C₂×#7714555 #72332290

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

Aliases

direct product

C₂×#7714555

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

91429 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)(16,18)(21,22)(23,26)(29,30);(3,4)(5,6)(9,13)(12,14)(15,16)(17,22)(18,20)(19,26)(21,25)(23,28)(24,29)(27,30);(15,17)(16,21)(18,22)(19,24)(20,25)(23,29)(26,30)(27,28);(1,2);(8,11)(12,14)(15,17)(16,22)(18,21)(19,27)(20,25)(23,29)(24,28)(26,30);(7,8,10,11)(9,12,13,14)(15,18,20,16)(17,22,25,21)(19,26,28,23)(24,30,27,29);(7,9)(8,12)(10,13)(11,14)(15,19,20,28)(16,23,18,26)(17,24,25,27)(21,29,22,30)

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72323293
maximal normal subgroup of	C₂×#72323293
maximal quotient of	C₂×#72323419

GAP

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

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

C2×#7714555 #72332290