C₂×#7752153 #72294567

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁹

Aliases

direct product

C₂×#7752153

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂⁶
commutator, Frattini, cosocle	C₂³	C₂⁷
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂³ ▷ C₁

90437 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 30
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 7 generators on 30 points:

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

Conjugacy classes & character table

196 conjugacy classes — view the full character table.

Relationships

maximal quotient of

C₂×#72293940, C₂×#72293950

GAP

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

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

C2×#7752153 #72294567