C₂×#8063108 #72332230

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

Aliases

direct product

C₂×#8063108

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

75599 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 30
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 7 generators on 30 points:

(4,6)(8,12)(10,13)(15,16)(17,21)(18,22)(19,30)(20,24)(23,26)(25,28)(27,29);(3,4)(5,6)(7,8)(9,13)(10,14)(11,12)(15,17)(16,21)(18,19)(20,28)(22,30)(23,29)(24,25)(26,27);(7,8)(9,13)(10,14)(11,12)(15,16,20,24)(17,21,28,25)(18,29,27,22)(19,23,26,30);(1,2);(8,12)(10,13)(15,16)(17,25)(18,22)(19,23)(20,24)(21,28)(26,30)(27,29);(7,9)(8,13)(10,12)(11,14)(15,18)(16,22)(17,26)(19,28)(20,27)(21,23)(24,29)(25,30);(7,10,11,13)(8,14,12,9)(15,19,20,26)(16,23,24,30)(17,27,28,18)(21,22,25,29)

Conjugacy classes & character table

190 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72323293
maximal normal subgroup of	C₂×#72323293
maximal quotient of	C₂×#72323397

GAP

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

gap> G := Group((4,6)(8,12)(10,13)(15,16)(17,21)(18,22)(19,30)(20,24)(23,26)(25,28)(27,29), (3,4)(5,6)(7,8)(9,13)(10,14)(11,12)(15,17)(16,21)(18,19)(20,28)(22,30)(23,29)(24,25)(26,27), (7,8)(9,13)(10,14)(11,12)(15,16,20,24)(17,21,28,25)(18,29,27,22)(19,23,26,30), (1,2), (8,12)(10,13)(15,16)(17,25)(18,22)(19,23)(20,24)(21,28)(26,30)(27,29), (7,9)(8,13)(10,12)(11,14)(15,18)(16,22)(17,26)(19,28)(20,27)(21,23)(24,29)(25,30), (7,10,11,13)(8,14,12,9)(15,19,20,26)(16,23,24,30)(17,27,28,18)(21,22,25,29));

C2×#8063108 #72332230