Abstract group 192.1276: $C_3:D_4^2$

• Label: 192.1276
• Order: \( 2^{6} \cdot 3 \)
• Exponent: \( 2^{2} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \)
• $\card{Z(G)}$: \( 2^{2} \)
• $\card{\Aut(G)}$: \( 2^{12} \cdot 3 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{8} \)
• Perm deg.: $11$
• Trans deg.: $48$
• Rank: $4$

Group information

Description:	$C_3:D_4^2$
Order:	\(192\)\(\medspace = 2^{6} \cdot 3 \)
Exponent:	\(12\)\(\medspace = 2^{2} \cdot 3 \)
Automorphism group:	$S_3\times C_2^2\wr D_4$, of order \(12288\)\(\medspace = 2^{12} \cdot 3 \)
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	12
Elements	1	67	2	60	38	24	192
Conjugacy classes	1	15	1	9	7	6	39
Divisions	1	15	1	9	7	5	38
Autjugacy classes	1	4	1	3	3	2	14

Dimension	1	2	4	8
Irr. complex chars.	16	16	7	0	39
Irr. rational chars.	16	16	5	1	38

Minimal presentations

Permutation degree:	$11$
Transitive degree:	$48$
Rank:	$4$
Inequivalent generating quadruples:	$32760$

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:	$\langle a, b, c, d \mid a^{2}=b^{2}=c^{4}=d^{12}=[a,b]=[a,c]=[c,d]=1, d^{a}=c^{2}d^{11}, c^{b}=c^{3}, d^{b}=d^{7} \rangle$
Permutation group:	Degree $11$ $\langle(1,2)(6,7), (1,2)(9,10), (3,4)(8,9)(10,11), (1,3,2,4)(8,10,11,9), (8,11)(9,10), (1,2)(3,4)(8,11)(9,10), (5,6,7)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{rrrrrr} 0 & 0 & 1 & -1 & 0 & 0 \\ 0 & 0 & 0 & -1 & 0 & 0 \\ 0 & -1 & 0 & 0 & 0 & 0 \\ -1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{array}\right), \left(\begin{array}{rrrrrr} 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & -1 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & -1 \end{array}\right), \left(\begin{array}{rrrrrr} 0 & 1 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & -1 & 0 & 0 \\ 0 & 0 & 0 & -1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & -1 \end{array}\right), \left(\begin{array}{rrrrrr} 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & -1 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 1 & 0 \end{array}\right) \right\rangle \subseteq \GL_{6}(\Z)$
	$\left\langle \left(\begin{array}{rr} 1 & 20 \\ 20 & 11 \end{array}\right), \left(\begin{array}{rr} 13 & 12 \\ 21 & 7 \end{array}\right), \left(\begin{array}{rr} 16 & 15 \\ 15 & 1 \end{array}\right), \left(\begin{array}{rr} 11 & 0 \\ 0 & 11 \end{array}\right), \left(\begin{array}{rr} 19 & 0 \\ 0 & 19 \end{array}\right), \left(\begin{array}{rr} 19 & 21 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 29 & 0 \\ 25 & 19 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/30\Z)$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$D_{12}$ $\,\rtimes\,$ $D_4$	$C_4^2$ $\,\rtimes\,$ $D_6$	$(C_4:D_4)$ $\,\rtimes\,$ $S_3$	$C_4$ $\,\rtimes\,$ $(S_3\times D_4)$	all 20
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_3$ . $D_4^2$	$C_2$ . $(D_4\times D_6)$	$C_2$ . $(D_4:D_6)$	$(C_2\times D_6)$ . $C_2^3$	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:	$C_{2}^{6}$
Commutator length:	$1$

Subgroups

There are 1376 subgroups in 428 conjugacy classes, 115 normal (13 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2^2$	$G/Z \simeq$ $C_2^2\times D_6$
Commutator:	$G' \simeq$ $C_2\times C_6$	$G/G' \simeq$ $C_2^4$
Frattini:	$\Phi \simeq$ $C_2^2$	$G/\Phi \simeq$ $C_2^2\times D_6$
Fitting:	$\operatorname{Fit} \simeq$ $C_{12}:D_4$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $C_3:D_4^2$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2\times C_6$	$G/\operatorname{soc} \simeq$ $C_2^4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $D_4^2$
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 192: $C_3:D_4^2$

Order 96: $D_4\times D_6$ x 4, $C_{12}:D_4$ x 4, $D_6:D_4$ x 4, $C_4\times D_{12}$ x 2, $C_{12}:D_4$

