C₂×#847893 #72294781

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²×C₄

Aliases

direct product

C₂×#847893

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

52873 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)(7,8,10,11)(9,13,14,12)(15,16,18,19)(17,21,22,20)(23,24,26,28)(25,29,30,27);(4,6)(8,11)(9,14)(16,19)(20,21)(23,24,26,28)(25,27,30,29);(3,4)(5,6)(9,14)(12,13)(15,17)(16,20)(18,22)(19,21)(23,25,26,30)(24,27,28,29);(7,8)(9,12)(10,11)(13,14)(15,16)(17,20)(18,19)(21,22)(23,24)(25,27)(26,28)(29,30);(7,9)(8,12)(10,14)(11,13)(15,17)(16,20)(18,22)(19,21)(23,25)(24,27)(26,30)(28,29);(1,2)

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72293946
maximal normal subgroup of	C₂×#72293946
maximal quotient of	C₂×#72293971

GAP

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

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

C2×#847893 #72294781