C₂²×#17102 #72293785

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁴×C₄²

Aliases

direct product

C₂²×#17102

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

109925 conjugacy classes of subgroups; only the 7125 normal subgroups are shown.

normal subgroups only

This group has 109925 conjugacy classes of subgroups — too many to draw. Only its 7125 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 6 generators on 20 points:

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

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#2108769, C₂²×#4122222, C₂²×#918662, …
maximal normal subgroup of	C₂²×#2108769, C₂²×#4122222, C₂²×#918662, …

GAP

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

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