Abstract group 192.479: $C_{12}.C_4^2$

• Label: 192.479
• Order: \( 2^{6} \cdot 3 \)
• Exponent: \( 2^{3} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{6} \)
• $\card{Z(G)}$: \( 2^{5} \)
• $\card{\Aut(G)}$: \( 2^{12} \cdot 3 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{11} \)
• Perm deg.: $17$
• Trans deg.: $192$
• Rank: $3$

Group information

Description:	$C_{12}.C_4^2$
Order:	\(192\)\(\medspace = 2^{6} \cdot 3 \)
Exponent:	\(24\)\(\medspace = 2^{3} \cdot 3 \)
Automorphism group:	$S_3\times C_2^5.D_4^2$, 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 $p = 2$, metabelian, and an A-group.

Group statistics

Order	1	2	3	4	6	8	12
Elements	1	7	2	24	14	96	48	192
Conjugacy classes	1	7	1	24	7	32	24	96
Divisions	1	7	1	12	7	8	12	48
Autjugacy classes	1	3	1	3	3	1	3	15

Dimension	1	2	4
Irr. complex chars.	64	32	0	96
Irr. rational chars.	8	20	20	48

Minimal presentations

Permutation degree:	$17$
Transitive degree:	$192$
Rank:	$3$
Inequivalent generating triples:	$168$

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:	$\langle a, b, c \mid a^{4}=b^{8}=c^{6}=[a,b]=[a,c]=1, c^{b}=c^{5} \rangle$
Permutation group:	Degree $17$ $\langle(1,2)(4,5,6,7)(8,9,11,13,10,12,14,15)(16,17), (4,6)(5,7)(8,10)(9,12)(11,14) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rrrr} 4 & 0 & 0 & 0 \\ 0 & 4 & 0 & 0 \\ 0 & 0 & 4 & 0 \\ 0 & 0 & 0 & 4 \end{array}\right), \left(\begin{array}{rrrr} 1 & 1 & 4 & 2 \\ 3 & 4 & 4 & 0 \\ 4 & 3 & 0 & 0 \\ 3 & 1 & 3 & 4 \end{array}\right), \left(\begin{array}{rrrr} 0 & 1 & 2 & 1 \\ 2 & 1 & 2 & 3 \\ 1 & 2 & 3 & 4 \\ 4 & 3 & 3 & 2 \end{array}\right), \left(\begin{array}{rrrr} 2 & 0 & 0 & 0 \\ 0 & 2 & 0 & 0 \\ 0 & 0 & 2 & 0 \\ 0 & 0 & 0 & 2 \end{array}\right), \left(\begin{array}{rrrr} 3 & 3 & 0 & 4 \\ 3 & 0 & 2 & 2 \\ 4 & 3 & 0 & 1 \\ 1 & 2 & 2 & 1 \end{array}\right), \left(\begin{array}{rrrr} 2 & 2 & 1 & 4 \\ 0 & 1 & 3 & 3 \\ 0 & 0 & 0 & 4 \\ 1 & 2 & 0 & 4 \end{array}\right), \left(\begin{array}{rrrr} 3 & 3 & 1 & 3 \\ 1 & 1 & 1 & 4 \\ 3 & 1 & 2 & 2 \\ 2 & 4 & 4 & 4 \end{array}\right) \right\rangle \subseteq \GL_{4}(\F_{5})$
Direct product:	$C_2$ $\, \times\, $ $C_4$ $\, \times\, $ $(C_3:C_8)$
Semidirect product:	$C_6$ $\,\rtimes\,$ $(C_4\times C_8)$	$(C_2\times C_{12})$ $\,\rtimes\,$ $C_8$	$C_{12}$ $\,\rtimes\,$ $(C_2\times C_8)$		more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_4^2$ . $D_6$	$C_{12}$ . $C_4^2$	$C_4$ . $(C_4\times D_6)$	$(C_4\times C_{12})$ . $C_4$	all 34

