C₂²×#34985 #72335697

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

Aliases

direct product

C₂²×#34985

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁷
prime radical	C₂⁵×D₄	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂³ ▷ C₁

110029 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:

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

Conjugacy classes & character table

256 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#7675416, C₂²×#7628174
maximal normal subgroup of	C₂²×#7675416, C₂²×#7628174
maximal quotient of	C₂²×#612537

GAP

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

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

C22×#34985 #72335697