C₂²×#10792 #72280631

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄² · Schur multiplier (H₂) = C₂¹³

Aliases

direct product

C₂²×#10792

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₄×C₂²⋊C₄
commutator	C₂×C₄	C₂³×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁵
socle	C₂³	C₂²⋊#95
prime radical	C₂⁴×#41	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×#117 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

39001 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:

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

Conjugacy classes & character table

232 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72239641, C₂²×#364600, C₂×#72239644, …
maximal normal subgroup of	C₂×#72239641, C₂²×#364600, C₂×#72239644, …

GAP

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

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

C22×#10792 #72280631