C₇×#8795 #13186011

order	1792
nilpotency class	3
isoclinism	isoclinic class of #8794
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₈

Aliases

direct product

C₇×#8795

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	#382
commutator	C₄²	C₂²×C₄×C₇
Frattini, cosocle	C₂×C₄²	C₂³×C₇
prime radical	C₂³×C₇	#124
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂×C₄²×C₇ ◁ G

270 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 126 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) = 47
transitive degree	μ_t(G) = 448

A minimal faithful permutation representation — 3 generators on 47 points:

(1,2,3,4,5,6,7)(9,14)(10,15)(11,12)(16,17)(18,26,30,22)(19,31,34,45)(20,25)(21,24)(23,33,39,42)(27,35)(28,43)(29,40,41,37)(32,46)(36,44)(38,47);(8,9,11,14,13,15,12,10)(16,18,27,41,21,30,20,29)(17,22,35,40,24,26,25,37)(19,32,42,44,34,28,33,47)(23,38,45,46,39,36,31,43);(8,10,11,9,13,14,12,15)(16,19)(17,23,24,39)(18,28)(20,33)(21,34)(22,36,26,38)(25,31,35,45)(27,42)(29,44)(30,32)(37,46,40,43)(41,47)

Conjugacy classes & character table

280 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 1783);