C₂²⋊C₄×#386 #72335676

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄² · Schur multiplier (H₂) = C₂¹²×C₄

Aliases

direct product

C₂²⋊C₄×#386

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂⁵	C₂⁵
commutator	C₂³	C₂³×C₄²
prime radical	C₂⁵×D₄	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₁

48549 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 20 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	#72335757, #72335778, #72335832
maximal normal subgroup of	#72335757, #72335778, #72335832

GAP

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

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

C22⋊C4×#386 #72335676