 & 0 & 0 & 0 & 0 \\ -1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 1 & 0 & 0 \\ 1 & -1 & -1 & -1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 1 & 0 \end{array}\right) \right\rangle \subseteq \GL_{6}(\Z)$
	$\left\langle \left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 6 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 11 & 0 \\ 12 & 11 \end{array}\right), \left(\begin{array}{rr} 5 & 9 \\ 0 & 23 \end{array}\right), \left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 5 & 1 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 13 & 0 \\ 0 & 13 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/24\Z)$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(Q_8:D_6)$ $\,\rtimes\,$ $C_2$ (2)	$(C_2\times Q_8)$ $\,\rtimes\,$ $D_6$	$(C_4:C_4)$ $\,\rtimes\,$ $D_6$	$(C_{12}.D_4)$ $\,\rtimes\,$ $C_2$	all 13
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$D_{12}$ . $D_4$ (4)	$C_4$ . $(S_3\times D_4)$ (2)	$C_2$ . $(Q_8:D_6)$	$C_2$ . $(D_6:D_4)$	all 21

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

Homology

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

Subgroups

There are 704 subgroups in 188 conjugacy classes, 47 normal (27 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2^2$	$G/Z \simeq$ $C_6:D_4$
Commutator:	$G' \simeq$ $C_2\times C_{12}$	$G/G' \simeq$ $C_2^3$
Frattini:	$\Phi \simeq$ $C_2\times C_4$	$G/\Phi \simeq$ $C_2\times D_6$
Fitting:	$\operatorname{Fit} \simeq$ $C_{12}.D_4$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $D_{12}.D_4$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2\times C_6$	$G/\operatorname{soc} \simeq$ $C_2\times D_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^2:\SD_{16}$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

No diagram available

Classes of subgroups up to automorphism

No diagram available

Normal subgroups

No diagram available

Normal subgroups up to automorphism

No diagram available

Classes of subgroups up to conjugation

Order 192: $D_{12}.D_4$

Order 96: $D_{12}:C_4$ x 2, $Q_8:D_6$ x 2, $C_{12}.D_4$, $C_2^2\times D_{12}$, $C_{12}.D_4$

Order 64: $C_2^2:\SD_{16}$

Order 48: $C_2\times D_{12}$ x 7, $Q_8:S_3$ x 4, $C_6:C_8$ x 2, $C_4:C_{12}$ x 2, $C_2^2\times D_6$, $C_6\times Q_8$, $C_2^2\times C_{12}$, $C_2^2:C_{12}$

Order 32: $D_4:C_4$ x 2, $C_2\times \SD_{16}$ x 2, $C_2^2:C_8$, $C_2^2\times D_4$, $C_2^2:Q_8$

Order 24: $D_{12}$ x 10, $C_2\times D_6$ x 10, $C_2\times C_{12}$ x 6, $C_3\times Q_8$ x 2, $C_3:C_8$ x 2, $C_2^2\times C_6$

Order 16: $C_2\times D_4$ x 7, $\SD_{16}$ x 4, $C_2\times C_8$ x 2, $C_4:C_4$ x 2, $C_2^2:C_4$, $C_2^4$, $C_2\times Q_8$, $C_2^2\times C_4$

Order 12: $D_6$ x 16, $C_2\times C_6$ x 5, $C_{12}$ x 5

Order 8: $C_2^3$ x 11, $D_4$ x 10, $C_2\times C_4$ x 6, $Q_8$ x 2, $C_8$ x 2

Order 6: $C_6$ x 5, $S_3$ x 4

Order 4: $C_2^2$ x 21, $C_4$ x 5

Order 3: $C_3$

Order 2: $C_2$ x 9

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 192: $D_{12}.D_4$

Order 96: $C_{12}.D_4$, $C_2^2\times D_{12}$, $C_{12}.D_4$, $D_{12}:C_4$, $Q_8:D_6$

Order 64: $C_2^2:\SD_{16}$

Order 48: $C_2\times D_{12}$ x 3, $C_4:C_{12}$ x 2, $C_6:C_8$, $C_2^2\times D_6$, $C_6\times Q_8$, $C_2^2\times C_{12}$, $C_2^2:C_{12}$, $Q_8:S_3$

Order 32: $D_4:C_4$, $C_2^2:C_8$, $C_2^2\times D_4$, $C_2\times \SD_{16}$, $C_2^2:Q_8$

Order 24: $C_2\times C_{12}$ x 5, $C_2\times D_6$ x 4, $D_{12}$ x 3, $C_2^2\times C_6$, $C_3\times Q_8$, $C_3:C_8$

Order 16: $C_2\times D_4$ x 3, $C_4:C_4$ x 2, $\SD_{16}$, $C_2\times C_8$, $C_2^2:C_4$, $C_2^4$, $C_2\times Q_8$, $C_2^2\times C_4$

