C₂×#72232360 #72338061

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁹

Aliases

direct product

C₂×#72232360

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁶
commutator, Frattini, cosocle	C₂⁴	C₂⁷
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

At least 8686 conjugacy classes of subgroups; only the 33672 normal subgroups are shown.

Full subgroup data →

Presentation

minimal degree	μ(G) = 30
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 7 generators on 30 points:

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

Conjugacy classes & character table

272 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂×#72232371
maximal normal subgroup of	C₂×#72232371

GAP

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

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

C2×#72232360 #72338061