C₂×D₄×#2846 #72286381

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂²⁴

Aliases

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

8406048 conjugacy classes of subgroups.

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 128

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

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

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

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

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