C₂²×C₂²⋊C₄² #72268554

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄² · Schur multiplier (H₂) = C₂¹⁶×C₄

Aliases

direct product

C₂²×C₂²⋊C₄²

Subgroups

	subgroup	G/H
center, socle	C₂⁶	C₂⁴
commutator	C₂²	C₂⁴×C₄²
Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂⁸	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

210741 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂²×C₂²⋊C₄×#120, C₂²×#648360, C₂²×#531607, …
maximal normal subgroup of	C₂²×C₂²⋊C₄×#120, C₂²×#648360, C₂²×#531607, …
maximal quotient of	C₂×C₂²⋊C₄×#379, C₂³×C₂²⋊C₄², C₂²×C₂²⋊C₄×#95

GAP

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

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

C22×C22⋊C42 #72268554