Abstract group 192.2: $C_{192}$

• Label: 192.2
• Order: \( 2^{6} \cdot 3 \)
• Exponent: \( 2^{6} \cdot 3 \)
• Abelian: yes
• $\card{\Aut(G)}$: \( 2^{6} \)
• Perm deg.: $67$
• Trans deg.: $192$
• Rank: $1$

Group information

Description:	$C_{192}$
Order:	\(192\)\(\medspace = 2^{6} \cdot 3 \)
Exponent:	\(192\)\(\medspace = 2^{6} \cdot 3 \)
Automorphism group:	$C_2^2\times C_{16}$, of order \(64\)\(\medspace = 2^{6} \)
Composition factors:	$C_2$ x 6, $C_3$
Nilpotency class:	$1$
Derived length:	$1$

This group is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 2,3$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group).

Group statistics

Order	1	2	3	4	6	8	12	16	24	32	48	64	96	192
Elements	1	1	2	2	2	4	4	8	8	16	16	32	32	64	192
Conjugacy classes	1	1	2	2	2	4	4	8	8	16	16	32	32	64	192
Divisions	1	1	1	1	1	1	1	1	1	1	1	1	1	1	14
Autjugacy classes	1	1	1	1	1	1	1	1	1	1	1	1	1	1	14

Dimension	1	2	4	8	16	32	64
Irr. complex chars.	192	0	0	0	0	0	0	192
Irr. rational chars.	2	3	2	2	2	2	1	14

Minimal presentations

Permutation degree:	$67$
Transitive degree:	$192$
Rank:	$1$
Inequivalent generators:	$1$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	1	2	64
Arbitrary	1	2	34

Constructions

Presentation:	$\langle a \mid a^{192}=1 \rangle$
Permutation group:	Degree $67$ $\langle(1,64,32,48,16,56,24,40,8,60,28,44,12,52,20,36,4,62,30,46,14,54,22,38,6,58,26,42,10,50,18,34,2,63,31,47,15,55,23,39,7,59,27,43,11,51,19,35,3,61,29,45,13,53,21,37,5,57,25,41,9,49,17,33) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 3 & 28 \\ 30 & 3 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{31})$
Direct product:	$C_{64}$ $\, \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:	$C_{96}$ . $C_2$	$C_{48}$ . $C_4$	$C_{32}$ . $C_6$	$C_{24}$ . $C_8$	all 10
Aut. group:	$\Aut(C_{193})$	$\Aut(C_{386})$

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

Homology

Primary decomposition:	$C_{64} \times C_{3}$
Schur multiplier:	$C_1$
Commutator length:	$0$

Subgroups

There are 14 subgroups, all normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_{192}$	$G/Z \simeq$ $C_1$
Commutator:	$G' \simeq$ $C_1$	$G/G' \simeq$ $C_{192}$
Frattini:	$\Phi \simeq$ $C_{32}$	$G/\Phi \simeq$ $C_6$
Fitting:	$\operatorname{Fit} \simeq$ $C_{192}$	$G/\operatorname{Fit} \simeq$ $C_1$
Radical:	$R \simeq$ $C_{192}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_6$	$G/\operatorname{soc} \simeq$ $C_{32}$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_{64}$
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: $C_{192}$

Order 96: $C_{96}$

Order 64: $C_{64}$

Order 48: $C_{48}$

Order 32: $C_{32}$

Order 24: $C_{24}$

Order 16: $C_{16}$

Order 12: $C_{12}$

Order 8: $C_8$

Order 6: $C_6$

Order 4: $C_4$

Order 3: $C_3$

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 192: $C_{192}$

Order 96: $C_{96}$

Order 64: $C_{64}$

Order 48: $C_{48}$

Order 32: $C_{32}$

Order 24: $C_{24}$

Order 16: $C_{16}$

Order 12: $C_{12}$

Order 8: $C_8$

Order 6: $C_6$

Order 4: $C_4$

Order 3: $C_3$

Order 2: $C_2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 192: $C_{192}$ ($C_1$)

Order 96: $C_{96}$ ($C_2$)

