C₇×#12463 #13189679

order	1792
nilpotency class	4
isoclinism	isoclinic class of #12463
automorphism	?
homology	Abelianization (H₁) = C₂³×C₇ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₇×#12463

Subgroups

	subgroup	G/H
center, socle	C₂×C₇	#1675
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₇
prime radical	C₇×#135	D₄
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇ ◁ C₂×C₄×C₇ ◁ C₄×C₇×C₈ ◁ G

172 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 172 conjugacy classes of subgroups — too many to draw. Only its 74 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)(8,9,15,21)(10,22,27,54)(11,28)(12,33,35,19)(13,18,37,49)(14,38,34,20)(16,39,43,24)(17,44)(23,55)(25,52,62,42)(26,41,60,53)(29,48)(30,64)(31,50)(32,45)(36,46)(40,57)(47,65)(51,66)(56,63)(58,68)(59,67)(61,70)(69,71);(8,10,13,25,14,26,35,39,15,27,37,62,34,60,12,24)(9,16,33,53,20,42,49,54,21,43,19,41,38,52,18,22)(11,23,36,61,30,57,44,66,32,59,65,71,31,58,29,56)(17,40,64,70,46,55,28,63,48,68,50,69,47,67,45,51);(8,11)(9,17)(10,23)(12,29)(13,36)(14,30)(15,32)(16,40)(18,45)(19,50)(20,46)(21,48)(22,51)(24,56)(25,61)(26,57)(27,59)(28,49)(31,34)(33,64)(35,44)(37,65)(38,47)(39,66)(41,69)(42,55)(43,68)(52,67)(53,70)(54,63)(58,60)(62,71)

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, 5451);