C₂×#72232286 #72338079

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

Aliases

direct product

C₂×#72232286

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 11570 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:

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

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

C2×#72232286 #72338079