C₂²×#10950 #72314833

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄² · Schur multiplier (H₂) = C₂¹¹×C₄

Aliases

direct product

C₂²×#10950

Subgroups

	subgroup	G/H
center	C₂²×C₄	#386
commutator	C₂×C₄	C₂³×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁵
socle	C₂³	#1437
prime radical	C₂⁴×#41	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂⁵×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

23809 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 36 points:

(6,12)(7,16,19,21)(9,23)(13,28,26,30)(14,20)(17,35,32,33)(18,29,24,34)(22,25);(5,6,8,20,11,12,10,14)(7,17,16,33,19,32,21,35)(9,15,22,36,23,27,25,31)(13,29,28,18,26,34,30,24);(1,2);(5,7)(6,13)(8,21)(9,24)(10,16)(11,19)(12,26)(14,28)(15,32)(17,27)(18,23)(20,30)(22,34)(25,29)(31,35)(33,36);(3,4)

Conjugacy classes & character table

232 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72226729, C₂²×#364600, C₂×#72239600, …
maximal normal subgroup of	C₂×#72226729, C₂²×#364600, C₂×#72239600, …

GAP

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

gap> G := Group((6,12)(7,16,19,21)(9,23)(13,28,26,30)(14,20)(17,35,32,33)(18,29,24,34)(22,25), (5,6,8,20,11,12,10,14)(7,17,16,33,19,32,21,35)(9,15,22,36,23,27,25,31)(13,29,28,18,26,34,30,24), (1,2), (5,7)(6,13)(8,21)(9,24)(10,16)(11,19)(12,26)(14,28)(15,32)(17,27)(18,23)(20,30)(22,34)(25,29)(31,35)(33,36), (3,4));

C22×#10950 #72314833