C₂²×#845297 #72289271

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂²¹×C₄

Aliases

direct product

C₂²×#845297

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

3042720 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:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂²×#72219693, C₂²×#72231456, C₂×#72296179
maximal normal subgroup of	C₂²×#72219693, C₂²×#72231456, C₂×#72296179
maximal quotient of	C₂×#72296179, C₂²×#72218795

GAP

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

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

C22×#845297 #72289271