C₂×#836223 #72294733

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

Aliases

direct product

C₂×#836223

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

35539 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:

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

Conjugacy classes & character table

196 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232681
maximal normal subgroup of	C₂×#72232681
maximal quotient of	C₂×#72293954

GAP

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

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

C2×#836223 #72294733