Abstract group 772.1: $C_{193}:C_4$

• Label: 772.1
• Order: \( 2^{2} \cdot 193 \)
• Exponent: \( 2^{2} \cdot 193 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{7} \cdot 3 \cdot 193 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{6} \cdot 3 \)
• Perm deg.: $197$
• Trans deg.: $772$
• Rank: $2$

Group information

Description:	$C_{193}:C_4$
Order:	\(772\)\(\medspace = 2^{2} \cdot 193 \)
Exponent:	\(772\)\(\medspace = 2^{2} \cdot 193 \)
Automorphism group:	$C_2\times F_{193}$, of order \(74112\)\(\medspace = 2^{7} \cdot 3 \cdot 193 \)
Composition factors:	$C_2$ x 2, $C_{193}$
Derived length:	$2$

This group is nonabelian, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), and hyperelementary for $p = 2$.

Group statistics

Order	1	2	4	193	386
Elements	1	1	386	192	192	772
Conjugacy classes	1	1	2	96	96	196
Divisions	1	1	1	1	1	5
Autjugacy classes	1	1	1	1	1	5

Dimension	1	2	192
Irr. complex chars.	4	192	0	196
Irr. rational chars.	2	1	2	5

Minimal presentations

Permutation degree:	$197$
Transitive degree:	$772$
Rank:	$2$
Inequivalent generating pairs:	$6$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	2	4	384
Arbitrary	2	4	194

Constructions

Presentation:	$\langle a, b \mid a^{4}=b^{193}=1, b^{a}=b^{192} \rangle$
Permutation group:	Degree $197$ $\langle(2,4)(3,7)(5,9)(6,12)(8,14)(10,16)(11,19)(13,21)(15,23)(17,25)(18,28)(20,30) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 1 & 1 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 112 & 0 \\ 0 & 81 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{193})$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$C_{193}$ $\,\rtimes\,$ $C_4$				more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_2$ . $D_{193}$	$C_{386}$ . $C_2$			more information

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

Homology

Abelianization:	$C_{4} $
Schur multiplier:	$C_1$
Commutator length:	$1$

Subgroups

There are 198 subgroups in 6 conjugacy classes, 5 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $D_{193}$
Commutator:	$G' \simeq$ $C_{193}$	$G/G' \simeq$ $C_4$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $D_{193}$
Fitting:	$\operatorname{Fit} \simeq$ $C_{386}$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $C_{193}:C_4$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{386}$	$G/\operatorname{soc} \simeq$ $C_2$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_4$
193-Sylow subgroup:	$P_{ 193 } \simeq$ $C_{193}$

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 772: $C_{193}:C_4$

Order 386: $C_{386}$

Order 193: $C_{193}$

Order 4: $C_4$

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 772: $C_{193}:C_4$

Order 386: $C_{386}$

Order 193: $C_{193}$

Order 4: $C_4$

Order 2: $C_2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 772: $C_{193}:C_4$ ($C_1$)

Order 386: $C_{386}$ ($C_2$)

Order 193: $C_{193}$ ($C_4$)

Order 2: $C_2$ ($D_{193}$)

Order 1: $C_1$ ($C_{193}:C_4$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 772: $C_{193}:C_4$ ($C_1$)

Order 386: $C_{386}$ ($C_2$)

Order 193: $C_{193}$ ($C_4$)

Order 2: $C_2$ ($D_{193}$)

Order 1: $C_1$ ($C_{193}:C_4$)

Series

Derived series	$C_{193}:C_4$	$\rhd$	$C_{193}$	$\rhd$	$C_1$
Chief series	$C_{193}:C_4$	$\rhd$	$C_{386}$	$\rhd$	$C_{193}$	$\rhd$	$C_1$
Lower central series	$C_{193}:C_4$	$\rhd$	$C_{193}$
Upper central series	$C_1$	$\lhd$	$C_2$

Supergroups

This group is a maximal subgroup of 7 larger groups in the database.

This group is a maximal quotient of 3 larger groups in the database.

Character theory

Complex character table

See the $196 \times 196$ 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	193A	386A
	Size	1	1	386	192	192
	2 P	1A	1A	2A	193A	193A
	193 P	1A	2A	4A	193A	386A
	Schur
772.1.1a		$1$	$1$	$1$	$1$	$1$
772.1.1b		$1$	$1$	$-1$	$1$	$1$
772.1.1c		$2$	$-2$	$0$	$2$	$-2$
772.1.2a		$192$	$192$	$0$	$-1$	$-1$
772.1.2b	2	$192$	$-192$	$0$	$-1$	$1$