C₂²×#93757 #72286358

order	2048
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₆ · Schur multiplier (H₂) = C₂⁷

Aliases

direct product

C₂²×#93757

Subgroups

	subgroup	G/H
center, socle	C₂⁴	#1333
commutator	C₁₆	C₂³×C₁₆
Frattini, cosocle	#1311	C₂⁴
prime radical	C₂³×D₁₆	C₈
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ C₂²×C₈² ◁ C₂²×#1311 ◁ G
LECS	G ▷ #1311 ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

2400 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 52
transitive degree	μ_t(G) = 1024

A minimal faithful permutation representation — 4 generators on 52 points:

(1,2);(3,4);(5,6,7,10,8,12,16,11,9,14,17,15,18,13,20,19)(22,27,28,41,29,42,39,44)(23,31,33,49)(24,35)(25,38)(30,47,45,43,46,32,36,51)(34,40,52,50)(37,48);(5,6)(7,15)(8,12)(9,14)(10,17)(11,20)(13,18)(16,19)(21,22)(23,32)(24,36)(25,39)(26,29)(27,40)(28,38)(30,48)(31,41)(33,47)(34,51)(35,46)(37,45)(42,50)(43,52)(44,49)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂²×#72224639, #72322541, #72322903, …
maximal normal subgroup of	C₂²×#72224639, #72322541, #72322903, …
maximal quotient of	#72323163

GAP

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

gap> G := Group((1,2), (3,4), (5,6,7,10,8,12,16,11,9,14,17,15,18,13,20,19)(22,27,28,41,29,42,39,44)(23,31,33,49)(24,35)(25,38)(30,47,45,43,46,32,36,51)(34,40,52,50)(37,48), (5,6)(7,15)(8,12)(9,14)(10,17)(11,20)(13,18)(16,19)(21,22)(23,32)(24,36)(25,39)(26,29)(27,40)(28,38)(30,48)(31,41)(33,47)(34,51)(35,46)(37,45)(42,50)(43,52)(44,49));

C22×#93757 #72286358