C₂×#3943961 #72337914

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

Aliases

direct product

C₂×#3943961

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂²×#2410	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

35411 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)(9,13)(10,11)(12,14)(16,17)(20,21)(23,24,27,30)(25,28,26,29);(3,4,6,5)(8,11)(9,14)(15,16)(17,19)(18,20)(21,22)(25,26)(28,29);(4,5)(9,14)(12,13)(16,17)(18,22)(23,25,27,26)(24,28,30,29);(4,5)(7,8)(9,13)(10,11)(12,14)(15,17,19,16)(18,20,22,21)(23,26,27,25)(24,29,30,28);(1,2);(3,5,6,4)(7,9)(8,12)(10,14)(11,13)(15,18)(16,20)(17,21)(19,22)(23,24)(25,28)(26,29)(27,30)

Conjugacy classes & character table

172 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232289, C₂×#72232295, C₂×#72232297
maximal normal subgroup of	C₂×#72232289, C₂×#72232295, C₂×#72232297

GAP

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

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

C2×#3943961 #72337914