C₂²×D₄×#124 #72324158

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁹×C₄

Aliases

direct product

C₂²×D₄×#124

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁷
prime radical	C₂³×D₄²	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂³ ▷ C₁

144565 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂²×D₄×#535, C₂²×D₄×#546
maximal normal subgroup of	C₂²×D₄×#535, C₂²×D₄×#546
maximal quotient of	C₂²×D₄×#390, C₂²×D₄×#394, C₂²×#649110, …

GAP

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

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

C22×D4×#124 #72324158