C₂×D₄×#380 #72324011

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

Aliases

direct product

C₂×D₄×#380

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

81057 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 18 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×D₄×#2847, C₂×D₄×#2408, C₂×D₄²×C₂²⋊C₄, …
maximal normal subgroup of	C₂×D₄×#2847, C₂×D₄×#2408, C₂×D₄²×C₂²⋊C₄, …
maximal quotient of	C₂²×D₄×#380, C₂×D₄×C₂²⋊#95, C₂×#72334306

GAP

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

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

C2×D4×#380 #72324011