C₂²×#7385 #72234352

order	1024
nilpotency class	5
isoclinism	isoclinic class of #1418
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁷

Aliases

direct product

C₂²×#7385

Subgroups

	subgroup	G/H
center, socle	C₂⁴	#357
commutator	C₁₆	C₂³×C₈
Frattini, cosocle	C₄×C₁₆	C₂⁴
prime radical	C₂³×D₁₆	C₄
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ C₂²×C₄×C₁₆ ◁ G
LECS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1657 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 44 points:

(5,6,8,11)(7,9,12,10)(14,21)(15,27)(16,34)(17,29)(18,19)(22,32)(23,40)(24,39)(25,38)(26,28)(30,33)(31,41)(35,44)(36,37)(42,43);(1,2);(3,4);(6,9,11,10)(13,14,22,40,16,35,15,28,18,24,30,37,29,36,31,39,20,26,41,44,17,23,33,21,19,25,27,43,34,42,32,38)

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#151085, #72322477, #72322516, …
maximal normal subgroup of	C₂²×#151085, #72322477, #72322516, …
maximal quotient of	#72321439, #72321470

GAP

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

gap> G := Group((5,6,8,11)(7,9,12,10)(14,21)(15,27)(16,34)(17,29)(18,19)(22,32)(23,40)(24,39)(25,38)(26,28)(30,33)(31,41)(35,44)(36,37)(42,43), (1,2), (3,4), (6,9,11,10)(13,14,22,40,16,35,15,28,18,24,30,37,29,36,31,39,20,26,41,44,17,23,33,21,19,25,27,43,34,42,32,38));

C22×#7385 #72234352