C₂×#72232295 #72338068

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁸

Aliases

direct product

C₂×#72232295

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 19916 conjugacy classes of subgroups; only the 33608 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:

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

Conjugacy classes & character table

308 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((10,14)(12,13)(17,21)(20,22)(25,29)(27,30), (7,8)(9,11)(10,13)(12,14)(23,24,26,28)(25,27,29,30), (3,4)(5,6)(8,9)(10,14)(15,16,18,19)(17,22,21,20)(23,24,26,28)(25,27,29,30), (4,5)(10,14)(12,13)(16,19)(17,21), (1,2), (3,4,6,5)(7,9,11,8)(10,12,14,13)(15,16)(17,22)(18,19)(20,21)(23,24)(25,30)(26,28)(27,29), (3,5,6,4)(7,10)(8,12)(9,13)(11,14)(15,17)(16,20)(18,21)(19,22)(23,25,26,29)(24,27,28,30));

C2×#72232295 #72338068