C₂×#1083773 #72294730

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³

Aliases

direct product

C₂×#1083773

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂×#14715	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

34535 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 30
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 30 points:

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

Conjugacy classes & character table

172 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232681
maximal normal subgroup of	C₂×#72232681
maximal quotient of	C₂×#72293952

GAP

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

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

C2×#1083773 #72294730