Order 64: $D_4^2$

Order 48: $S_3\times D_4$ x 16, $C_6:D_4$ x 8, $C_6\times D_4$ x 6, $C_2^2\times D_6$ x 4, $C_4\times D_6$ x 4, $C_6.D_4$ x 4, $D_6:C_4$ x 4, $C_2\times D_{12}$ x 2, $C_{12}:C_4$ x 2, $C_4\times C_{12}$

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

Order 24: $C_2\times D_6$ x 24, $C_3:D_4$ x 16, $C_3\times D_4$ x 10, $D_{12}$ x 8, $C_4\times S_3$ x 8, $C_6:C_4$ x 4, $C_2^2\times C_6$ x 4, $C_2\times C_{12}$ x 3

Order 16: $C_2\times D_4$ x 32, $C_2^2:C_4$ x 8, $C_2^4$ x 4, $C_2^2\times C_4$ x 4, $C_4:C_4$ x 2, $C_4^2$

Order 12: $D_6$ x 32, $C_2\times C_6$ x 13, $C_{12}$ x 5, $C_3:C_4$ x 4

Order 8: $D_4$ x 34, $C_2^3$ x 28, $C_2\times C_4$ x 15

Order 6: $S_3$ x 8, $C_6$ x 7

Order 4: $C_2^2$ x 45, $C_4$ x 9

Order 3: $C_3$

Order 2: $C_2$ x 15

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 192: $C_3:D_4^2$

Order 96: $C_4\times D_{12}$, $D_4\times D_6$, $C_{12}:D_4$, $C_{12}:D_4$, $D_6:D_4$

Order 64: $D_4^2$

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

Order 32: $C_2^2\times D_4$, $C_4:D_4$, $C_4:D_4$, $C_2^2\wr C_2$, $C_4\times D_4$

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

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

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

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

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

Order 4: $C_2^2$ x 7, $C_4$ x 3

Order 3: $C_3$

Order 2: $C_2$ x 4

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 192: $C_3:D_4^2$ ($C_1$)

Order 96: $D_4\times D_6$ ($C_2$) x 4, $C_{12}:D_4$ ($C_2$) x 4, $D_6:D_4$ ($C_2$) x 4, $C_4\times D_{12}$ ($C_2$) x 2, $C_{12}:D_4$ ($C_2$)

Order 48: $C_6:D_4$ ($C_2^2$) x 8, $C_6\times D_4$ ($C_2^2$) x 6, $C_2^2\times D_6$ ($C_2^2$) x 4, $C_4\times D_6$ ($C_2^2$) x 4, $C_6.D_4$ ($C_2^2$) x 4, $D_6:C_4$ ($C_2^2$) x 4, $C_2\times D_{12}$ ($C_2^2$) x 2, $C_{12}:C_4$ ($C_2^2$) x 2, $C_4\times C_{12}$ ($C_2^2$)

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

Order 24: $D_{12}$ ($D_4$) x 8, $C_6:C_4$ ($C_2^3$) x 4, $C_2^2\times C_6$ ($C_2^3$) x 4, $C_2\times D_6$ ($C_2^3$) x 4, $C_2\times C_{12}$ ($C_2^3$) x 3

Order 16: $C_2\times D_4$ ($D_6$) x 6, $C_4^2$ ($D_6$)

Order 12: $D_6$ ($C_2\times D_4$) x 8, $C_{12}$ ($C_2\times D_4$) x 4, $C_2\times C_6$ ($C_2^4$)

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

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

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

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

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

Order 1: $C_1$ ($C_3:D_4^2$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 192: $C_3:D_4^2$ ($C_1$)

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

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

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

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

Order 16: $C_2\times D_4$ ($D_6$) x 2, $C_4^2$ ($D_6$)

Order 12: $C_2\times C_6$ ($C_2^4$), $D_6$ ($C_2\times D_4$), $C_{12}$ ($C_2\times D_4$)

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

Order 6: $C_6$ ($D_4:C_2^2$), $C_6$ ($C_2^2\times D_4$)

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

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

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

Order 1: $C_1$ ($C_3:D_4^2$)

Series

Derived series

$C_3:D_4^2$

$\rhd$

$C_2\times C_6$

$\rhd$

$C_1$

Chief series

$C_3:D_4^2$

$\rhd$

$D_4\times D_6$

$\rhd$

$C_2^2\times D_6$

$\rhd$

$C_2\times D_6$

$\rhd$

$C_2\times C_6$

$\rhd$

$C_6$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$C_3:D_4^2$

$\rhd$

$C_2\times C_6$

$\rhd$

$C_3$

Upper central series

$C_1$

$\lhd$

$C_2^2$

$\lhd$

$C_4:D_4$

Supergroups

This group is a maximal subgroup of 94 larger groups in the database.

This group is a maximal quotient of 102 larger groups in the database.

Character theory

Complex character table

See the $39 \times 39$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $38 \times 38$ rational character table.