C₂×D₄×#381 #72334517

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹⁶

Aliases

direct product

C₂×D₄×#381

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator	C₂³	C₂⁵×C₄
Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂⁵×D₄	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

75717 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 22 points:

(4,6)(8,10)(11,12,14,13)(17,21)(19,22);(3,4)(5,6)(7,8,9,10)(11,13,14,12)(15,16,18,20)(17,22,21,19);(4,6);(3,4)(5,6);(1,2);(15,17)(16,19)(18,21)(20,22)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×D₄×#2849, C₂×D₄×#2613, C₂×D₄×#2308
maximal normal subgroup of	C₂×D₄×#2849, C₂×D₄×#2613, C₂×D₄×#2308
maximal quotient of	C₂×D₄×C₂²⋊#95

GAP

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

gap> G := Group((4,6)(8,10)(11,12,14,13)(17,21)(19,22), (3,4)(5,6)(7,8,9,10)(11,13,14,12)(15,16,18,20)(17,22,21,19), (4,6), (3,4)(5,6), (1,2), (15,17)(16,19)(18,21)(20,22));

C2×D4×#381 #72334517