C₂×#4606354 #72337846

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

Aliases

direct product

C₂×#4606354

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

37807 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:

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

Conjugacy classes & character table

136 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232279, C₂×#72232308, C₂×#72232330
maximal normal subgroup of	C₂×#72232279, C₂×#72232308, C₂×#72232330

GAP

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

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

C2×#4606354 #72337846