C₂×#72232180 #72278652

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

Aliases

direct product

C₂×#72232180

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 8438 conjugacy classes of subgroups; only the 41408 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);(8,10)(15,16,18,17);(3,4,5,6)(12,14)(16,17);(3,4)(5,6)(11,12)(13,14)(15,17,18,16);(3,4,5,6)(7,8)(9,10);(3,4)(5,6);(1,2)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×#72298104, C₂×#72232191
maximal normal subgroup of	C₂×#72298104, C₂×#72232191
maximal quotient of	C₂×#72278177

GAP

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

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

C2×#72232180 #72278652