Abstract group 56882.1: $C_{56882}$

• Label: 56882.1
• Order: \( 2 \cdot 7 \cdot 17 \cdot 239 \)
• Exponent: \( 2 \cdot 7 \cdot 17 \cdot 239 \)
• Abelian: yes
• $\card{\Aut(G)}$: \( 2^{6} \cdot 3 \cdot 7 \cdot 17 \)
• Perm deg.: $265$
• Trans deg.: $56882$
• Rank: $1$

Group information

Description:	$C_{56882}$
Order:	\(56882\)\(\medspace = 2 \cdot 7 \cdot 17 \cdot 239 \)
Exponent:	\(56882\)\(\medspace = 2 \cdot 7 \cdot 17 \cdot 239 \)
Automorphism group:	$C_2^2\times C_{5712}$, of order \(22848\)\(\medspace = 2^{6} \cdot 3 \cdot 7 \cdot 17 \)
Composition factors:	$C_2$, $C_7$, $C_{17}$, $C_{239}$
Nilpotency class:	$1$
Derived length:	$1$

This group is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 2,7,17,239$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group).

Group statistics

Order	1	2	7	14	17	34	119	238	239	478	1673	3346	4063	8126	28441	56882
Elements	1	1	6	6	16	16	96	96	238	238	1428	1428	3808	3808	22848	22848	56882
Conjugacy classes	1	1	6	6	16	16	96	96	238	238	1428	1428	3808	3808	22848	22848	56882
Divisions	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	16
Autjugacy classes	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	16

Minimal presentations

Permutation degree:	$265$
Transitive degree:	$56882$
Rank:	$1$
Inequivalent generators:	$1$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	1	not computed	not computed
Arbitrary	1	not computed	not computed

Constructions

Presentation:	$\langle a \mid a^{56882}=1 \rangle$
Permutation group:	Degree $265$ $\langle(1,2)(3,6,9,5,8,4,7)(10,16,22,11,17,23,12,18,24,13,19,25,14,20,26,15,21) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 1 & 1 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 238 & 0 \\ 0 & 238 \end{array}\right), \left(\begin{array}{rr} 7 & 0 \\ 0 & 7 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{239})$
Direct product:	$C_2$ $\, \times\, $ $C_7$ $\, \times\, $ $C_{17}$ $\, \times\, $ $C_{239}$
Semidirect product:	not isomorphic to a non-trivial semidirect product
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product

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

Homology

Primary decomposition:	$C_{2} \times C_{7} \times C_{17} \times C_{239}$
Schur multiplier:	$C_1$
Commutator length:	$0$

Subgroups

There are 16 subgroups, all normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_{56882}$	$G/Z \simeq$ $C_1$
Commutator:	$G' \simeq$ $C_1$	$G/G' \simeq$ $C_{56882}$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $C_{56882}$
Fitting:	$\operatorname{Fit} \simeq$ $C_{56882}$	$G/\operatorname{Fit} \simeq$ $C_1$
Radical:	$R \simeq$ $C_{56882}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{56882}$	$G/\operatorname{soc} \simeq$ $C_1$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2$
7-Sylow subgroup:	$P_{ 7 } \simeq$ $C_7$
17-Sylow subgroup:	$P_{ 17 } \simeq$ $C_{17}$
239-Sylow subgroup:	$P_{ 239 } \simeq$ $C_{239}$

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 56882: $C_{56882}$

Order 28441: $C_{28441}$

Order 8126: $C_{8126}$

Order 4063: $C_{4063}$

Order 3346: $C_{3346}$

Order 1673: $C_{1673}$

Order 478: $C_{478}$

Order 239: $C_{239}$

Order 238: $C_{238}$

Order 119: $C_{119}$

Order 34: $C_{34}$

Order 17: $C_{17}$

Order 14: $C_{14}$

Order 7: $C_7$

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 56882: $C_{56882}$

Order 28441: $C_{28441}$

Order 8126: $C_{8126}$

Order 4063: $C_{4063}$

Order 3346: $C_{3346}$

Order 1673: $C_{1673}$

Order 478: $C_{478}$

Order 239: $C_{239}$

Order 238: $C_{238}$

Order 119: $C_{119}$

Order 34: $C_{34}$

Order 17: $C_{17}$

Order 14: $C_{14}$

Order 7: $C_7$

Order 2: $C_2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 56882: $C_{56882}$ ($C_1$)

Order 28441: $C_{28441}$ ($C_2$)

Order 8126: $C_{8126}$ ($C_7$)

Order 4063: $C_{4063}$ ($C_{14}$)

Order 3346: $C_{3346}$ ($C_{17}$)

Order 1673: $C_{1673}$ ($C_{34}$)

Order 478: $C_{478}$ ($C_{119}$)

Order 239: $C_{239}$ ($C_{238}$)

Order 238: $C_{238}$ ($C_{239}$)

Order 119: $C_{119}$ ($C_{478}$)

Order 34: $C_{34}$ ($C_{1673}$)

Order 17: $C_{17}$ ($C_{3346}$)

Order 14: $C_{14}$ ($C_{4063}$)

Order 7: $C_7$ ($C_{8126}$)

Order 2: $C_2$ ($C_{28441}$)

Order 1: $C_1$ ($C_{56882}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 56882: $C_{56882}$ ($C_1$)

Order 28441: $C_{28441}$ ($C_2$)

Order 8126: $C_{8126}$ ($C_7$)

Order 4063: $C_{4063}$ ($C_{14}$)

Order 3346: $C_{3346}$ ($C_{17}$)

Order 1673: $C_{1673}$ ($C_{34}$)

Order 478: $C_{478}$ ($C_{119}$)

Order 239: $C_{239}$ ($C_{238}$)

Order 238: $C_{238}$ ($C_{239}$)

Order 119: $C_{119}$ ($C_{478}$)

Order 34: $C_{34}$ ($C_{1673}$)

Order 17: $C_{17}$ ($C_{3346}$)

Order 14: $C_{14}$ ($C_{4063}$)

Order 7: $C_7$ ($C_{8126}$)

Order 2: $C_2$ ($C_{28441}$)

Order 1: $C_1$ ($C_{56882}$)

Series

Derived series	$C_{56882}$	$\rhd$	$C_1$
Chief series	$C_{56882}$	$\rhd$	$C_{28441}$	$\rhd$	$C_{4063}$	$\rhd$	$C_{239}$	$\rhd$	$C_1$
Lower central series	$C_{56882}$	$\rhd$	$C_1$
Upper central series	$C_1$	$\lhd$	$C_{56882}$

Supergroups

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

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

Character theory

Complex character table

The $56882 \times 56882$ character table is not available for this group.

Rational character table

The $16 \times 16$ rational character table is not available for this group.