C₂×#72278070 #72296446

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

Aliases

direct product

C₂×#72278070

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 12734 conjugacy classes of subgroups; only the 38160 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:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×#72298104
maximal normal subgroup of	C₂×#72298104
maximal quotient of	C₂×#72278229

GAP

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

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

C2×#72278070 #72296446