C₂×#2109325 #72294541

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

Aliases

direct product

C₂×#2109325

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂²×#2403	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

42969 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:

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

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72293937
maximal normal subgroup of	C₂×#72293937
maximal quotient of	C₂×#72293977

GAP

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

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

C2×#2109325 #72294541