Abstract group 192.245: $C_8\times D_{12}$

• Label: 192.245
• Order: \( 2^{6} \cdot 3 \)
• Exponent: \( 2^{3} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{5} \)
• $\card{Z(G)}$: \( 2^{4} \)
• $\card{\Aut(G)}$: \( 2^{9} \cdot 3 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{7} \)
• Perm deg.: $15$
• Trans deg.: $96$
• Rank: $3$

Group information

Description:	$C_8\times D_{12}$
Order:	\(192\)\(\medspace = 2^{6} \cdot 3 \)
Exponent:	\(24\)\(\medspace = 2^{3} \cdot 3 \)
Automorphism group:	$C_6.(D_4\times C_2^5)$, 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	27	2	36	6	64	24	32	192
Conjugacy classes	1	7	1	12	3	20	12	16	72
Divisions	1	7	1	7	3	5	5	3	32
Autjugacy classes	1	4	1	5	3	3	4	2	23

Dimension	1	2	4	8	16
Irr. complex chars.	32	40	0	0	0	72
Irr. rational chars.	8	10	9	4	1	32

Minimal presentations

Permutation degree:	$15$
Transitive degree:	$96$
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	3	4	8

Constructions

Presentation:	$\langle a, b, c \mid a^{2}=b^{8}=c^{12}=[a,b]=[b,c]=1, c^{a}=c^{11} \rangle$
Permutation group:	Degree $15$ $\langle(1,2)(3,4)(6,7), (1,3,4,2)(8,9)(10,12)(11,13)(14,15), (1,4)(2,3)(8,10,11,14,9,12,13,15) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 7 & 0 \\ 0 & 7 \end{array}\right), \left(\begin{array}{rr} 0 & 7 \\ 1 & 0 \end{array}\right), \left(\begin{array}{rr} 16 & 15 \\ 15 & 1 \end{array}\right), \left(\begin{array}{rr} 11 & 0 \\ 0 & 11 \end{array}\right), \left(\begin{array}{rr} 3 & 20 \\ 10 & 3 \end{array}\right), \left(\begin{array}{rr} 19 & 0 \\ 0 & 19 \end{array}\right), \left(\begin{array}{rr} 24 & 25 \\ 25 & 24 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/30\Z)$
Direct product:	$C_8$ $\, \times\, $ $D_{12}$
Semidirect product:	$C_{24}$ $\,\rtimes\,$ $D_4$ (2)	$(C_4\times C_8)$ $\,\rtimes\,$ $S_3$	$C_4$ $\,\rtimes\,$ $(S_3\times C_8)$ (2)	$(C_8\times D_6)$ $\,\rtimes\,$ $C_2$ (2)	all 10
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_4^2$ . $D_6$	$(D_6:C_4)$ . $C_4$ (2)	$(C_2\times C_8)$ . $D_6$ (2)	$C_6$ . $(C_4\times D_4)$	all 28

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

Homology

Abelianization:	$C_{2}^{2} \times C_{8} $
Schur multiplier:	$C_{2}^{3}$
Commutator length:	$1$

Subgroups

There are 312 subgroups in 134 conjugacy classes, 63 normal (33 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2\times C_8$	$G/Z \simeq$ $D_6$
Commutator:	$G' \simeq$ $C_6$	$G/G' \simeq$ $C_2^2\times C_8$
Frattini:	$\Phi \simeq$ $C_2\times C_4$	$G/\Phi \simeq$ $C_2\times D_6$
Fitting:	$\operatorname{Fit} \simeq$ $C_4\times C_{24}$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $C_8\times D_{12}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2\times C_6$	$G/\operatorname{soc} \simeq$ $C_2^2\times C_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_8\times D_4$
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: $C_8\times D_{12}$

Order 96: $D_6:C_8$ x 2, $C_8\times D_6$ x 2, $C_4\times D_{12}$, $C_4\times C_{24}$, $C_{12}:C_8$

Order 64: $C_8\times D_4$

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

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

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

Order 16: $C_2\times C_8$ x 8, $C_2^2:C_4$ x 2, $C_2^2\times C_4$ x 2, $C_4:C_4$, $C_4^2$, $C_2\times D_4$

