C₂×#3629213 #72296402

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁰×C₄²

Aliases

direct product

C₂×#3629213

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

32925 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:

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

Conjugacy classes & character table

148 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72231820
maximal normal subgroup of	C₂×#72231820
maximal quotient of	C₂×#72278117

GAP

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

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

C2×#3629213 #72296402