C₇×#13656 #13190872

order	1792
nilpotency class	5
isoclinism	isoclinic class of #2263
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇×#13656

Subgroups

	subgroup	G/H
center	C₄×C₇	C₂×D₁₆
commutator	C₁₆	C₂²×C₄×C₇
Frattini, cosocle	C₂×C₁₆	C₂³×C₇
socle	C₂×C₇	C₄×D₁₆
prime radical	C₇×#2262	C₂
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₇ ◁ C₂×C₇×C₈ ◁ C₄×C₇×C₈ ◁ C₄×C₇×C₁₆ ◁ G

184 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 86 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)(8,9,13,19,15,22,37,21)(10,23,26,54,28,55,27,61)(11,20,30,46,31,17,14,38)(12,32,34,64,36,45,69,65)(16,40,42,60,44,59,52,25)(18,29,48,62,50,63,49,35)(24,56,57,71,58,53,39,47)(33,51,66,70,68,41,67,43);(8,10)(9,16)(11,24)(12,33)(13,26)(14,39)(15,28)(17,41)(18,47)(19,42)(20,51)(21,52)(22,44)(23,32)(25,35)(27,37)(29,40)(30,57)(31,58)(34,66)(36,68)(38,43)(45,55)(46,70)(48,56)(49,53)(50,71)(54,64)(59,63)(60,62)(61,65)(67,69);(8,11,13,30,15,31,37,14)(9,17,19,38,22,20,21,46)(10,24,26,57,28,58,27,39)(12,29,34,62,36,63,69,35)(16,41,42,43,44,51,52,70)(18,45,48,65,50,32,49,64)(23,53,54,47,55,56,61,71)(25,33,40,66,60,68,59,67)

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, 6644);