C₂×#7678839 #72332284

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

Aliases

direct product

C₂×#7678839

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 6120 conjugacy classes of subgroups; only the 32077 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)(9,13)(11,14)(18,26)(22,27)(24,28)(29,30);(3,4)(5,6)(15,16)(17,21)(18,27)(19,23)(20,25)(22,26)(24,29)(28,30);(15,17)(16,20)(18,24)(19,25)(21,23)(22,29)(26,28)(27,30);(1,2);(9,13)(11,14)(15,17)(16,21)(18,28)(19,25)(20,23)(22,29)(24,26)(27,30);(7,8,10,12)(9,11,13,14)(15,16)(17,20)(18,22)(19,23)(21,25)(24,29)(26,27)(28,30);(7,9)(8,11)(10,13)(12,14)(15,18,19,26)(16,22,23,27)(17,24,25,28)(20,29,21,30)

Conjugacy classes & character table

220 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72323293
maximal normal subgroup of	C₂×#72323293
maximal quotient of	C₂×#72323403

GAP

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

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

C2×#7678839 #72332284