C₂×#7719290 #72332287

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

Aliases

direct product

C₂×#7719290

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

75313 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)(9,13)(12,14)(18,26)(22,30)(24,27)(28,29);(3,4)(5,6)(8,11)(12,14)(15,16)(17,21)(18,22)(19,23)(20,25)(24,29)(26,30)(27,28);(15,17)(16,20)(18,24)(19,25)(21,23)(22,28)(26,27)(29,30);(1,2);(9,13)(12,14)(15,17)(16,21)(18,27)(19,25)(20,23)(22,28)(24,26)(29,30);(7,8,10,11)(9,12,13,14)(15,16)(17,20)(18,22)(19,23)(21,25)(24,28)(26,30)(27,29);(7,9)(8,12)(10,13)(11,14)(15,18,19,26)(16,22,23,30)(17,24,25,27)(20,28,21,29)

Conjugacy classes & character table

196 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72323293
maximal normal subgroup of	C₂×#72323293
maximal quotient of	C₂×#72323409

GAP

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

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

C2×#7719290 #72332287