C₂²×#612465 #72321311

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

Aliases

direct product

C₂²×#612465

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

1170144 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂²×#120×#121
maximal normal subgroup of	C₂²×#120×#121
maximal quotient of	C₂²×#72278330

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

C22×#612465 #72321311