C₂×#72279977 #72302779

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂²⁰

Aliases

direct product

C₂×#72279977

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 9790 conjugacy classes of subgroups; only the 40672 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:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×#72280587, C₂²×#72279977
maximal normal subgroup of	C₂×#72280587, C₂²×#72279977
maximal quotient of	C₂×#120×#392, C₂²×#72279977

GAP

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

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

C2×#72279977 #72302779