Abstract group 37056.be: $C_{193}:C_{192}$

• Label: 37056.be
• Order: \( 2^{6} \cdot 3 \cdot 193 \)
• Exponent: \( 2^{6} \cdot 3 \cdot 193 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{6} \cdot 3 \)
• $\card{Z(G)}$: \( 2^{5} \cdot 3 \)
• $\card{\Aut(G)}$: \( 2^{12} \cdot 3 \cdot 193 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{11} \cdot 3 \)
• Perm deg.: $260$
• Trans deg.: $37056$
• Rank: $2$

Group information

Description:	$C_{193}:C_{192}$
Order:	\(37056\)\(\medspace = 2^{6} \cdot 3 \cdot 193 \)
Exponent:	\(37056\)\(\medspace = 2^{6} \cdot 3 \cdot 193 \)
Automorphism group:	$C_{193}.C_{12}.C_8^2.C_2^4$, of order \(2371584\)\(\medspace = 2^{12} \cdot 3 \cdot 193 \)
Composition factors:	$C_2$ x 6, $C_3$, $C_{193}$
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	3	4	6	8	12	16	24	32	48	64	96	192	193	386	579	772	1158	1544	2316	3088	4632	6176	9264	18528
Elements	1	1	2	2	2	4	4	8	8	16	16	6176	32	12352	192	192	384	384	384	768	768	1536	1536	3072	3072	6144	37056
Conjugacy classes	1	1	2	2	2	4	4	8	8	16	16	32	32	64	96	96	192	192	192	384	384	768	768	1536	1536	3072	9408
Divisions	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	26
Autjugacy classes	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	26

Minimal presentations

Permutation degree:	$260$
Transitive degree:	$37056$
Rank:	$2$
Inequivalent generating pairs:	$384$

Minimal degrees of linear representations for this group have not been computed

Constructions

Presentation:	$\langle a, b \mid a^{192}=b^{193}=1, b^{a}=b^{192} \rangle$
Permutation group:	Degree $260$ $\langle(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(18,19)(20,21)(22,23)(24,25) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 1 & 1 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 174 & 0 \\ 0 & 19 \end{array}\right), \left(\begin{array}{rr} 192 & 0 \\ 0 & 192 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{193})$
Direct product:	$C_3$ $\, \times\, $ $(C_{193}:C_{64})$
Semidirect product:	$C_{579}$ $\,\rtimes\,$ $C_{64}$	$C_{193}$ $\,\rtimes\,$ $C_{192}$			more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_{96}$ . $D_{193}$	$C_{9264}$ . $C_4$	$C_{6176}$ . $C_6$	$C_{4632}$ . $C_8$	all 20

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

Homology

Abelianization:	$C_{192} \simeq C_{64} \times C_{3}$
Schur multiplier:	$C_1$
Commutator length:	not computed

Subgroups

There are 412 subgroups in 28 conjugacy classes, 26 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_{96}$	$G/Z \simeq$ $D_{193}$
Commutator:	$G' \simeq$ $C_{193}$	$G/G' \simeq$ $C_{192}$
Frattini:	$\Phi \simeq$ $C_{32}$	$G/\Phi \simeq$ $C_3\times D_{193}$
Fitting:	$\operatorname{Fit} \simeq$ $C_{18528}$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $C_{193}:C_{192}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{1158}$	$G/\operatorname{soc} \simeq$ $C_{32}$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_{64}$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3$
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 37056: $C_{193}:C_{192}$

Order 18528: $C_{18528}$

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

Order 9264: $C_{9264}$

Order 6176: $C_{6176}$

Order 4632: $C_{4632}$

Order 3088: $C_{3088}$

Order 2316: $C_{2316}$

Order 1544: $C_{1544}$

Order 1158: $C_{1158}$

Order 772: $C_{772}$

Order 579: $C_{579}$

Order 386: $C_{386}$

Order 193: $C_{193}$

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 37056: $C_{193}:C_{192}$

Order 18528: $C_{18528}$

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

Order 9264: $C_{9264}$

Order 6176: $C_{6176}$

Order 4632: $C_{4632}$

Order 3088: $C_{3088}$

Order 2316: $C_{2316}$

Order 1544: $C_{1544}$

Order 1158: $C_{1158}$

Order 772: $C_{772}$

Order 579: $C_{579}$

Order 386: $C_{386}$

Order 193: $C_{193}$

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 37056: $C_{193}:C_{192}$ ($C_1$)

