C₂²×#612424 #72268591

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂²⁵

Aliases

direct product

C₂²×#612424

Subgroups

	subgroup	G/H
center, socle	C₂⁶	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁷
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

8866344 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 7 generators on 20 points:

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

Conjugacy classes & character table

464 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂²×#120², C₂×#72286737, C₂²×#72232804, …
maximal normal subgroup of	C₂²×#120², C₂×#72286737, C₂²×#72232804, …
maximal quotient of	C₂×#72286737, #72287919, C₂×#72286701, …

GAP

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

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

C22×#612424 #72268591