H(5,𝔽₂)×#379 #72337325

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶×C₄ · Schur multiplier (H₂) = C₂²³

Aliases

direct product

H(5,𝔽₂)×#379

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂⁴	C₂⁷
commutator	C₂³	C₂⁶×C₄
prime radical	C₂⁵×H(5,𝔽₂)	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

4646344 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 7 generators on 20 points:

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

Conjugacy classes & character table

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

Relationships

maximal quotient of

#72337359, #72337394

GAP

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

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

H(5,𝔽2)×#379 #72337325