C₇×#7309 #13184525

order	1792
nilpotency class	4
isoclinism	isoclinic class of #7307
automorphism	?
homology	Abelianization (H₁) = C₄²×C₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇×#7309

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	#342
commutator	C₂²×C₄	C₄²×C₇
Frattini, cosocle	C₂²×#41	C₂²×C₇
prime radical	C₂²×C₇×#41	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂²×C₄×C₇ ◁ C₂²×C₇×#41 ◁ G

470 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 470 conjugacy classes of subgroups — too many to draw. Only its 78 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 — 2 generators on 47 points:

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

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, 297);