Abstract group 192.175: $C_2\times C_{96}$

• Label: 192.175
• Order: \( 2^{6} \cdot 3 \)
• Exponent: \( 2^{5} \cdot 3 \)
• Abelian: yes
• $\card{\Aut(G)}$: \( 2^{7} \)
• Perm deg.: $37$
• Trans deg.: $192$
• Rank: $2$

Group information

Description:	$C_2\times C_{96}$
Order:	\(192\)\(\medspace = 2^{6} \cdot 3 \)
Exponent:	\(96\)\(\medspace = 2^{5} \cdot 3 \)
Automorphism group:	$C_8.C_2^4$, of order \(128\)\(\medspace = 2^{7} \)
Composition factors:	$C_2$ x 6, $C_3$
Nilpotency class:	$1$
Derived length:	$1$

This group is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), elementary for $p = 2$ (hence hyperelementary), and metacyclic.

Group statistics

Order	1	2	3	4	6	8	12	16	24	32	48	96
Elements	1	3	2	4	6	8	8	16	16	32	32	64	192
Conjugacy classes	1	3	2	4	6	8	8	16	16	32	32	64	192
Divisions	1	3	1	2	3	2	2	2	2	2	2	2	24
Autjugacy classes	1	2	1	2	2	2	2	2	2	1	2	1	20

Dimension	1	2	4	8	16	32
Irr. complex chars.	192	0	0	0	0	0	192
Irr. rational chars.	4	6	4	4	4	2	24

Minimal presentations

Permutation degree:	$37$
Transitive degree:	$192$
Rank:	$2$
Inequivalent generating pairs:	$96$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	2	3	18

Constructions

Presentation:	Abelian group $\langle a, b \mid a^{2}=b^{96}=1 \rangle$
Permutation group:	Degree $37$ $\langle(3,34,18,26,10,30,14,22,6,32,16,24,8,28,12,20,4,33,17,25,9,29,13,21,5,31,15,23,7,27,11,19) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 6 & 0 \\ 0 & 17 \end{array}\right), \left(\begin{array}{rr} 96 & 0 \\ 0 & 96 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{97})$
Direct product:	$C_2$ $\, \times\, $ $C_{32}$ $\, \times\, $ $C_3$
Semidirect product:	not isomorphic to a non-trivial semidirect product
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_{48}$ . $C_4$	$C_{24}$ . $C_8$	$C_8$ . $C_{24}$	$C_4$ . $C_{48}$	all 22
Aut. group:	$\Aut(C_{291})$	$\Aut(C_{388})$	$\Aut(C_{582})$

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

Homology

Primary decomposition:	$C_{2} \times C_{32} \times C_{3}$
Schur multiplier:	$C_{2}$
Commutator length:	$0$

Subgroups

There are 34 subgroups, all normal (26 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2\times C_{96}$	$G/Z \simeq$ $C_1$
Commutator:	$G' \simeq$ $C_1$	$G/G' \simeq$ $C_2\times C_{96}$
Frattini:	$\Phi \simeq$ $C_{16}$	$G/\Phi \simeq$ $C_2\times C_6$
Fitting:	$\operatorname{Fit} \simeq$ $C_2\times C_{96}$	$G/\operatorname{Fit} \simeq$ $C_1$
Radical:	$R \simeq$ $C_2\times C_{96}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2\times C_6$	$G/\operatorname{soc} \simeq$ $C_{16}$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2\times C_{32}$
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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Classes of subgroups up to conjugation

Order 192: $C_2\times C_{96}$

Order 96: $C_{96}$ x 2, $C_2\times C_{48}$

Order 64: $C_2\times C_{32}$

Order 48: $C_{48}$ x 2, $C_2\times C_{24}$

Order 32: $C_{32}$ x 2, $C_2\times C_{16}$

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

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

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

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

Order 6: $C_6$ x 3

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

Order 3: $C_3$

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 192: $C_2\times C_{96}$

Order 96: $C_2\times C_{48}$, $C_{96}$

Order 64: $C_2\times C_{32}$

Order 48: $C_{48}$ x 2, $C_2\times C_{24}$

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

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

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

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

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

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: $C_2\times C_{96}$ ($C_1$)

Order 96: $C_{96}$ ($C_2$) x 2, $C_2\times C_{48}$ ($C_2$)

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

Order 48: $C_2\times C_{24}$ ($C_4$), $C_{48}$ ($C_2^2$), $C_{48}$ ($C_4$)

Order 32: $C_{32}$ ($C_6$) x 2, $C_2\times C_{16}$ ($C_6$)

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

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

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

Order 8: $C_2\times C_4$ ($C_{24}$), $C_8$ ($C_2\times C_{12}$), $C_8$ ($C_{24}$)

Order 6: $C_6$ ($C_{32}$) x 2, $C_6$ ($C_2\times C_{16}$)

Order 4: $C_2^2$ ($C_{48}$), $C_4$ ($C_2\times C_{24}$), $C_4$ ($C_{48}$)

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

Order 2: $C_2$ ($C_{96}$) x 2, $C_2$ ($C_2\times C_{48}$)

Order 1: $C_1$ ($C_2\times C_{96}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 192: $C_2\times C_{96}$ ($C_1$)

Order 96: $C_2\times C_{48}$ ($C_2$), $C_{96}$ ($C_2$)

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

Order 48: $C_2\times C_{24}$ ($C_4$), $C_{48}$ ($C_2^2$), $C_{48}$ ($C_4$)

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

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

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

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

Order 8: $C_2\times C_4$ ($C_{24}$), $C_8$ ($C_2\times C_{12}$), $C_8$ ($C_{24}$)

Order 6: $C_6$ ($C_2\times C_{16}$), $C_6$ ($C_{32}$)

Order 4: $C_2^2$ ($C_{48}$), $C_4$ ($C_2\times C_{24}$), $C_4$ ($C_{48}$)

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

Order 2: $C_2$ ($C_2\times C_{48}$), $C_2$ ($C_{96}$)

Order 1: $C_1$ ($C_2\times C_{96}$)

Series

Derived series

$C_2\times C_{96}$

$\rhd$

$C_1$

Chief series

$C_2\times C_{96}$

$\rhd$

$C_2\times C_{48}$

$\rhd$

$C_{48}$

$\rhd$

$C_{24}$

$\rhd$

$C_{12}$

$\rhd$

$C_6$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$C_2\times C_{96}$

$\rhd$

$C_1$

Upper central series

$C_1$

$\lhd$

$C_2\times C_{96}$

Supergroups

This group is a maximal subgroup of 92 larger groups in the database.

This group is a maximal quotient of 56 larger groups in the database.

Character theory

Complex character table

See the $192 \times 192$ character table (warning: may be slow to load). Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $24 \times 24$ rational character table.