C₂×#4941683 #72337997

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³

Aliases

direct product

C₂×#4941683

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂²×#2433	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

37417 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:

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

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232342, C₂×#72232297
maximal normal subgroup of	C₂×#72232342, C₂×#72232297

GAP

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

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

C2×#4941683 #72337997