C₇×#7387 #13184603

order	1792
nilpotency class	5
isoclinism	isoclinic class of D₃₂
automorphism	?
homology	Abelianization (H₁) = C₂×C₇×C₈ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₇×#7387

Subgroups

	subgroup	G/H
center	C₇×C₈	D₁₆
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 ◁ C₇×C₈ ◁ C₂×C₇×C₈ ◁ C₄×C₇×C₈ ◁ C₄×C₇×C₁₆ ◁ G

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

normal subgroups only

This group has 120 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)(8,9,12,18,14,19,34,47,15,21,35,48,38,20,13,36)(10,22,24,50,25,51,56,70,27,52,57,71,26,58,37,43)(11,28,30,44,32,59,65,46,33,61,66,45,67,60,31,17)(16,29,39,53,41,62,68,55,42,64,69,54,49,63,40,23);(8,10)(9,16)(11,29)(12,24)(13,37)(14,25)(15,27)(17,43)(18,39)(19,41)(20,49)(21,42)(22,28)(23,31)(26,38)(30,53)(32,62)(33,64)(34,56)(35,57)(36,40)(44,50)(45,71)(46,70)(47,68)(48,69)(51,59)(52,61)(54,66)(55,65)(58,60)(63,67)

Conjugacy classes & character table

532 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 375);