Abstract group 111168.p: $C_{1158}.C_{96}$

• Label: 111168.p
• Order: \( 2^{6} \cdot 3^{2} \cdot 193 \)
• Exponent: \( 2^{6} \cdot 3 \cdot 193 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{6} \cdot 3^{2} \)
• $\card{Z(G)}$: \( 2^{5} \cdot 3 \)
• $\card{\Aut(G)}$: \( 2^{12} \cdot 3^{2} \cdot 193 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{11} \cdot 3 \)
• Perm deg.: $260$
• Trans deg.: $37056$
• Rank: $2$

Group information

Description:	$C_{1158}.C_{96}$
Order:	\(111168\)\(\medspace = 2^{6} \cdot 3^{2} \cdot 193 \)
Exponent:	\(37056\)\(\medspace = 2^{6} \cdot 3 \cdot 193 \)
Automorphism group:	$C_{579}.C_{12}.C_8^2.C_2^4$, of order \(7114752\)\(\medspace = 2^{12} \cdot 3^{2} \cdot 193 \)
Composition factors:	$C_2$ x 6, $C_3$ x 2, $C_{193}$
Derived length:	$2$

This group is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), and an A-group.

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	1160	2	1160	4	2320	8	4640	16	9280	6176	18560	49408	192	192	384	384	384	768	768	1536	1536	3072	3072	6144	111168
Conjugacy classes	1	1	8	2	8	4	16	8	32	16	64	32	128	256	32	32	64	64	64	128	128	256	256	512	512	1024	3648
Divisions	1	1	4	1	4	1	4	1	4	1	4	1	4	4	1	1	1	1	1	1	1	1	1	1	1	1	47
Autjugacy classes	1	1	3	1	3	1	3	1	3	1	3	1	3	3	1	1	1	1	1	1	1	1	1	1	1	1	40

Minimal presentations

Permutation degree:	$260$
Transitive degree:	$37056$
Rank:	$2$
Inequivalent generating pairs:	not computed

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	6	not computed	not computed
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	$\langle a, b \mid a^{192}=b^{579}=1, b^{a}=b^{85} \rangle$
Permutation group:	Degree $260$ $\langle(2,4,6,3,8,7)(5,11,13,9,15,14)(10,20,16,18,22,17)(12,24,26,19,28,27)(21,36,29,34,38,30) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 1 & 1 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 52 & 0 \\ 0 & 71 \end{array}\right), \left(\begin{array}{rr} 85 & 0 \\ 0 & 109 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{193})$
Direct product:	not computed
Semidirect product:	not isomorphic to a non-trivial semidirect product
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Possibly split product:	$C_{18528}$ . $C_6$	$C_{9264}$ . $C_{12}$	$C_{4632}$ . $C_{24}$	$C_{2316}$ . $C_{48}$	all 35

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

Homology

Abelianization:	$C_{3} \times C_{192} \simeq C_{64} \times C_{3}^{2}$
Schur multiplier:	$C_{3}$
Commutator length:	$1$

Subgroups

There are 5844 subgroups in 84 conjugacy classes, 54 normal (33 characteristic).

Characteristic subgroups are shown in this color. Normal (but not characteristic) subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_{96}$	$G/Z \simeq$ $C_{193}:C_6$
Commutator:	$G' \simeq$ $C_{193}$	$G/G' \simeq$ $C_3\times C_{192}$
Frattini:	$\Phi \simeq$ $C_{32}$	$G/\Phi \simeq$ $C_{579}:C_6$
Fitting:	$\operatorname{Fit} \simeq$ $C_{18528}$	$G/\operatorname{Fit} \simeq$ $C_6$
Radical:	$R \simeq$ $C_{1158}.C_{96}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{1158}$	$G/\operatorname{soc} \simeq$ $C_{96}$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_{64}$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^2$
193-Sylow subgroup:	$P_{ 193 } \simeq$ $C_{193}$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