Order 64: $C_{64}$ ($C_3$)

Order 48: $C_{48}$ ($C_4$)

Order 32: $C_{32}$ ($C_6$)

Order 24: $C_{24}$ ($C_8$)

Order 16: $C_{16}$ ($C_{12}$)

Order 12: $C_{12}$ ($C_{16}$)

Order 8: $C_8$ ($C_{24}$)

Order 6: $C_6$ ($C_{32}$)

Order 4: $C_4$ ($C_{48}$)

Order 3: $C_3$ ($C_{64}$)

Order 2: $C_2$ ($C_{96}$)

Order 1: $C_1$ ($C_{192}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 192: $C_{192}$ ($C_1$)

Order 96: $C_{96}$ ($C_2$)

Order 64: $C_{64}$ ($C_3$)

Order 48: $C_{48}$ ($C_4$)

Order 32: $C_{32}$ ($C_6$)

Order 24: $C_{24}$ ($C_8$)

Order 16: $C_{16}$ ($C_{12}$)

Order 12: $C_{12}$ ($C_{16}$)

Order 8: $C_8$ ($C_{24}$)

Order 6: $C_6$ ($C_{32}$)

Order 4: $C_4$ ($C_{48}$)

Order 3: $C_3$ ($C_{64}$)

Order 2: $C_2$ ($C_{96}$)

Order 1: $C_1$ ($C_{192}$)

Series

Derived series

$C_{192}$

$\rhd$

$C_1$

Chief series

$C_{192}$

$\rhd$

$C_{96}$

$\rhd$

$C_{48}$

$\rhd$

$C_{24}$

$\rhd$

$C_{12}$

$\rhd$

$C_6$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$C_{192}$

$\rhd$

$C_1$

Upper central series

$C_1$

$\lhd$

$C_{192}$

Supergroups

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

This group is a maximal quotient of 39 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

		1A	2A	3A	4A	6A	8A	12A	16A	24A	32A	48A	64A	96A	192A
	Size	1	1	2	2	2	4	4	8	8	16	16	32	32	64
	2 P	1A	1A	3A	2A	3A	4A	6A	8A	12A	16A	24A	32A	48A	96A
	3 P	1A	2A	1A	4A	2A	8A	4A	16A	8A	32A	16A	64A	32A	64A
192.2.1a		$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
192.2.1b		$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$-1$	$1$	$-1$
192.2.1c		$2$	$2$	$-1$	$2$	$-1$	$2$	$-1$	$2$	$-1$	$2$	$-1$	$2$	$-1$	$-1$
192.2.1d		$2$	$2$	$2$	$2$	$2$	$2$	$2$	$2$	$2$	$-2$	$2$	$0$	$-2$	$0$
192.2.1e		$2$	$2$	$-1$	$2$	$-1$	$2$	$-1$	$2$	$-1$	$2$	$-1$	$-2$	$-1$	$1$
192.2.1f		$4$	$4$	$4$	$4$	$4$	$4$	$4$	$-4$	$4$	$0$	$-4$	$0$	$0$	$0$
192.2.1g		$4$	$4$	$-2$	$4$	$-2$	$4$	$-2$	$4$	$-2$	$-4$	$-2$	$0$	$2$	$0$
192.2.1h		$8$	$8$	$8$	$8$	$8$	$-8$	$8$	$0$	$-8$	$0$	$0$	$0$	$0$	$0$
192.2.1i		$8$	$8$	$-4$	$8$	$-4$	$8$	$-4$	$-8$	$-4$	$0$	$4$	$0$	$0$	$0$
192.2.1j		$16$	$16$	$16$	$-16$	$16$	$0$	$-16$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
192.2.1k		$16$	$16$	$-8$	$16$	$-8$	$-16$	$-8$	$0$	$8$	$0$	$0$	$0$	$0$	$0$
192.2.1l		$32$	$-32$	$32$	$0$	$-32$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
192.2.1m		$32$	$32$	$-16$	$-32$	$-16$	$0$	$16$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
192.2.1n		$64$	$-64$	$-32$	$0$	$32$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$