C₇×#7447 #13184663

order	1792
nilpotency class	5
isoclinism	isoclinic class of #7447
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₇ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₇×#7447

Subgroups

	subgroup	G/H
center, socle	C₂×C₇	#1361
commutator	C₂×C₁₆	C₂×C₄×C₇
Frattini, cosocle	C₄×C₁₆	C₂²×C₇
prime radical	C₂×C₇×D₁₆	C₄
LCS	G ▷ C₂×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇ ◁ C₂×C₄×C₇ ◁ C₄×C₇×C₈ ◁ C₄×C₇×C₁₆ ◁ G

130 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 130 conjugacy classes of subgroups — too many to draw. Only its 50 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) = 71
transitive degree	μ_t(G) = 448

A minimal faithful permutation representation — 2 generators on 71 points:

(1,2,3,4,5,6,7)(8,9,12,19,37,22,39,20,16,23,13,21,15,42,14,40)(10,24,26,48,43,56,27,61,29,57,41,71,28,62,38,55)(11,30,32,52,66,50,68,63,36,64,33,18,35,49,34,51)(17,31,44,60,46,69,53,59,47,65,70,25,54,58,45,67);(8,10)(9,17)(11,31)(12,27)(13,38)(14,41)(15,43)(16,29)(18,48)(19,53)(20,44)(21,45)(22,54)(23,47)(24,50)(25,34)(26,39)(28,37)(30,56)(32,59)(33,67)(35,69)(36,65)(40,70)(42,46)(49,57)(51,55)(52,71)(58,66)(60,68)(61,63)(62,64)

Conjugacy classes & character table

259 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 435);