Abstract group 12352.268: $C_{193}:C_{64}$

• Label: 12352.268
• Order: \( 2^{6} \cdot 193 \)
• Exponent: \( 2^{6} \cdot 193 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{6} \)
• $\card{Z(G)}$: \( 2^{5} \)
• $\card{\Aut(G)}$: \( 2^{11} \cdot 3 \cdot 193 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{10} \cdot 3 \)
• Perm deg.: $257$
• Trans deg.: $12352$
• Rank: $2$

Group information

Description:	$C_{193}:C_{64}$
Order:	\(12352\)\(\medspace = 2^{6} \cdot 193 \)
Exponent:	\(12352\)\(\medspace = 2^{6} \cdot 193 \)
Automorphism group:	$C_5^4.\OD_{16}$, of order \(1185792\)\(\medspace = 2^{11} \cdot 3 \cdot 193 \)
Composition factors:	$C_2$ x 6, $C_{193}$
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	8	16	32	64	193	386	772	1544	3088	6176
Elements	1	1	2	4	8	16	6176	192	192	384	768	1536	3072	12352
Conjugacy classes	1	1	2	4	8	16	32	96	96	192	384	768	1536	3136
Divisions	1	1	1	1	1	1	1	1	1	1	1	1	1	13
Autjugacy classes	1	1	1	1	1	1	1	1	1	1	1	1	1	13

Minimal presentations

Permutation degree:	$257$
Transitive degree:	$12352$
Rank:	$2$
Inequivalent generating pairs:	$96$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	2	not computed	not computed
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	$\langle a, b \mid a^{64}=b^{193}=1, b^{a}=b^{192} \rangle$
Permutation group:	Degree $257$ $\langle(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(18,19)(20,30)(21,31)(22,32) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 1 & 1 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 192 & 0 \\ 0 & 192 \end{array}\right), \left(\begin{array}{rr} 104 & 0 \\ 0 & 89 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{193})$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$C_{193}$ $\,\rtimes\,$ $C_{64}$				more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_{32}$ . $D_{193}$	$C_{6176}$ . $C_2$	$C_{3088}$ . $C_4$	$C_{1544}$ . $C_8$	all 10

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

Homology

Abelianization:	$C_{64} $
Schur multiplier:	$C_1$
Commutator length:	$1$

Subgroups

There are 206 subgroups in 14 conjugacy classes, 13 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_{32}$	$G/Z \simeq$ $D_{193}$
Commutator:	$G' \simeq$ $C_{193}$	$G/G' \simeq$ $C_{64}$
Frattini:	$\Phi \simeq$ $C_{32}$	$G/\Phi \simeq$ $D_{193}$
Fitting:	$\operatorname{Fit} \simeq$ $C_{6176}$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $C_{193}:C_{64}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{386}$	$G/\operatorname{soc} \simeq$ $C_{32}$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_{64}$
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 12352: $C_{193}:C_{64}$

Order 6176: $C_{6176}$

Order 3088: $C_{3088}$

Order 1544: $C_{1544}$

Order 772: $C_{772}$

Order 386: $C_{386}$

Order 193: $C_{193}$

Order 64: $C_{64}$

Order 32: $C_{32}$

Order 16: $C_{16}$

Order 8: $C_8$

Order 4: $C_4$

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 12352: $C_{193}:C_{64}$

Order 6176: $C_{6176}$

Order 3088: $C_{3088}$

Order 1544: $C_{1544}$

Order 772: $C_{772}$

Order 386: $C_{386}$

Order 193: $C_{193}$

Order 64: $C_{64}$

Order 32: $C_{32}$

Order 16: $C_{16}$

Order 8: $C_8$

Order 4: $C_4$

Order 2: $C_2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 12352: $C_{193}:C_{64}$ ($C_1$)

Order 6176: $C_{6176}$ ($C_2$)

Order 3088: $C_{3088}$ ($C_4$)

Order 1544: $C_{1544}$ ($C_8$)

Order 772: $C_{772}$ ($C_{16}$)

Order 386: $C_{386}$ ($C_{32}$)

Order 193: $C_{193}$ ($C_{64}$)

Order 32: $C_{32}$ ($D_{193}$)

Order 16: $C_{16}$ ($C_{193}:C_4$)

Order 8: $C_8$ ($C_{193}:C_8$)

Order 4: $C_4$ ($C_{193}:C_{16}$)

Order 2: $C_2$ ($C_{193}:C_{32}$)

Order 1: $C_1$ ($C_{193}:C_{64}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 12352: $C_{193}:C_{64}$ ($C_1$)

Order 6176: $C_{6176}$ ($C_2$)

Order 3088: $C_{3088}$ ($C_4$)

Order 1544: $C_{1544}$ ($C_8$)

Order 772: $C_{772}$ ($C_{16}$)

Order 386: $C_{386}$ ($C_{32}$)

Order 193: $C_{193}$ ($C_{64}$)

Order 32: $C_{32}$ ($D_{193}$)

Order 16: $C_{16}$ ($C_{193}:C_4$)

Order 8: $C_8$ ($C_{193}:C_8$)

Order 4: $C_4$ ($C_{193}:C_{16}$)

Order 2: $C_2$ ($C_{193}:C_{32}$)

Order 1: $C_1$ ($C_{193}:C_{64}$)

Series

Derived series

$C_{193}:C_{64}$

$\rhd$

$C_{193}$

$\rhd$

$C_1$

Chief series

$C_{193}:C_{64}$

$\rhd$

$C_{6176}$

$\rhd$

$C_{3088}$

$\rhd$

$C_{1544}$

$\rhd$

$C_{772}$

$\rhd$

$C_{386}$

$\rhd$

$C_{193}$

$\rhd$

$C_1$

Lower central series

$C_{193}:C_{64}$

$\rhd$

$C_{193}$

Upper central series

$C_1$

$\lhd$

$C_{32}$

Supergroups

This group is a maximal subgroup of 2 larger groups in the database.

This group is a maximal quotient of 1 larger groups in the database.

Character theory

Complex character table

The $3136 \times 3136$ character table is not available for this group.

Rational character table

The $13 \times 13$ rational character table is not available for this group.