C₇×#9421 #13186637

order	1792
nilpotency class	3
isoclinism	isoclinic class of #9415
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇×#9421

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	#386
commutator	C₄²	C₂²×C₄×C₇
Frattini, cosocle	C₂×C₄²	C₂³×C₇
prime radical	C₇×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂×C₄²×C₇ ◁ G

488 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 488 conjugacy classes of subgroups — too many to draw. Only its 118 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) = 896

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

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

Conjugacy classes & character table

238 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 2409);