C₂²×#7463 #72234356

order	1024
nilpotency class	5
isoclinism	isoclinic class of #1426
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₂²×#7463

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	#367
commutator	C₁₆	C₂³×C₈
Frattini, cosocle	C₄×C₁₆	C₂⁴
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

753 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 1024

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

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

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#151085, #72286907, #72286954, …
maximal normal subgroup of	C₂²×#151085, #72286907, #72286954, …
maximal quotient of	#72321484

GAP

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

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

C22×#7463 #72234356