C₂×#72232365 #72338087

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁶×C₄²

Aliases

direct product

C₂×#72232365

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 6316 conjugacy classes of subgroups; only the 32808 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:

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

Conjugacy classes & character table

248 conjugacy classes — view the full character table.

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

C2×#72232365 #72338087