#120×#379 #72286705

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹⁹

Aliases

direct product

#120×#379

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂⁵	C₂⁶
commutator	C₂⁴	C₂⁵×C₄
prime radical	C₂⁵×#120	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₁

2750804 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 128

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

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	#120×#2846, #72337359
maximal normal subgroup of	#120×#2846, #72337359

GAP

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

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