C₂×#72232099 #72325686

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

Aliases

direct product

C₂×#72232099

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 13662 conjugacy classes of subgroups; only the 38656 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)(11,12)(13,14)(16,17)(18,22);(7,8)(9,10)(12,14)(15,16)(17,19)(18,21)(20,22);(11,12)(13,14)(16,17)(20,21);(4,6)(11,12)(13,14)(15,16,19,17)(18,21,22,20);(1,2);(11,12,13,14)(15,17,19,16)(18,20,22,21);(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,18)(16,20)(17,21)(19,22)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×#72232191
maximal normal subgroup of	C₂×#72232191

GAP

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

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

C2×#72232099 #72325686