Order 12: $D_6$ x 8, $C_{12}$ x 5, $C_3:C_4$ x 2, $C_2\times C_6$

Order 8: $C_2\times C_4$ x 9, $C_8$ x 5, $D_4$ x 4, $C_2^3$ x 2

Order 6: $S_3$ x 4, $C_6$ x 3

Order 4: $C_2^2$ x 9, $C_4$ x 7

Order 3: $C_3$

Order 2: $C_2$ x 7

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 192: $C_8\times D_{12}$

Order 96: $C_4\times D_{12}$, $C_4\times C_{24}$, $D_6:C_8$, $C_{12}:C_8$, $C_8\times D_6$

Order 64: $C_8\times D_4$

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

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

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

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

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

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

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

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

Order 3: $C_3$

Order 2: $C_2$ x 4

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 192: $C_8\times D_{12}$ ($C_1$)

Order 96: $D_6:C_8$ ($C_2$) x 2, $C_8\times D_6$ ($C_2$) x 2, $C_4\times D_{12}$ ($C_2$), $C_4\times C_{24}$ ($C_2$), $C_{12}:C_8$ ($C_2$)

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

Order 32: $C_4\times C_8$ ($S_3$)

Order 24: $D_{12}$ ($C_8$) x 4, $C_2\times C_{12}$ ($C_2\times C_4$) x 2, $C_6:C_4$ ($C_2\times C_4$) x 2, $C_{24}$ ($D_4$) x 2, $C_2\times D_6$ ($C_2\times C_4$) x 2, $C_2\times C_{12}$ ($C_2^3$)

Order 16: $C_2\times C_8$ ($D_6$) x 2, $C_4^2$ ($D_6$)

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

Order 8: $C_2\times C_4$ ($C_4\times S_3$) x 2, $C_8$ ($D_{12}$) x 2, $C_2\times C_4$ ($C_2\times D_6$)

Order 6: $C_6$ ($\OD_{16}:C_2$), $C_6$ ($C_2^2\times C_8$), $C_6$ ($C_4\times D_4$)

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

Order 3: $C_3$ ($C_8\times D_4$)

Order 2: $C_2$ ($C_4\times D_{12}$), $C_2$ ($C_8.D_6$), $C_2$ ($C_8\times D_6$)

Order 1: $C_1$ ($C_8\times D_{12}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 192: $C_8\times D_{12}$ ($C_1$)

Order 96: $C_4\times D_{12}$ ($C_2$), $C_4\times C_{24}$ ($C_2$), $D_6:C_8$ ($C_2$), $C_{12}:C_8$ ($C_2$), $C_8\times D_6$ ($C_2$)

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

Order 32: $C_4\times C_8$ ($S_3$)

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

Order 16: $C_2\times C_8$ ($D_6$) x 2, $C_4^2$ ($D_6$)

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

Order 8: $C_2\times C_4$ ($C_4\times S_3$) x 2, $C_2\times C_4$ ($C_2\times D_6$), $C_8$ ($D_{12}$)

Order 6: $C_6$ ($\OD_{16}:C_2$), $C_6$ ($C_2^2\times C_8$), $C_6$ ($C_4\times D_4$)

Order 4: $C_2^2$ ($C_4\times D_6$), $C_4$ ($S_3\times C_8$), $C_4$ ($D_{12}:C_2$), $C_4$ ($C_2\times D_{12}$)

Order 3: $C_3$ ($C_8\times D_4$)

Order 2: $C_2$ ($C_4\times D_{12}$), $C_2$ ($C_8.D_6$), $C_2$ ($C_8\times D_6$)

Order 1: $C_1$ ($C_8\times D_{12}$)

Series

Derived series

$C_8\times D_{12}$

$\rhd$

$C_6$

$\rhd$

$C_1$

Chief series

$C_8\times D_{12}$

$\rhd$

$C_8\times D_6$

$\rhd$

$C_4\times D_6$

$\rhd$

$C_2\times C_{12}$

$\rhd$

$C_{12}$

$\rhd$

$C_6$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$C_8\times D_{12}$

$\rhd$

$C_6$

$\rhd$

$C_3$

Upper central series

$C_1$

$\lhd$

$C_2\times C_8$

$\lhd$

$C_4\times C_8$

Supergroups

This group is a maximal subgroup of 178 larger groups in the database.

This group is a maximal quotient of 48 larger groups in the database.

Character theory

Complex character table

See the $72 \times 72$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $32 \times 32$ rational character table.