Order 18528: $C_{18528}$ ($C_2$)

Order 12352: $C_{193}:C_{64}$ ($C_3$)

Order 9264: $C_{9264}$ ($C_4$)

Order 6176: $C_{6176}$ ($C_6$)

Order 4632: $C_{4632}$ ($C_8$)

Order 3088: $C_{3088}$ ($C_{12}$)

Order 2316: $C_{2316}$ ($C_{16}$)

Order 1544: $C_{1544}$ ($C_{24}$)

Order 1158: $C_{1158}$ ($C_{32}$)

Order 772: $C_{772}$ ($C_{48}$)

Order 579: $C_{579}$ ($C_{64}$)

Order 386: $C_{386}$ ($C_{96}$)

Order 193: $C_{193}$ ($C_{192}$)

Order 96: $C_{96}$ ($D_{193}$)

Order 48: $C_{48}$ ($C_{193}:C_4$)

Order 32: $C_{32}$ ($C_3\times D_{193}$)

Order 24: $C_{24}$ ($C_{193}:C_8$)

Order 16: $C_{16}$ ($C_{193}:C_{12}$)

Order 12: $C_{12}$ ($C_{193}:C_{16}$)

Order 8: $C_8$ ($C_{193}:C_{24}$)

Order 6: $C_6$ ($C_{193}:C_{32}$)

Order 4: $C_4$ ($C_{193}:C_{48}$)

Order 3: $C_3$ ($C_{193}:C_{64}$)

Order 2: $C_2$ ($C_{193}:C_{96}$)

Order 1: $C_1$ ($C_{193}:C_{192}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 37056: $C_{193}:C_{192}$ ($C_1$)

Order 18528: $C_{18528}$ ($C_2$)

Order 12352: $C_{193}:C_{64}$ ($C_3$)

Order 9264: $C_{9264}$ ($C_4$)

Order 6176: $C_{6176}$ ($C_6$)

Order 4632: $C_{4632}$ ($C_8$)

Order 3088: $C_{3088}$ ($C_{12}$)

Order 2316: $C_{2316}$ ($C_{16}$)

Order 1544: $C_{1544}$ ($C_{24}$)

Order 1158: $C_{1158}$ ($C_{32}$)

Order 772: $C_{772}$ ($C_{48}$)

Order 579: $C_{579}$ ($C_{64}$)

Order 386: $C_{386}$ ($C_{96}$)

Order 193: $C_{193}$ ($C_{192}$)

Order 96: $C_{96}$ ($D_{193}$)

Order 48: $C_{48}$ ($C_{193}:C_4$)

Order 32: $C_{32}$ ($C_3\times D_{193}$)

Order 24: $C_{24}$ ($C_{193}:C_8$)

Order 16: $C_{16}$ ($C_{193}:C_{12}$)

Order 12: $C_{12}$ ($C_{193}:C_{16}$)

Order 8: $C_8$ ($C_{193}:C_{24}$)

Order 6: $C_6$ ($C_{193}:C_{32}$)

Order 4: $C_4$ ($C_{193}:C_{48}$)

Order 3: $C_3$ ($C_{193}:C_{64}$)

Order 2: $C_2$ ($C_{193}:C_{96}$)

Order 1: $C_1$ ($C_{193}:C_{192}$)

Series

Derived series

$C_{193}:C_{192}$

$\rhd$

$C_{193}$

$\rhd$

$C_1$

Chief series

$C_{193}:C_{192}$

$\rhd$

$C_{18528}$

$\rhd$

$C_{9264}$

$\rhd$

$C_{4632}$

$\rhd$

$C_{2316}$

$\rhd$

$C_{1158}$

$\rhd$

$C_{579}$

$\rhd$

$C_{193}$

$\rhd$

$C_1$

Lower central series

$C_{193}:C_{192}$

$\rhd$

$C_{193}$

Upper central series

$C_1$

$\lhd$

$C_{96}$

Supergroups

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

This group is a maximal quotient of 0 larger groups in the database.

Character theory

Complex character table

The $9408 \times 9408$ character table is not available for this group.

Rational character table

The $26 \times 26$ rational character table is not available for this group.