C₂²⋊C₄×#2846 #72286749

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

Aliases

	subgroup	G/H
center, Frattini, socle, cosocle	C₂⁵	C₂⁶
commutator	C₂⁴	C₂⁵×C₄
prime radical	C₂³×#2846	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₁

2777978 conjugacy classes of subgroups.

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

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

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

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

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

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

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