C₂×#1913213 #72337826

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

Aliases

direct product

C₂×#1913213

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

34755 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)(7,8)(9,13)(10,11)(12,14)(15,16,18,19)(17,21,22,20)(23,24,27,29)(25,30,26,28);(8,11)(12,13)(15,16)(17,21)(18,19)(20,22)(23,25,27,26)(24,28,29,30);(4,6)(9,14)(12,13)(16,19)(20,21)(25,26)(28,30);(3,4)(5,6)(7,9,10,14)(8,12,11,13)(15,16)(17,20)(18,19)(21,22)(23,26,27,25)(24,28,29,30);(1,2);(7,8)(9,13)(10,11)(12,14)(15,17)(16,20)(18,22)(19,21)(23,24)(25,28)(26,30)(27,29)

Conjugacy classes & character table

148 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232281, C₂×#72232291, C₂×#72232286
maximal normal subgroup of	C₂×#72232281, C₂×#72232291, C₂×#72232286

GAP

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

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

C2×#1913213 #72337826