C₂²×#49546 #72325833

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

Aliases

direct product

C₂²×#49546

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁷
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂³ ▷ C₁

106793 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:

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

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#7733277, C₂²×#7849737
maximal normal subgroup of	C₂²×#7733277, C₂²×#7849737

GAP

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

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

C22×#49546 #72325833