C₂²×#36111 #72324847

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁹

Aliases

direct product

C₂²×#36111

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁷
prime radical	C₂²×#2408	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂³ ▷ C₁

98277 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:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂²×#7779659
maximal normal subgroup of	C₂²×#7779659

GAP

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

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

C22×#36111 #72324847