C₇×#60400 #13237616

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

Aliases

direct product

C₇×#60400

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

4732 conjugacy classes of subgroups; only the 1114 normal subgroups are shown.

normal subgroups only

This group has 4732 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)(14,25)(26,36)(27,37)(38,39);(8,9)(10,14)(11,15)(12,16)(13,17)(18,26)(19,27)(20,25)(21,28)(22,29)(23,30)(24,31)(32,38)(33,36)(34,39)(35,37);(8,10)(9,14)(11,18)(12,19)(13,20)(15,26)(16,27)(17,25)(21,32)(22,33)(23,34)(24,35)(28,38)(29,36)(30,39)(31,37);(8,11,13,22)(9,15,17,29)(10,18,20,33)(12,23,24,21)(14,26,25,36)(16,30,31,28)(19,34,35,32)(27,39,37,38);(8,12,13,24)(9,16,17,31)(10,19,20,35)(11,21,22,23)(14,27,25,37)(15,28,29,30)(18,32,33,34)(26,38,36,39)

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, 53388);