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

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

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

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

Order 3: $C_3$

Order 2: $C_2$ x 5

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 192: $D_{12}.D_4$ ($C_1$)

Order 96: $D_{12}:C_4$ ($C_2$) x 2, $Q_8:D_6$ ($C_2$) x 2, $C_{12}.D_4$ ($C_2$), $C_2^2\times D_{12}$ ($C_2$), $C_{12}.D_4$ ($C_2$)

Order 48: $C_6:C_8$ ($C_2^2$) x 2, $C_2\times D_{12}$ ($C_2^2$) x 2, $C_6\times Q_8$ ($C_2^2$), $C_2^2\times C_{12}$ ($C_2^2$), $C_4:C_{12}$ ($C_2^2$)

Order 32: $C_2^2:Q_8$ ($S_3$)

Order 24: $D_{12}$ ($D_4$) x 4, $C_2\times C_{12}$ ($C_2^3$), $C_2\times C_{12}$ ($D_4$), $C_2^2\times C_6$ ($D_4$)

Order 16: $C_4:C_4$ ($D_6$), $C_2\times Q_8$ ($D_6$), $C_2^2\times C_4$ ($D_6$)

Order 12: $C_2\times C_6$ ($\SD_{16}$) x 2, $C_{12}$ ($C_2\times D_4$) x 2, $C_2\times C_6$ ($C_2\times D_4$)

Order 8: $C_2^3$ ($C_3:D_4$), $C_2\times C_4$ ($C_3:D_4$), $C_2\times C_4$ ($C_2\times D_6$)

Order 6: $C_6$ ($D_8:C_2$), $C_6$ ($C_2\times \SD_{16}$), $C_6$ ($C_2^2\wr C_2$)

Order 4: $C_2^2$ ($Q_8:S_3$) x 2, $C_4$ ($S_3\times D_4$) x 2, $C_2^2$ ($C_6:D_4$)

Order 3: $C_3$ ($C_2^2:\SD_{16}$)

Order 2: $C_2$ ($D_4:D_6$), $C_2$ ($Q_8:D_6$), $C_2$ ($D_6:D_4$)

Order 1: $C_1$ ($D_{12}.D_4$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 192: $D_{12}.D_4$ ($C_1$)

Order 96: $C_{12}.D_4$ ($C_2$), $C_2^2\times D_{12}$ ($C_2$), $C_{12}.D_4$ ($C_2$), $D_{12}:C_4$ ($C_2$), $Q_8:D_6$ ($C_2$)

Order 48: $C_6:C_8$ ($C_2^2$), $C_6\times Q_8$ ($C_2^2$), $C_2^2\times C_{12}$ ($C_2^2$), $C_2\times D_{12}$ ($C_2^2$), $C_4:C_{12}$ ($C_2^2$)

Order 32: $C_2^2:Q_8$ ($S_3$)

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

Order 16: $C_4:C_4$ ($D_6$), $C_2\times Q_8$ ($D_6$), $C_2^2\times C_4$ ($D_6$)

Order 12: $C_2\times C_6$ ($\SD_{16}$), $C_2\times C_6$ ($C_2\times D_4$), $C_{12}$ ($C_2\times D_4$)

Order 8: $C_2^3$ ($C_3:D_4$), $C_2\times C_4$ ($C_3:D_4$), $C_2\times C_4$ ($C_2\times D_6$)

Order 6: $C_6$ ($D_8:C_2$), $C_6$ ($C_2\times \SD_{16}$), $C_6$ ($C_2^2\wr C_2$)

Order 4: $C_2^2$ ($C_6:D_4$), $C_2^2$ ($Q_8:S_3$), $C_4$ ($S_3\times D_4$)

Order 3: $C_3$ ($C_2^2:\SD_{16}$)

Order 2: $C_2$ ($D_4:D_6$), $C_2$ ($Q_8:D_6$), $C_2$ ($D_6:D_4$)

Order 1: $C_1$ ($D_{12}.D_4$)

Series

Derived series

$D_{12}.D_4$

$\rhd$

$C_2\times C_{12}$

$\rhd$

$C_1$

Chief series

$D_{12}.D_4$

$\rhd$

$Q_8:D_6$

$\rhd$

$C_6\times Q_8$

$\rhd$

$C_2\times C_{12}$

$\rhd$

$C_{12}$

$\rhd$

$C_6$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$D_{12}.D_4$

$\rhd$

$C_2\times C_{12}$

$\rhd$

$C_6$

$\rhd$

$C_3$

Upper central series

$C_1$

$\lhd$

$C_2^2$

$\lhd$

$C_2^2\times C_4$

$\lhd$

$C_2^2:Q_8$

Supergroups

This group is a maximal subgroup of 112 larger groups in the database.

This group is a maximal quotient of 47 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 $28 \times 28$ rational character table.