C₂×#72293937 #72294553

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

Aliases

direct product

C₂×#72293937

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 19884 conjugacy classes of subgroups; only the 32576 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)(9,10)(11,14)(12,13)(15,16,19,22)(17,18,21,20)(23,24)(25,28)(26,27)(29,30);(7,9,11,13)(8,12,14,10)(15,16)(17,18)(19,22)(20,21)(25,29)(28,30);(4,6)(7,10,11,12)(8,13,14,9)(15,16)(17,20)(18,21)(19,22)(24,27)(25,29);(3,4)(5,6)(15,17)(16,20)(18,22)(19,21)(23,24)(25,28)(26,27)(29,30);(7,10)(8,9)(11,12)(13,14)(15,17)(16,18)(19,21)(20,22);(7,9)(8,10)(11,13)(12,14)(15,18)(16,21)(17,22)(19,20)(23,25)(24,28)(26,29)(27,30);(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)(9,10)(11,14)(12,13)(15,16,19,22)(17,18,21,20)(23,24)(25,28)(26,27)(29,30), (7,9,11,13)(8,12,14,10)(15,16)(17,18)(19,22)(20,21)(25,29)(28,30), (4,6)(7,10,11,12)(8,13,14,9)(15,16)(17,20)(18,21)(19,22)(24,27)(25,29), (3,4)(5,6)(15,17)(16,20)(18,22)(19,21)(23,24)(25,28)(26,27)(29,30), (7,10)(8,9)(11,12)(13,14)(15,17)(16,18)(19,21)(20,22), (7,9)(8,10)(11,13)(12,14)(15,18)(16,21)(17,22)(19,20)(23,25)(24,28)(26,29)(27,30), (1,2));

C2×#72293937 #72294553