C₂×#7727971 #72332228

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

Aliases

direct product

C₂×#7727971

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 6398 conjugacy classes of subgroups; only the 31821 normal subgroups are shown.

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:

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

Conjugacy classes & character table

196 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72323293
maximal normal subgroup of	C₂×#72323293
maximal quotient of	C₂×#72323389

GAP

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

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

C2×#7727971 #72332228