C₂×#72297292 #72334557

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

Aliases

direct product

C₂×#72297292

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 10594 conjugacy classes of subgroups; only the 43128 normal subgroups are shown.

Full subgroup data →

Presentation

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 7 generators on 22 points:

(8,10)(12,14)(15,16)(17,18)(19,20)(21,22);(4,6)(8,10)(11,12)(13,14)(16,19)(17,22);(3,4)(5,6)(7,8)(9,10)(12,14)(15,17)(16,21)(18,19)(20,22);(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,17)(16,21)(18,19)(20,22);(1,2);(8,10)(15,18)(16,17)(19,22)(20,21);(3,4)(5,6)(7,8)(9,10)(15,19,20,16)(17,21,22,18)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×D₄×#34980, C₂×#72335129
maximal normal subgroup of	C₂×D₄×#34980, C₂×#72335129

GAP

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

gap> G := Group((8,10)(12,14)(15,16)(17,18)(19,20)(21,22), (4,6)(8,10)(11,12)(13,14)(16,19)(17,22), (3,4)(5,6)(7,8)(9,10)(12,14)(15,17)(16,21)(18,19)(20,22), (3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,17)(16,21)(18,19)(20,22), (1,2), (8,10)(15,18)(16,17)(19,22)(20,21), (3,4)(5,6)(7,8)(9,10)(15,19,20,16)(17,21,22,18));

C2×#72297292 #72334557