C₇×#13368 #13190584

order	1792
nilpotency class	4
isoclinism	isoclinic class of #13316
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇×#13368

Subgroups

	subgroup	G/H
center, socle	C₂²×C₇	#447
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₇
prime radical	C₂³×C₇	C₂×D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂²×C₄×C₇ ◁ C₂×C₇×#97 ◁ G

370 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 114 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)(8,9,11,13)(10,14,15,12)(16,17)(18,26)(19,30)(20,35)(21,24)(22,27)(23,34)(25,33)(28,46)(29,43)(31,42)(32,41)(36,40)(37,44)(38,47)(39,45);(8,10,11,15)(9,12,13,14)(19,31)(22,37)(23,39)(27,44)(30,42)(32,47)(34,45)(38,41);(16,18,21,29)(17,22,24,37)(19,32,31,47)(20,36,33,28)(23,40,39,46)(25,41,35,38)(26,42,43,30)(27,45,44,34)

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, 6356);