C₂×#72232299 #72338072

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

Aliases

direct product

C₂×#72232299

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 20970 conjugacy classes of subgroups; only the 32920 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:

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

Conjugacy classes & character table

260 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((3,4,6,5)(8,9)(10,14)(15,16,19,17)(18,20,22,21)(23,24)(25,29)(26,27)(28,30), (7,8)(9,11)(10,13)(12,14)(15,16)(17,19)(18,21)(20,22)(23,24,26,27)(25,28,30,29), (3,4)(5,6)(10,14)(12,13)(24,27)(28,29), (4,5)(8,9)(10,14)(16,17)(20,21)(23,24)(25,28)(26,27)(29,30), (1,2), (3,4,6,5)(7,9,11,8)(10,12,14,13)(15,17,19,16)(18,21,22,20)(23,24)(25,29)(26,27)(28,30), (3,5,6,4)(7,10)(8,12)(9,13)(11,14)(15,18)(16,20)(17,21)(19,22)(23,25,26,30)(24,28,27,29));

C2×#72232299 #72338072