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

Homology

Abelianization:	$C_{2} \times C_{4} \times C_{8} $
Schur multiplier:	$C_{2}^{2} \times C_{4}$
Commutator length:	$1$

Subgroups

There are 216 subgroups in 162 conjugacy classes, 135 normal (17 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2\times C_4^2$	$G/Z \simeq$ $S_3$
Commutator:	$G' \simeq$ $C_3$	$G/G' \simeq$ $C_2\times C_4\times C_8$
Frattini:	$\Phi \simeq$ $C_2\times C_4$	$G/\Phi \simeq$ $C_2\times D_6$
Fitting:	$\operatorname{Fit} \simeq$ $C_2\times C_4\times C_{12}$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $C_{12}.C_4^2$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^2\times C_6$	$G/\operatorname{soc} \simeq$ $C_2\times C_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2\times C_4\times C_8$
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

No diagram available

Normal subgroups

No diagram available: subgroups only stored up to automorphism

Normal subgroups up to automorphism

No diagram available

Classes of subgroups up to conjugation

Order 192: $C_{12}.C_4^2$

Order 96: $C_{12}:C_8$ x 4, $C_{12}.C_2^3$ x 2, $C_2\times C_4\times C_{12}$

Order 64: $C_2\times C_4\times C_8$

Order 48: $C_6:C_8$ x 12, $C_4\times C_{12}$ x 4, $C_2^2\times C_{12}$ x 3

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

Order 24: $C_2\times C_{12}$ x 18, $C_3:C_8$ x 8, $C_2^2\times C_6$

Order 16: $C_2\times C_8$ x 12, $C_4^2$ x 4, $C_2^2\times C_4$ x 3

Order 12: $C_{12}$ x 12, $C_2\times C_6$ x 7

Order 8: $C_2\times C_4$ x 18, $C_8$ x 8, $C_2^3$

Order 6: $C_6$ x 7

Order 4: $C_4$ x 12, $C_2^2$ x 7

Order 3: $C_3$

Order 2: $C_2$ x 7

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 192: $C_{12}.C_4^2$

Order 96: $C_{12}:C_8$, $C_2\times C_4\times C_{12}$, $C_{12}.C_2^3$

Order 64: $C_2\times C_4\times C_8$

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

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

Order 24: $C_2\times C_{12}$ x 5, $C_2^2\times C_6$, $C_3:C_8$

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

Order 12: $C_2\times C_6$ x 3, $C_{12}$ x 3

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

Order 6: $C_6$ x 3

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

Order 3: $C_3$

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 192: $C_{12}.C_4^2$ ($C_1$)

Order 96: $C_{12}:C_8$ ($C_2$) x 4, $C_{12}.C_2^3$ ($C_2$) x 2, $C_2\times C_4\times C_{12}$ ($C_2$)

Order 48: $C_6:C_8$ ($C_4$) x 8, $C_6:C_8$ ($C_2^2$) x 4, $C_2^2\times C_{12}$ ($C_4$) x 2, $C_4\times C_{12}$ ($C_2^2$) x 2, $C_4\times C_{12}$ ($C_4$) x 2, $C_2^2\times C_{12}$ ($C_2^2$)

Order 32: $C_2\times C_4^2$ ($S_3$)

Order 24: $C_2\times C_{12}$ ($C_2\times C_4$) x 9, $C_2\times C_{12}$ ($C_8$) x 8, $C_3:C_8$ ($C_2\times C_4$) x 8, $C_2\times C_{12}$ ($C_2^3$), $C_2^2\times C_6$ ($C_2\times C_4$)

Order 16: $C_4^2$ ($D_6$) x 2, $C_4^2$ ($C_3:C_4$) x 2, $C_2^2\times C_4$ ($C_3:C_4$) x 2, $C_2^2\times C_4$ ($D_6$)

