C₇×#7480 #13184696

order	1792
nilpotency class	5
isoclinism	isoclinic class of #1418
automorphism	?
homology	Abelianization (H₁) = C₄²×C₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇×#7480

Subgroups

	subgroup	G/H
center	C₄×C₇	#357
commutator	C₁₆	C₄²×C₇
Frattini, cosocle	C₂²×C₁₆	C₂²×C₇
socle	C₂×C₇	#1380
prime radical	C₂³×C₇	SD₃₂
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₇ ◁ C₂²×C₄×C₇ ◁ C₂²×C₇×C₈ ◁ C₂²×C₇×C₁₆ ◁ G

166 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 166 conjugacy classes of subgroups — too many to draw. Only its 98 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)(8,9,12,18,15,22,36,21)(10,23,25,55,28,47,59,57)(11,29,32,53,34,52,70,19)(13,37,35,49,39,48,33,17)(14,40,31,20,38,50,69,51)(16,41,42,30,46,67,45,24)(26,61,60,54,63,44,68,71)(27,64,58,56,62,43,66,65);(8,10,13,26,38,62,34,24,36,59,33,68,31,58,32,67,15,28,39,63,14,27,11,30,12,25,35,60,69,66,70,41)(9,16,19,43,51,61,49,23,18,42,29,65,40,54,48,55,22,46,53,64,20,44,17,47,21,45,52,56,50,71,37,57)

Conjugacy classes & character table

364 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 468);