C₂²×#35703 #72325831

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

Aliases

direct product

C₂²×#35703

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

106773 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:

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

Conjugacy classes & character table

256 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#7631563, C₂²×#7849737
maximal normal subgroup of	C₂²×#7631563, C₂²×#7849737

GAP

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

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

C22×#35703 #72325831