Abstract group $C_{388}$

• Label: 388.2
• Order: \( 2^{2} \cdot 97 \)
• Exponent: \( 2^{2} \cdot 97 \)
• Abelian: yes
• $\card{\operatorname{Aut}(G)}$: \( 2^{6} \cdot 3 \)
• Perm deg.: $101$
• Trans deg.: $388$
• Rank: $1$

Group information

Description:	$C_{388}$
Order:	\(388\)\(\medspace = 2^{2} \cdot 97 \)
Exponent:	\(388\)\(\medspace = 2^{2} \cdot 97 \)
Automorphism group:	$C_2\times C_{96}$, of order \(192\)\(\medspace = 2^{6} \cdot 3 \) (generators)
Outer automorphisms:	$C_2\times C_{96}$, of order \(192\)\(\medspace = 2^{6} \cdot 3 \)
Composition factors:	$C_2$ x 2, $C_{97}$
Nilpotency class:	$1$
Derived length:	$1$

This group is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 2,97$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group).

Group statistics

Order	1	2	4	97	194	388
Elements	1	1	2	96	96	192	388
Conjugacy classes	1	1	2	96	96	192	388
Divisions	1	1	1	1	1	1	6
Autjugacy classes	1	1	1	1	1	1	6

Dimension	1	2	96	192
Irr. complex chars.	388	0	0	0	388
Irr. rational chars.	2	1	2	1	6

Minimal Presentations

Permutation degree:	$101$
Transitive degree:	$388$
Rank:	$1$
Inequivalent generators:	$1$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	1	2	192
Arbitrary	1	2	98

Constructions

Presentation:	$\langle a \mid a^{388}=1 \rangle$
Permutation group:	Degree $101$ $\langle(1,4,2,3), (5,101,100,99,98,97,96,95,94,93,92,91,90,89,88,87,86,85,84,83,82,81,80,79,78,77,76,75,74,73,72,71,70,69,68,67,66,65,64,63,62,61,60,59,58,57,56,55,54,53,52,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 1 & 1 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 22 & 0 \\ 0 & 22 \end{array}\right), \left(\begin{array}{rr} 96 & 0 \\ 0 & 96 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{97})$
Direct product:	$C_4$ $\, \times\, $ $C_{97}$
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:	$C_{194}$ . $C_2$	$C_2$ . $C_{194}$			more information
Aut. group:	$\Aut(C_{389})$	$\Aut(C_{778})$

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

Homology

Primary decomposition:	$C_{4} \times C_{97}$
Schur multiplier:	$C_1$
Commutator length:	$0$

Subgroups

There are 6 subgroups, all normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_{388}$	$G/Z \simeq$ $C_1$
Commutator:	$G' \simeq$ $C_1$	$G/G' \simeq$ $C_{388}$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $C_{194}$
Fitting:	$\operatorname{Fit} \simeq$ $C_{388}$	$G/\operatorname{Fit} \simeq$ $C_1$
Radical:	$R \simeq$ $C_{388}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{194}$	$G/\operatorname{soc} \simeq$ $C_2$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_4$
97-Sylow subgroup:	$P_{ 97 } \simeq$ $C_{97}$

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 388: $C_{388}$

Order 194: $C_{194}$

Order 97: $C_{97}$

Order 4: $C_4$

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 388: $C_{388}$

Order 194: $C_{194}$

Order 97: $C_{97}$

Order 4: $C_4$

Order 2: $C_2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 388: $C_{388}$ ($C_1$)

Order 194: $C_{194}$ ($C_2$)

Order 97: $C_{97}$ ($C_4$)

Order 4: $C_4$ ($C_{97}$)

Order 2: $C_2$ ($C_{194}$)

Order 1: $C_1$ ($C_{388}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 388: $C_{388}$ ($C_1$)

Order 194: $C_{194}$ ($C_2$)

Order 97: $C_{97}$ ($C_4$)

Order 4: $C_4$ ($C_{97}$)

Order 2: $C_2$ ($C_{194}$)

Order 1: $C_1$ ($C_{388}$)

Series

Derived series	$C_{388}$	$\rhd$	$C_1$
Chief series	$C_{388}$	$\rhd$	$C_{194}$	$\rhd$	$C_{97}$	$\rhd$	$C_1$
Lower central series	$C_{388}$	$\rhd$	$C_1$
Upper central series	$C_1$	$\lhd$	$C_{388}$

Supergroups

This group is a maximal subgroup of 24 larger groups in the database.

This group is a maximal quotient of 17 larger groups in the database.

Character theory

Complex character table

See the $388 \times 388$ 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	4A	97A	194A	388A
	Size	1	1	2	96	96	192
	2 P	1A	1A	2A	97A	97A	194A
	97 P	1A	2A	4A	97A	194A	388A
388.2.1a		$1$	$1$	$1$	$1$	$1$	$1$
388.2.1b		$1$	$1$	$-1$	$1$	$1$	$-1$
388.2.1c		$2$	$-2$	$0$	$2$	$-2$	$0$
388.2.1d		$96$	$96$	$96$	$-1$	$-1$	$-1$
388.2.1e		$96$	$96$	$-96$	$-1$	$-1$	$1$
388.2.1f		$192$	$-192$	$0$	$-2$	$2$	$0$