C₂²×#14977 #72296936

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

Aliases

direct product

C₂²×#14977

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

75861 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:

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

Conjugacy classes & character table

256 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#1337713, C₂²×#3860970, C₂²×#4033889, …
maximal normal subgroup of	C₂²×#1337713, C₂²×#3860970, C₂²×#4033889, …
maximal quotient of	C₂²×#154635

GAP

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

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

C22×#14977 #72296936