C₇×#13648 #13190864

order	1792
nilpotency class	3
isoclinism	isoclinic class of AGL(1,ℤ/8)
automorphism	?
homology	Abelianization (H₁) = C₂²×C₇×C₁₆ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇×#13648

Subgroups

	subgroup	G/H
center	C₇×C₁₆	C₂×D₄
commutator	C₄	C₂²×C₇×C₁₆
Frattini, cosocle	C₂×C₁₆	C₂³×C₇
socle	C₂×C₇	D₄×C₁₆
prime radical	C₇×#135	C₈
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₇×C₁₆ ◁ C₂×C₇×C₃₂ ◁ G

184 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 118 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)(8,9,13,20,14,21,38,53,15,19,39,51,41,52,44,69,16,22,40,54,42,24,56,58,12,34,36,65,37,57,62,68)(10,23,26,55,27,32,60,64,25,17,43,45,59,33,18,48,28,46,50,70,61,30,49,63,35,47,66,71,67,11,29,31);(8,10)(9,17)(11,24)(12,35)(13,26)(14,27)(15,25)(16,28)(18,44)(19,46)(20,45)(21,33)(22,47)(23,34)(29,62)(30,52)(31,58)(32,57)(36,66)(37,67)(38,60)(39,43)(40,50)(41,59)(42,61)(48,53)(49,56)(51,70)(54,71)(55,65)(63,69)(64,68);(8,11,13,31,14,23,38,55,15,32,39,64,41,17,44,45,16,33,40,48,42,46,56,70,12,30,36,63,37,47,62,71)(9,18,20,28,21,50,53,61,19,49,51,35,52,66,69,67,22,29,54,10,24,26,58,27,34,60,65,25,57,43,68,59)

Conjugacy classes & character table

616 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 6636);