C₂²×#15172 #72320809

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

Aliases

direct product

C₂²×#15172

Subgroups

	subgroup	G/H
center, socle	C₂⁶	C₂⁴
commutator, Frattini, cosocle	C₂⁴	C₂⁶
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

96429 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);(9,10,12,11)(18,21)(19,23);(13,14,16,15)(19,23)(22,24);(6,8)(10,11)(14,15)(17,18)(19,22)(20,21)(23,24);(1,2)(3,4);(5,6,7,8)(9,11,12,10)(13,15,16,14)(17,19)(18,22)(20,23)(21,24)

Conjugacy classes & character table

256 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#823456, C₂²×#1409001, C₂²×#790151, …
maximal normal subgroup of	C₂²×#823456, C₂²×#1409001, C₂²×#790151, …
maximal quotient of	C₂²×#154647, C₂²×#158361

GAP

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

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

C22×#15172 #72320809