C₇×#13630 #13190846

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×#13630

Subgroups

	subgroup	G/H
center	C₇×C₁₆	C₂×D₄
commutator	C₄	C₂²×C₇×C₁₆
Frattini, cosocle	C₂×C₁₆	C₂³×C₇
socle	C₂×C₇	C₂×C₄⋊C₁₆
prime radical	C₂³×C₇	M₃₂
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₇×C₁₆ ◁ C₂²×C₇×C₁₆ ◁ G

198 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 154 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)(8,9,12,18,13,19,30,41,14,17,31,39,33,40,56,62,15,20,32,42,34,43,57,63,11,27,28,53,29,54,55,69)(10,21,24,45,25,46,50,66,26,36,51,59,52,60,68,70,22,44,47,64,48,65,67,71,23,16,35,37,49,38,58,61);(8,10,11,23,15,22,14,26)(9,16,17,21,20,36,27,44)(12,24,28,35,32,47,31,51)(13,25,29,49,34,48,33,52)(18,37,39,45,42,59,53,64)(19,38,40,46,43,60,54,65)(30,50,55,58,57,67,56,68)(41,61,62,66,63,70,69,71);(10,22)(16,36)(21,44)(23,26)(24,47)(25,48)(35,51)(37,59)(38,60)(45,64)(46,65)(49,52)(50,67)(58,68)(61,70)(66,71)

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, 6618);