C₂²×C₂²⋊C₄×H(5,𝔽₂) #72324195

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷×C₄ · Schur multiplier (H₂) = C₂²⁸

Aliases

direct product

C₂²×C₂²⋊C₄×H(5,𝔽₂)

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁶
commutator	C₂²	C₂⁷×C₄
Frattini, cosocle	C₂³	C₂⁸
prime radical	C₂⁵×H(5,𝔽₂)	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5711688 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:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂²×#120×H(5,𝔽₂)
maximal normal subgroup of	C₂²×#120×H(5,𝔽₂)

GAP

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

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

C22×C22⋊C4×H(5,𝔽2) #72324195