C₂×#2822413 #72337865

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹¹×C₄

Aliases

direct product

C₂×#2822413

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂²×#2403	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

42917 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 30
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 30 points:

(3,4,6,5)(7,8,11,9)(10,13,14,12)(16,20)(17,22)(23,24)(25,27)(26,28)(29,30);(3,4,6,5)(10,14)(12,13)(15,16,19,20)(17,18,22,21)(23,24,26,28)(25,29,30,27);(4,5)(8,9)(12,13)(16,20)(17,22)(23,24,26,28)(25,29,30,27);(4,5)(7,9,11,8)(10,12,14,13)(15,17,19,22)(16,21,20,18)(23,24)(25,29)(26,28)(27,30);(1,2);(3,5,6,4)(7,10)(8,12)(9,13)(11,14)(15,18,19,21)(16,22,20,17)(23,25,26,30)(24,27,28,29)

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232283, C₂×#72232336, C₂×#72232297
maximal normal subgroup of	C₂×#72232283, C₂×#72232336, C₂×#72232297

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

gap> G := Group((3,4,6,5)(7,8,11,9)(10,13,14,12)(16,20)(17,22)(23,24)(25,27)(26,28)(29,30), (3,4,6,5)(10,14)(12,13)(15,16,19,20)(17,18,22,21)(23,24,26,28)(25,29,30,27), (4,5)(8,9)(12,13)(16,20)(17,22)(23,24,26,28)(25,29,30,27), (4,5)(7,9,11,8)(10,12,14,13)(15,17,19,22)(16,21,20,18)(23,24)(25,29)(26,28)(27,30), (1,2), (3,5,6,4)(7,10)(8,12)(9,13)(11,14)(15,18,19,21)(16,22,20,17)(23,25,26,30)(24,27,28,29));

C2×#2822413 #72337865