C₂²×#120² #72268598

order	4096
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁸ · Schur multiplier (H₂) = C₂³⁰

Aliases

direct product

C₂²×#120²

Subgroups

	subgroup	G/H
center, socle	C₂⁶	C₂⁶
commutator, Frattini, cosocle	C₂⁴	C₂⁸
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

82197047 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 8 generators on 20 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂³×#120²
maximal normal subgroup of	C₂³×#120²
maximal quotient of	C₂²×#120×#392, C₂³×#120²
automorphism group of	#3514, #3515, #17547, …

GAP

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

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

C22×#1202 #72268598