No diagram available: inclusions not computed

Classes of subgroups up to automorphism

No diagram available: inclusions not computed

Normal subgroups

No diagram available: inclusions not computed

Normal subgroups up to automorphism

No diagram available: inclusions not computed

Classes of subgroups up to conjugation

Order 111168: $C_{1158}.C_{96}$

Order 55584: $C_{96}\times C_{193}:C_3$

Order 37056: $C_{193}:C_{192}$ x 3, $C_{193}:C_{192}$

Order 27792: $C_{48}\times C_{193}:C_3$

Order 18528: $C_{32}\times C_{193}:C_3$ x 3, $C_{18528}$

Order 13896: $C_{4632}:C_3$

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

Order 9264: $C_{193}:C_{48}$ x 3, $C_{9264}$

Order 6948: $C_{2316}:C_3$

Order 6176: $C_{6176}$

Order 4632: $C_{1544}:C_3$ x 3, $C_{4632}$

Order 3474: $C_{1158}:C_3$

Order 3088: $C_{3088}$

Order 2316: $C_{193}:C_{12}$ x 3, $C_{2316}$

Order 1737: $C_{579}:C_3$

Order 1544: $C_{1544}$

Order 1158: $C_{193}:C_6$ x 3, $C_{1158}$

Order 772: $C_{772}$

Order 579: $C_{193}:C_3$ x 3, $C_{579}$

Order 576: $C_3\times C_{192}$

Order 386: $C_{386}$

Order 288: $C_3\times C_{96}$

Order 193: $C_{193}$

Order 192: $C_{192}$ x 4

Order 144: $C_3\times C_{48}$

Order 96: $C_{96}$ x 4

Order 72: $C_3\times C_{24}$

Order 64: $C_{64}$

Order 48: $C_{48}$ x 4

Order 36: $C_3\times C_{12}$

Order 32: $C_{32}$

Order 24: $C_{24}$ x 4

Order 18: $C_3\times C_6$

Order 16: $C_{16}$

Order 12: $C_{12}$ x 4

Order 9: $C_3^2$

Order 8: $C_8$

Order 6: $C_6$ x 4

Order 4: $C_4$

Order 3: $C_3$ x 4

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 111168: $C_{1158}.C_{96}$

Order 55584: $C_{96}\times C_{193}:C_3$

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

Order 27792: $C_{48}\times C_{193}:C_3$

Order 13896: $C_{4632}:C_3$

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

Order 6948: $C_{2316}:C_3$

Order 6176: $C_{6176}$

Order 3474: $C_{1158}:C_3$

Order 3088: $C_{3088}$

Order 1737: $C_{579}:C_3$

Order 1544: $C_{1544}$

Order 772: $C_{772}$

Order 576: $C_3\times C_{192}$

Order 386: $C_{386}$

Order 288: $C_3\times C_{96}$

Order 193: $C_{193}$

Order 144: $C_3\times C_{48}$

Order 72: $C_3\times C_{24}$

Order 64: $C_{64}$

Order 36: $C_3\times C_{12}$

Order 32: $C_{32}$

Order 18: $C_3\times C_6$

Order 16: $C_{16}$

Order 9: $C_3^2$

Order 8: $C_8$

Order 4: $C_4$

Order 2: $C_2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 111168: $C_{1158}.C_{96}$ ($C_1$)

Order 55584: $C_{96}\times C_{193}:C_3$ ($C_2$)

Order 37056: $C_{193}:C_{192}$ ($C_3$) x 3, $C_{193}:C_{192}$ ($C_3$)

Order 27792: $C_{48}\times C_{193}:C_3$ ($C_4$)

Order 18528: $C_{32}\times C_{193}:C_3$ ($C_6$) x 3, $C_{18528}$ ($C_6$)

Order 13896: $C_{4632}:C_3$ ($C_8$)

Order 12352: $C_{193}:C_{64}$ ($C_3^2$)

Order 9264: $C_{193}:C_{48}$ ($C_{12}$) x 3, $C_{9264}$ ($C_{12}$)

Order 6948: $C_{2316}:C_3$ ($C_{16}$)

Order 6176: $C_{6176}$ ($C_3\times C_6$)

Order 4632: $C_{1544}:C_3$ ($C_{24}$) x 3, $C_{4632}$ ($C_{24}$)

