C₂²×#15304 #72296893

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁴×C₄

Aliases

direct product

C₂²×#15304

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

70973 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:

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

Conjugacy classes & character table

232 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#2822351, C₂²×#1427668, C₂²×#1337713, …
maximal normal subgroup of	C₂²×#2822351, C₂²×#1427668, C₂²×#1337713, …
maximal quotient of	C₂²×#156707

GAP

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

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

C22×#15304 #72296893