C₇×#7388 #13184604

order	1792
nilpotency class	5
isoclinism	isoclinic class of #1418
automorphism	?
homology	Abelianization (H₁) = C₂×C₇×C₈ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₇×#7388

Subgroups

	subgroup	G/H
center	C₄×C₇	#357
commutator	C₁₆	C₂×C₇×C₈
Frattini, cosocle	C₄×C₁₆	C₂²×C₇
socle	C₂×C₇	#1331
prime radical	C₇×#508	C₄
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₇ ◁ C₂×C₇×C₈ ◁ C₄×C₇×C₈ ◁ C₄×C₇×C₁₆ ◁ G

118 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

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

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, 376);