C₂×#7642350 #72339328

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁹

Aliases

direct product

C₂×#7642350

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂⁶
commutator, Frattini, cosocle	C₂³	C₂⁷
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂³ ▷ C₁

109669 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 128

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

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

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72234675
maximal normal subgroup of	C₂×#72234675
maximal quotient of	C₂×#72334925, C₂×#72237462, C₂×#72237464

GAP

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

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

C2×#7642350 #72339328