C₂²×#37714 #72324827

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

Aliases

direct product

C₂²×#37714

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

101645 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:

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

Conjugacy classes & character table

232 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#7768137, C₂²×#7763623
maximal normal subgroup of	C₂²×#7768137, C₂²×#7763623
maximal quotient of	C₂²×#525856

GAP

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

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

C22×#37714 #72324827