Order 12: $C_{12}$ ($C_2\times C_8$) x 8, $C_2\times C_6$ ($C_2\times C_8$) x 4, $C_2\times C_6$ ($C_4^2$) x 2, $C_{12}$ ($C_4^2$) x 2, $C_{12}$ ($C_2^2\times C_4$) x 2, $C_2\times C_6$ ($C_2^2\times C_4$)

Order 8: $C_2\times C_4$ ($C_3:C_8$) x 8, $C_2\times C_4$ ($C_6:C_4$) x 5, $C_2\times C_4$ ($C_4\times S_3$) x 4, $C_2^3$ ($C_6:C_4$), $C_2\times C_4$ ($C_2\times D_6$)

Order 6: $C_6$ ($C_4\times C_8$) x 4, $C_6$ ($C_2^2\times C_8$) x 2, $C_6$ ($C_2\times C_4^2$)

Order 4: $C_4$ ($C_6:C_8$) x 8, $C_2^2$ ($C_6:C_8$) x 4, $C_2^2$ ($C_{12}:C_4$) x 2, $C_4$ ($C_4\times D_6$) x 2, $C_4$ ($C_{12}:C_4$) x 2, $C_2^2$ ($C_6.C_2^3$)

Order 3: $C_3$ ($C_2\times C_4\times C_8$)

Order 2: $C_2$ ($C_{12}:C_8$) x 4, $C_2$ ($C_{12}.C_2^3$) x 2, $C_2$ ($C_6:C_4^2$)

Order 1: $C_1$ ($C_{12}.C_4^2$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 192: $C_{12}.C_4^2$ ($C_1$)

Order 96: $C_{12}:C_8$ ($C_2$), $C_2\times C_4\times C_{12}$ ($C_2$), $C_{12}.C_2^3$ ($C_2$)

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

Order 32: $C_2\times C_4^2$ ($S_3$)

Order 24: $C_2\times C_{12}$ ($C_2\times C_4$) x 3, $C_2\times C_{12}$ ($C_2^3$), $C_2\times C_{12}$ ($C_8$), $C_2^2\times C_6$ ($C_2\times C_4$), $C_3:C_8$ ($C_2\times C_4$)

Order 16: $C_4^2$ ($D_6$), $C_4^2$ ($C_3:C_4$), $C_2^2\times C_4$ ($D_6$), $C_2^2\times C_4$ ($C_3:C_4$)

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

Order 8: $C_2\times C_4$ ($C_6:C_4$) x 2, $C_2^3$ ($C_6:C_4$), $C_2\times C_4$ ($C_4\times S_3$), $C_2\times C_4$ ($C_2\times D_6$), $C_2\times C_4$ ($C_3:C_8$)

Order 6: $C_6$ ($C_2^2\times C_8$), $C_6$ ($C_4\times C_8$), $C_6$ ($C_2\times C_4^2$)

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

Order 3: $C_3$ ($C_2\times C_4\times C_8$)

Order 2: $C_2$ ($C_{12}:C_8$), $C_2$ ($C_6:C_4^2$), $C_2$ ($C_{12}.C_2^3$)

Order 1: $C_1$ ($C_{12}.C_4^2$)

Series

Derived series

$C_{12}.C_4^2$

$\rhd$

$C_3$

$\rhd$

$C_1$

Chief series

$C_{12}.C_4^2$

$\rhd$

$C_{12}:C_8$

$\rhd$

$C_4\times C_{12}$

$\rhd$

$C_2\times C_{12}$

$\rhd$

$C_2\times C_6$

$\rhd$

$C_6$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$C_{12}.C_4^2$

$\rhd$

$C_3$

Upper central series

$C_1$

$\lhd$

$C_2\times C_4^2$

Supergroups

This group is a maximal subgroup of 196 larger groups in the database.

This group is a maximal quotient of 36 larger groups in the database.

Character theory

Complex character table

See the $96 \times 96$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $48 \times 48$ rational character table.