C₂²×#7332 #72231868

order	1024
nilpotency class	2
isoclinism	isoclinic class of H(3,ℤ/4)
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₆ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₂²×#7332

Subgroups

	subgroup	G/H
center	C₂³×C₈	C₄²
commutator	C₄	C₂²×C₄×C₁₆
Frattini, cosocle	C₄×C₁₆	C₂⁴
socle, prime radical	C₂⁴	#346
LCS	G ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

973 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:

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

Conjugacy classes & character table

352 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂²×#93668, #72231908, #72231911, …
maximal normal subgroup of	C₂²×#93668, #72231908, #72231911, …
maximal quotient of	#72323161

GAP

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

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

C22×#7332 #72231868