C₂²×#14662 #72293243

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹²×C₄³

Aliases

direct product

C₂²×#14662

Subgroups

	subgroup	G/H
center, socle	C₂⁶	C₂⁴
commutator	C₂³	C₂⁵×C₄
Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂⁵×D₄	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

59445 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 6 generators on 28 points:

(3,4);(13,14,16,18)(15,17,19,20)(21,22,25,23)(24,26,28,27);(6,8)(10,12)(15,19)(17,20)(22,23)(26,27);(5,6)(7,8)(9,10,11,12)(13,14)(15,20)(16,18)(17,19)(21,23,25,22)(24,26,28,27);(13,15,16,19)(14,17,18,20)(21,24)(22,26)(23,27)(25,28);(1,2)(3,4)

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#581603, C₂²×#4421184, C₂³×#14662
maximal normal subgroup of	C₂²×#581603, C₂²×#4421184, C₂³×#14662
maximal quotient of	C₂³×#14662

GAP

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

gap> G := Group((3,4), (13,14,16,18)(15,17,19,20)(21,22,25,23)(24,26,28,27), (6,8)(10,12)(15,19)(17,20)(22,23)(26,27), (5,6)(7,8)(9,10,11,12)(13,14)(15,20)(16,18)(17,19)(21,23,25,22)(24,26,28,27), (13,15,16,19)(14,17,18,20)(21,24)(22,26)(23,27)(25,28), (1,2)(3,4));

C22×#14662 #72293243