C₂×#72278732 #72297074

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁵×C₄

Aliases

direct product

C₂×#72278732

Subgroups

	subgroup	G/H
center, socle	C₂⁶	C₂⁵
commutator, Frattini, cosocle	C₂⁵	C₂⁶
LCS	G ▷ C₂⁵ ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁵ ▷ C₁

At least 16652 conjugacy classes of subgroups; only the 7392 normal subgroups are shown.

Full subgroup data →

Presentation

minimal degree	μ(G) = 30
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 6 generators on 30 points:

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

Conjugacy classes & character table

236 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72278822
maximal normal subgroup of	C₂×#72278822

GAP

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

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

C2×#72278732 #72297074