C₂×D₄×#2433 #72299628

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂²¹×C₄

Aliases

direct product

C₂×D₄×#2433

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

1158796 conjugacy classes of subgroups.

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:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×#72298104, C₂×D₄²×#120
maximal normal subgroup of	C₂×#72298104, C₂×D₄²×#120

GAP

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

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

C2×D4×#2433 #72299628