C₇×#13608 #13190824

order	1792
nilpotency class	5
isoclinism	isoclinic class of #13605
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇×#13608

Subgroups

	subgroup	G/H
center, socle	C₂×C₇	#2127
commutator	C₄²	C₂²×C₄×C₇
Frattini, cosocle	#124	C₂³×C₇
prime radical	C₂×C₇×#41	D₄
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇ ◁ C₂²×C₇ ◁ C₂²×C₄×C₇ ◁ C₇×#536 ◁ G

308 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 86 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 — 3 generators on 71 points:

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

Conjugacy classes & character table

175 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 6596);