Order 3474: $C_{1158}:C_3$ ($C_{32}$)

Order 3088: $C_{3088}$ ($C_3\times C_{12}$)

Order 2316: $C_{193}:C_{12}$ ($C_{48}$) x 3, $C_{2316}$ ($C_{48}$)

Order 1737: $C_{579}:C_3$ ($C_{64}$)

Order 1544: $C_{1544}$ ($C_3\times C_{24}$)

Order 1158: $C_{193}:C_6$ ($C_{96}$) x 3, $C_{1158}$ ($C_{96}$)

Order 772: $C_{772}$ ($C_3\times C_{48}$)

Order 579: $C_{193}:C_3$ ($C_{192}$) x 3, $C_{579}$ ($C_{192}$)

Order 386: $C_{386}$ ($C_3\times C_{96}$)

Order 193: $C_{193}$ ($C_3\times C_{192}$)

Order 96: $C_{96}$ ($C_{193}:C_6$)

Order 48: $C_{48}$ ($C_{193}:C_{12}$)

Order 32: $C_{32}$ ($C_{579}:C_6$)

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

Order 16: $C_{16}$ ($C_{579}:C_{12}$)

Order 12: $C_{12}$ ($C_{193}:C_{48}$)

Order 8: $C_8$ ($C_{579}:C_{24}$)

Order 6: $C_6$ ($C_{193}:C_{96}$)

Order 4: $C_4$ ($C_{579}:C_{48}$)

Order 3: $C_3$ ($C_{193}:C_{192}$)

Order 2: $C_2$ ($C_{1158}.C_{48}$)

Order 1: $C_1$ ($C_{1158}.C_{96}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 111168: $C_{1158}.C_{96}$ ($C_1$)

Order 55584: $C_{96}\times C_{193}:C_3$ ($C_2$)

Order 37056: $C_{193}:C_{192}$ ($C_3$)

Order 27792: $C_{48}\times C_{193}:C_3$ ($C_4$)

Order 13896: $C_{4632}:C_3$ ($C_8$)

Order 12352: $C_{193}:C_{64}$ ($C_3^2$)

Order 6948: $C_{2316}:C_3$ ($C_{16}$)

Order 6176: $C_{6176}$ ($C_3\times C_6$)

Order 3474: $C_{1158}:C_3$ ($C_{32}$)

Order 3088: $C_{3088}$ ($C_3\times C_{12}$)

Order 1737: $C_{579}:C_3$ ($C_{64}$)

Order 1544: $C_{1544}$ ($C_3\times C_{24}$)

Order 772: $C_{772}$ ($C_3\times C_{48}$)

Order 386: $C_{386}$ ($C_3\times C_{96}$)

Order 193: $C_{193}$ ($C_3\times C_{192}$)

Order 32: $C_{32}$ ($C_{579}:C_6$)

Order 16: $C_{16}$ ($C_{579}:C_{12}$)

Order 8: $C_8$ ($C_{579}:C_{24}$)

Order 4: $C_4$ ($C_{579}:C_{48}$)

Order 2: $C_2$ ($C_{1158}.C_{48}$)

Order 1: $C_1$ ($C_{1158}.C_{96}$)

Series

Derived series

$C_{1158}.C_{96}$

$\rhd$

$C_{193}$

$\rhd$

$C_1$

Chief series

$C_{1158}.C_{96}$

$\rhd$

$C_{96}\times C_{193}:C_3$

$\rhd$

$C_{48}\times C_{193}:C_3$

$\rhd$

$C_{4632}:C_3$

$\rhd$

$C_{2316}:C_3$

$\rhd$

$C_{1158}:C_3$

$\rhd$

$C_{579}:C_3$

$\rhd$

$C_{579}$

$\rhd$

$C_{193}$

$\rhd$

$C_1$

Lower central series

$C_{1158}.C_{96}$

$\rhd$

$C_{193}$

Upper central series

$C_1$

$\lhd$

$C_{96}$

Supergroups

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

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

Character theory

Complex character table

The $3648 \times 3648$ character table is not available for this group.

Rational character table

The $47 \times 47$ rational character table is not available for this group.