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G = C5×C17⋊C4order 340 = 22·5·17

Direct product of C5 and C17⋊C4

direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary

Aliases: C5×C17⋊C4, C17⋊C20, C855C4, D17.C10, (C5×D17).2C2, SmallGroup(340,5)

Series: Derived Chief Lower central Upper central

C1C17 — C5×C17⋊C4
C1C17D17C5×D17 — C5×C17⋊C4
C17 — C5×C17⋊C4
C1C5

Generators and relations for C5×C17⋊C4
 G = < a,b,c | a5=b17=c4=1, ab=ba, ac=ca, cbc-1=b4 >

17C2
17C4
17C10
17C20

Smallest permutation representation of C5×C17⋊C4
On 85 points
Generators in S85
(1 69 52 35 18)(2 70 53 36 19)(3 71 54 37 20)(4 72 55 38 21)(5 73 56 39 22)(6 74 57 40 23)(7 75 58 41 24)(8 76 59 42 25)(9 77 60 43 26)(10 78 61 44 27)(11 79 62 45 28)(12 80 63 46 29)(13 81 64 47 30)(14 82 65 48 31)(15 83 66 49 32)(16 84 67 50 33)(17 85 68 51 34)
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17)(18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34)(35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51)(52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68)(69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85)
(2 14 17 5)(3 10 16 9)(4 6 15 13)(7 11 12 8)(19 31 34 22)(20 27 33 26)(21 23 32 30)(24 28 29 25)(36 48 51 39)(37 44 50 43)(38 40 49 47)(41 45 46 42)(53 65 68 56)(54 61 67 60)(55 57 66 64)(58 62 63 59)(70 82 85 73)(71 78 84 77)(72 74 83 81)(75 79 80 76)

G:=sub<Sym(85)| (1,69,52,35,18)(2,70,53,36,19)(3,71,54,37,20)(4,72,55,38,21)(5,73,56,39,22)(6,74,57,40,23)(7,75,58,41,24)(8,76,59,42,25)(9,77,60,43,26)(10,78,61,44,27)(11,79,62,45,28)(12,80,63,46,29)(13,81,64,47,30)(14,82,65,48,31)(15,83,66,49,32)(16,84,67,50,33)(17,85,68,51,34), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17)(18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34)(35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51)(52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68)(69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85), (2,14,17,5)(3,10,16,9)(4,6,15,13)(7,11,12,8)(19,31,34,22)(20,27,33,26)(21,23,32,30)(24,28,29,25)(36,48,51,39)(37,44,50,43)(38,40,49,47)(41,45,46,42)(53,65,68,56)(54,61,67,60)(55,57,66,64)(58,62,63,59)(70,82,85,73)(71,78,84,77)(72,74,83,81)(75,79,80,76)>;

G:=Group( (1,69,52,35,18)(2,70,53,36,19)(3,71,54,37,20)(4,72,55,38,21)(5,73,56,39,22)(6,74,57,40,23)(7,75,58,41,24)(8,76,59,42,25)(9,77,60,43,26)(10,78,61,44,27)(11,79,62,45,28)(12,80,63,46,29)(13,81,64,47,30)(14,82,65,48,31)(15,83,66,49,32)(16,84,67,50,33)(17,85,68,51,34), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17)(18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34)(35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51)(52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68)(69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85), (2,14,17,5)(3,10,16,9)(4,6,15,13)(7,11,12,8)(19,31,34,22)(20,27,33,26)(21,23,32,30)(24,28,29,25)(36,48,51,39)(37,44,50,43)(38,40,49,47)(41,45,46,42)(53,65,68,56)(54,61,67,60)(55,57,66,64)(58,62,63,59)(70,82,85,73)(71,78,84,77)(72,74,83,81)(75,79,80,76) );

G=PermutationGroup([(1,69,52,35,18),(2,70,53,36,19),(3,71,54,37,20),(4,72,55,38,21),(5,73,56,39,22),(6,74,57,40,23),(7,75,58,41,24),(8,76,59,42,25),(9,77,60,43,26),(10,78,61,44,27),(11,79,62,45,28),(12,80,63,46,29),(13,81,64,47,30),(14,82,65,48,31),(15,83,66,49,32),(16,84,67,50,33),(17,85,68,51,34)], [(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17),(18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34),(35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51),(52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68),(69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85)], [(2,14,17,5),(3,10,16,9),(4,6,15,13),(7,11,12,8),(19,31,34,22),(20,27,33,26),(21,23,32,30),(24,28,29,25),(36,48,51,39),(37,44,50,43),(38,40,49,47),(41,45,46,42),(53,65,68,56),(54,61,67,60),(55,57,66,64),(58,62,63,59),(70,82,85,73),(71,78,84,77),(72,74,83,81),(75,79,80,76)])

40 conjugacy classes

class 1  2 4A4B5A5B5C5D10A10B10C10D17A17B17C17D20A···20H85A···85P
order12445555101010101717171720···2085···85
size1171717111117171717444417···174···4

40 irreducible representations

dim11111144
type+++
imageC1C2C4C5C10C20C17⋊C4C5×C17⋊C4
kernelC5×C17⋊C4C5×D17C85C17⋊C4D17C17C5C1
# reps112448416

Matrix representation of C5×C17⋊C4 in GL4(𝔽1021) generated by

589000
058900
005890
000589
,
16489540251
10077
01023
001733
,
8285742795
584765587774
263501239996
378996462210
G:=sub<GL(4,GF(1021))| [589,0,0,0,0,589,0,0,0,0,589,0,0,0,0,589],[164,1,0,0,895,0,1,0,40,0,0,1,251,77,23,733],[828,584,263,378,5,765,501,996,742,587,239,462,795,774,996,210] >;

C5×C17⋊C4 in GAP, Magma, Sage, TeX

C_5\times C_{17}\rtimes C_4
% in TeX

G:=Group("C5xC17:C4");
// GroupNames label

G:=SmallGroup(340,5);
// by ID

G=gap.SmallGroup(340,5);
# by ID

G:=PCGroup([4,-2,-5,-2,-17,40,4163,523]);
// Polycyclic

G:=Group<a,b,c|a^5=b^17=c^4=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^4>;
// generators/relations

Export

Subgroup lattice of C5×C17⋊C4 in TeX

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