C₂×D₄×#2408 #72324021

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂²²

Aliases

direct product

C₂×D₄×#2408

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁶
commutator, Frattini, cosocle	C₂⁴	C₂⁷
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

At least 7502 conjugacy classes of subgroups; only the 49224 normal subgroups are shown.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 7 generators on 18 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×D₄²×#120
maximal normal subgroup of	C₂×D₄²×#120

GAP

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

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

C2×D4×#2408 #72324021