C₂²×#34218 #72320636

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶×C₄ · Schur multiplier (H₂) = C₂¹⁹

Aliases

direct product

C₂²×#34218

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator	C₂²	C₂⁶×C₄
Frattini, cosocle	C₂³	C₂⁷
prime radical	C₂³×#525	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂³ ▷ C₁

114725 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 7 generators on 24 points:

(3,4);(9,10,13,12)(11,14,15,16)(17,18,21,24)(19,20,23,22);(5,6,7,8)(9,11)(10,14)(12,16)(13,15)(17,19)(18,22)(20,24)(21,23);(1,2)(3,4);(17,19)(18,20)(21,23)(22,24);(9,11)(10,14)(12,16)(13,15)(17,19)(18,20)(21,23)(22,24);(9,12,13,10)(11,14,15,16)(17,20)(18,23)(19,24)(21,22)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂²×#7744932
maximal normal subgroup of	C₂²×#7744932
maximal quotient of	C₂²×#513587, C₂²×#519339, C₂²×#519346

GAP

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

gap> G := Group((3,4), (9,10,13,12)(11,14,15,16)(17,18,21,24)(19,20,23,22), (5,6,7,8)(9,11)(10,14)(12,16)(13,15)(17,19)(18,22)(20,24)(21,23), (1,2)(3,4), (17,19)(18,20)(21,23)(22,24), (9,11)(10,14)(12,16)(13,15)(17,19)(18,20)(21,23)(22,24), (9,12,13,10)(11,14,15,16)(17,20)(18,23)(19,24)(21,22));

C22×#34218 #72320636