Abstract group 192.937: $\OD_{32}:C_6$

• Label: 192.937
• Order: \( 2^{6} \cdot 3 \)
• Exponent: \( 2^{4} \cdot 3 \)
• Nilpotent: yes
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{5} \cdot 3 \)
• $\card{Z(G)}$: \( 2^{4} \cdot 3 \)
• $\card{\Aut(G)}$: \( 2^{7} \cdot 3 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{5} \cdot 3 \)
• Perm deg.: $35$
• Trans deg.: $96$
• Rank: $3$

Group information

Description:	$\OD_{32}:C_6$
Order:	\(192\)\(\medspace = 2^{6} \cdot 3 \)
Exponent:	\(48\)\(\medspace = 2^{4} \cdot 3 \)
Automorphism group:	$C_2^5.D_6$, of order \(384\)\(\medspace = 2^{7} \cdot 3 \)
Composition factors:	$C_2$ x 6, $C_3$
Nilpotency class:	$2$
Derived length:	$2$

This group is nonabelian, elementary for $p = 2$ (hence nilpotent, solvable, supersolvable, monomial, and hyperelementary), and metabelian.

Group statistics

Order	1	2	3	4	6	8	12	16	24	48
Elements	1	7	2	8	14	16	16	32	32	64	192
Conjugacy classes	1	4	2	5	8	10	10	20	20	40	120
Divisions	1	4	1	4	4	4	4	4	4	4	34
Autjugacy classes	1	2	1	2	2	2	2	2	2	2	18

Dimension	1	2	4	8	16	32
Irr. complex chars.	96	24	0	0	0	0	120
Irr. rational chars.	8	12	8	4	1	1	34

Minimal presentations

Permutation degree:	$35$
Transitive degree:	$96$
Rank:	$3$
Inequivalent generating triples:	$5824$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	2	4	32
Arbitrary	2	4	18

Constructions

Presentation:	$\langle a, b, c \mid a^{2}=b^{2}=c^{48}=[a,c]=[b,c]=1, b^{a}=bc^{24} \rangle$
Permutation group:	Degree $35$ $\langle(1,24,8,20,4,22,6,18,2,23,7,19,3,21,5,17)(9,32,16,28,12,30,14,26,10,31,15,27,11,29,13,25) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 177 & 0 \\ 0 & 64 \end{array}\right), \left(\begin{array}{rr} 178 & 0 \\ 0 & 178 \end{array}\right), \left(\begin{array}{rr} 0 & 1 \\ 1 & 0 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{241})$
Direct product:	$C_3$ $\, \times\, $ $(\OD_{32}:C_2)$
Semidirect product:	$\OD_{32}$ $\,\rtimes\,$ $C_6$ (3)	$(C_2\times C_{48})$ $\,\rtimes\,$ $C_2$ (3)	$(C_2\times C_{16})$ $\,\rtimes\,$ $C_6$ (3)	$(C_3\times \OD_{32})$ $\,\rtimes\,$ $C_2$ (3)	more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$Q_8$ . $C_{24}$	$D_4$ . $C_{24}$ (3)	$C_{48}$ . $C_2^2$ (4)	$C_{24}$ . $C_2^3$	all 28

Elements of the group are displayed as matrices in $\GL_{2}(\F_{241})$.

Homology

Abelianization:	$C_{2}^{2} \times C_{24} \simeq C_{2}^{2} \times C_{8} \times C_{3}$
Schur multiplier:	$C_{2}^{2}$
Commutator length:	$1$

Subgroups

There are 90 subgroups in 84 conjugacy classes, 78 normal (18 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_{48}$	$G/Z \simeq$ $C_2^2$
Commutator:	$G' \simeq$ $C_2$	$G/G' \simeq$ $C_2^2\times C_{24}$
Frattini:	$\Phi \simeq$ $C_8$	$G/\Phi \simeq$ $C_2^2\times C_6$
Fitting:	$\operatorname{Fit} \simeq$ $\OD_{32}:C_6$	$G/\operatorname{Fit} \simeq$ $C_1$
Radical:	$R \simeq$ $\OD_{32}:C_6$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_6$	$G/\operatorname{soc} \simeq$ $C_2^2\times C_8$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $\OD_{32}:C_2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

Classes of subgroups up to automorphism

Normal subgroups

Normal subgroups up to automorphism

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Subgroup information

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Classes of subgroups up to conjugation

Order 192: $\OD_{32}:C_6$

Order 96: $C_3\times \OD_{32}$ x 3, $C_2\times C_{48}$ x 3, $\OD_{16}:C_6$

Order 64: $\OD_{32}:C_2$

Order 48: $C_{48}$ x 4, $C_3\times \OD_{16}$ x 3, $C_2\times C_{24}$ x 3, $D_4:C_6$

Order 32: $\OD_{32}$ x 3, $C_2\times C_{16}$ x 3, $\OD_{16}:C_2$

Order 24: $C_{24}$ x 4, $C_2\times C_{12}$ x 3, $C_3\times D_4$ x 3, $C_3\times Q_8$

