C₂×#72237044 #72294445

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

Aliases

direct product

C₂×#72237044

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 11354 conjugacy classes of subgroups; only the 32656 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:

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

Conjugacy classes & character table

248 conjugacy classes — view the full character table.

Relationships

maximal quotient of

C₂×#72293988

GAP

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

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

C2×#72237044 #72294445