C₂²×#151085 #72234357

order	2048
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₈ · Schur multiplier (H₂) = C₂¹⁰

Aliases

direct product

C₂²×#151085

Subgroups

	subgroup	G/H
center, socle	C₂⁴	#2177
commutator	C₁₆	C₂⁴×C₈
Frattini, cosocle	C₄×C₁₆	C₂⁵
prime radical	C₂³×D₃₂	C₄
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ C₂²×C₈² ◁ C₂²×#1311 ◁ G
LECS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

4424 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 44 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂²×#72224639, #72286964, #72287264, …
maximal normal subgroup of	C₂²×#72224639, #72286964, #72287264, …
maximal quotient of	#72321544

GAP

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

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

C22×#151085 #72234357