Order 16: $C_{16}$ x 4, $\OD_{16}$ x 3, $C_2\times C_8$ x 3, $D_4:C_2$

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

Order 8: $C_8$ x 4, $D_4$ x 3, $C_2\times C_4$ x 3, $Q_8$

Order 6: $C_6$ x 4

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

Order 3: $C_3$

Order 2: $C_2$ x 4

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 192: $\OD_{32}:C_6$

Order 96: $C_3\times \OD_{32}$, $C_2\times C_{48}$, $\OD_{16}:C_6$

Order 64: $\OD_{32}:C_2$

Order 48: $C_{48}$ x 2, $D_4:C_6$, $C_3\times \OD_{16}$, $C_2\times C_{24}$

Order 32: $\OD_{16}:C_2$, $\OD_{32}$, $C_2\times C_{16}$

Order 24: $C_{24}$ x 2, $C_2\times C_{12}$, $C_3\times Q_8$, $C_3\times D_4$

Order 16: $C_{16}$ x 2, $\OD_{16}$, $C_2\times C_8$, $D_4:C_2$

Order 12: $C_{12}$ x 2, $C_2\times C_6$

Order 8: $C_8$ x 2, $Q_8$, $D_4$, $C_2\times C_4$

Order 6: $C_6$ x 2

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

Order 3: $C_3$

Order 2: $C_2$ x 2

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 192: $\OD_{32}:C_6$ ($C_1$)

Order 96: $C_3\times \OD_{32}$ ($C_2$) x 3, $C_2\times C_{48}$ ($C_2$) x 3, $\OD_{16}:C_6$ ($C_2$)

Order 64: $\OD_{32}:C_2$ ($C_3$)

Order 48: $C_{48}$ ($C_2^2$) x 4, $C_3\times \OD_{16}$ ($C_4$) x 3, $C_2\times C_{24}$ ($C_2^2$) x 3, $D_4:C_6$ ($C_4$)

Order 32: $\OD_{32}$ ($C_6$) x 3, $C_2\times C_{16}$ ($C_6$) x 3, $\OD_{16}:C_2$ ($C_6$)

Order 24: $C_2\times C_{12}$ ($C_2\times C_4$) x 3, $C_{24}$ ($C_2\times C_4$) x 3, $C_3\times D_4$ ($C_8$) x 3, $C_{24}$ ($C_2^3$), $C_3\times Q_8$ ($C_8$)

Order 16: $C_{16}$ ($C_2\times C_6$) x 4, $\OD_{16}$ ($C_{12}$) x 3, $C_2\times C_8$ ($C_2\times C_6$) x 3, $D_4:C_2$ ($C_{12}$)

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

Order 8: $D_4$ ($C_{24}$) x 3, $C_2\times C_4$ ($C_2\times C_{12}$) x 3, $C_8$ ($C_2\times C_{12}$) x 3, $Q_8$ ($C_{24}$), $C_8$ ($C_2^2\times C_6$)

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

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

Order 3: $C_3$ ($\OD_{32}:C_2$)

Order 2: $C_2$ ($C_2^2\times C_{24}$)

Order 1: $C_1$ ($\OD_{32}:C_6$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 192: $\OD_{32}:C_6$ ($C_1$)

Order 96: $C_3\times \OD_{32}$ ($C_2$), $C_2\times C_{48}$ ($C_2$), $\OD_{16}:C_6$ ($C_2$)

Order 64: $\OD_{32}:C_2$ ($C_3$)

Order 48: $C_{48}$ ($C_2^2$) x 2, $D_4:C_6$ ($C_4$), $C_3\times \OD_{16}$ ($C_4$), $C_2\times C_{24}$ ($C_2^2$)

Order 32: $\OD_{16}:C_2$ ($C_6$), $\OD_{32}$ ($C_6$), $C_2\times C_{16}$ ($C_6$)

Order 24: $C_2\times C_{12}$ ($C_2\times C_4$), $C_{24}$ ($C_2^3$), $C_{24}$ ($C_2\times C_4$), $C_3\times Q_8$ ($C_8$), $C_3\times D_4$ ($C_8$)

Order 16: $C_{16}$ ($C_2\times C_6$) x 2, $\OD_{16}$ ($C_{12}$), $C_2\times C_8$ ($C_2\times C_6$), $D_4:C_2$ ($C_{12}$)

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

Order 8: $Q_8$ ($C_{24}$), $D_4$ ($C_{24}$), $C_2\times C_4$ ($C_2\times C_{12}$), $C_8$ ($C_2\times C_{12}$), $C_8$ ($C_2^2\times C_6$)

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

Order 4: $C_2^2$ ($C_2\times C_{24}$), $C_4$ ($C_2^2\times C_{12}$), $C_4$ ($C_2\times C_{24}$)

Order 3: $C_3$ ($\OD_{32}:C_2$)

Order 2: $C_2$ ($C_2^2\times C_{24}$)

Order 1: $C_1$ ($\OD_{32}:C_6$)

Series

Derived series

$\OD_{32}:C_6$

$\rhd$

$C_2$

$\rhd$

$C_1$

Chief series

$\OD_{32}:C_6$

$\rhd$

$C_2\times C_{48}$

$\rhd$

$C_2\times C_{24}$

$\rhd$

$C_{24}$

$\rhd$

$C_{12}$

$\rhd$

$C_6$

$\rhd$

$C_2$

$\rhd$

$C_1$

Lower central series

$\OD_{32}:C_6$

$\rhd$

$C_2$

$\rhd$

$C_1$

Upper central series

$C_1$

$\lhd$

$C_{48}$

$\lhd$

$\OD_{32}:C_6$

Supergroups

This group is a maximal subgroup of 51 larger groups in the database.

This group is a maximal quotient of 33 larger groups in the database.

Character theory

Complex character table

See the $120 \times 120$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $34 \times 34$ rational character table.