C₂×#7633130 #72324372

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

Aliases

direct product

C₂×#7633130

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

97977 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:

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

Conjugacy classes & character table

244 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232032
maximal normal subgroup of	C₂×#72232032
maximal quotient of	C₂×#72232042

GAP

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

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

C2×#7633130 #72324372