C₂²×#15317 #72321263

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³×C₄²

Aliases

direct product

C₂²×#15317

Subgroups

	subgroup	G/H
center, socle	C₂⁶	C₂⁴
commutator, Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂³×#392	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

78557 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 6 generators on 24 points:

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

Conjugacy classes & character table

232 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#778474, C₂²×#778498, C₂²×#686216
maximal normal subgroup of	C₂²×#778474, C₂²×#778498, C₂²×#686216
maximal quotient of	C₂²×#158362

GAP

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

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

C22×#15317 #72321263