C₇×#60402 #13237618

order	1792
nilpotency class	3
isoclinism	isoclinic class of #60399
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₇ · Schur multiplier (H₂) = C₂¹⁰

Aliases

direct product

C₇×#60402

Subgroups

	subgroup	G/H
center, socle	C₂×C₇	C₂²×#120
commutator, Frattini, cosocle	C₂³	C₂⁵×C₇
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇ ◁ C₂²×C₇×Q₈ ◁ G

4778 conjugacy classes of subgroups; only the 1114 normal subgroups are shown.

normal subgroups only

This group has 4778 conjugacy classes of subgroups — too many to draw. Only its 1114 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 39
transitive degree	μ_t(G) = 224

A minimal faithful permutation representation — 5 generators on 39 points:

(1,2,3,4,5,6,7)(8,9,14,13)(10,15,21,20)(11,16,24,23)(12,17,25,26)(18,30,32,27)(19,28,33,34)(22,29,35,36)(31,38,39,37);(8,10)(9,15)(11,18)(12,19)(13,20)(14,21)(16,27)(17,28)(22,31)(23,30)(24,32)(25,33)(26,34)(29,37)(35,39)(36,38);(8,11)(9,16)(10,18)(12,22)(13,23)(14,24)(15,27)(17,29)(19,31)(20,30)(21,32)(25,35)(26,36)(28,37)(33,39)(34,38);(8,12,14,25)(9,17,13,26)(10,19,21,33)(11,22,24,35)(15,28,20,34)(16,29,23,36)(18,31,32,39)(27,37,30,38);(8,13,14,9)(10,20,21,15)(11,23,24,16)(12,17,25,26)(18,30,32,27)(19,28,33,34)(22,29,35,36)(31,37,39,38)

Conjugacy classes & character table

406 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 53390);