C₂×#7941013 #72304494

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

Aliases

direct product

C₂×#7941013

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

At least 6404 conjugacy classes of subgroups; only the 31853 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:

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

Conjugacy classes & character table

190 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72304329
maximal normal subgroup of	C₂×#72304329

GAP

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

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

C2×#7941013 #72304494