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## G = C15×C7⋊C3order 315 = 32·5·7

### Direct product of C15 and C7⋊C3

Aliases: C15×C7⋊C3, C105⋊C3, C21⋊C15, C35⋊C32, C7⋊(C3×C15), SmallGroup(315,3)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C7 — C15×C7⋊C3
 Chief series C1 — C7 — C35 — C5×C7⋊C3 — C15×C7⋊C3
 Lower central C7 — C15×C7⋊C3
 Upper central C1 — C15

Generators and relations for C15×C7⋊C3
G = < a,b,c | a15=b7=c3=1, ab=ba, ac=ca, cbc-1=b4 >

Smallest permutation representation of C15×C7⋊C3
On 105 points
Generators in S105
(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 86 87 88 89 90)(91 92 93 94 95 96 97 98 99 100 101 102 103 104 105)
(1 59 19 36 93 68 77)(2 60 20 37 94 69 78)(3 46 21 38 95 70 79)(4 47 22 39 96 71 80)(5 48 23 40 97 72 81)(6 49 24 41 98 73 82)(7 50 25 42 99 74 83)(8 51 26 43 100 75 84)(9 52 27 44 101 61 85)(10 53 28 45 102 62 86)(11 54 29 31 103 63 87)(12 55 30 32 104 64 88)(13 56 16 33 105 65 89)(14 57 17 34 91 66 90)(15 58 18 35 92 67 76)
(16 105 56)(17 91 57)(18 92 58)(19 93 59)(20 94 60)(21 95 46)(22 96 47)(23 97 48)(24 98 49)(25 99 50)(26 100 51)(27 101 52)(28 102 53)(29 103 54)(30 104 55)(31 87 63)(32 88 64)(33 89 65)(34 90 66)(35 76 67)(36 77 68)(37 78 69)(38 79 70)(39 80 71)(40 81 72)(41 82 73)(42 83 74)(43 84 75)(44 85 61)(45 86 62)

G:=sub<Sym(105)| (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)(76,77,78,79,80,81,82,83,84,85,86,87,88,89,90)(91,92,93,94,95,96,97,98,99,100,101,102,103,104,105), (1,59,19,36,93,68,77)(2,60,20,37,94,69,78)(3,46,21,38,95,70,79)(4,47,22,39,96,71,80)(5,48,23,40,97,72,81)(6,49,24,41,98,73,82)(7,50,25,42,99,74,83)(8,51,26,43,100,75,84)(9,52,27,44,101,61,85)(10,53,28,45,102,62,86)(11,54,29,31,103,63,87)(12,55,30,32,104,64,88)(13,56,16,33,105,65,89)(14,57,17,34,91,66,90)(15,58,18,35,92,67,76), (16,105,56)(17,91,57)(18,92,58)(19,93,59)(20,94,60)(21,95,46)(22,96,47)(23,97,48)(24,98,49)(25,99,50)(26,100,51)(27,101,52)(28,102,53)(29,103,54)(30,104,55)(31,87,63)(32,88,64)(33,89,65)(34,90,66)(35,76,67)(36,77,68)(37,78,69)(38,79,70)(39,80,71)(40,81,72)(41,82,73)(42,83,74)(43,84,75)(44,85,61)(45,86,62)>;

G:=Group( (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)(76,77,78,79,80,81,82,83,84,85,86,87,88,89,90)(91,92,93,94,95,96,97,98,99,100,101,102,103,104,105), (1,59,19,36,93,68,77)(2,60,20,37,94,69,78)(3,46,21,38,95,70,79)(4,47,22,39,96,71,80)(5,48,23,40,97,72,81)(6,49,24,41,98,73,82)(7,50,25,42,99,74,83)(8,51,26,43,100,75,84)(9,52,27,44,101,61,85)(10,53,28,45,102,62,86)(11,54,29,31,103,63,87)(12,55,30,32,104,64,88)(13,56,16,33,105,65,89)(14,57,17,34,91,66,90)(15,58,18,35,92,67,76), (16,105,56)(17,91,57)(18,92,58)(19,93,59)(20,94,60)(21,95,46)(22,96,47)(23,97,48)(24,98,49)(25,99,50)(26,100,51)(27,101,52)(28,102,53)(29,103,54)(30,104,55)(31,87,63)(32,88,64)(33,89,65)(34,90,66)(35,76,67)(36,77,68)(37,78,69)(38,79,70)(39,80,71)(40,81,72)(41,82,73)(42,83,74)(43,84,75)(44,85,61)(45,86,62) );

G=PermutationGroup([(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),(76,77,78,79,80,81,82,83,84,85,86,87,88,89,90),(91,92,93,94,95,96,97,98,99,100,101,102,103,104,105)], [(1,59,19,36,93,68,77),(2,60,20,37,94,69,78),(3,46,21,38,95,70,79),(4,47,22,39,96,71,80),(5,48,23,40,97,72,81),(6,49,24,41,98,73,82),(7,50,25,42,99,74,83),(8,51,26,43,100,75,84),(9,52,27,44,101,61,85),(10,53,28,45,102,62,86),(11,54,29,31,103,63,87),(12,55,30,32,104,64,88),(13,56,16,33,105,65,89),(14,57,17,34,91,66,90),(15,58,18,35,92,67,76)], [(16,105,56),(17,91,57),(18,92,58),(19,93,59),(20,94,60),(21,95,46),(22,96,47),(23,97,48),(24,98,49),(25,99,50),(26,100,51),(27,101,52),(28,102,53),(29,103,54),(30,104,55),(31,87,63),(32,88,64),(33,89,65),(34,90,66),(35,76,67),(36,77,68),(37,78,69),(38,79,70),(39,80,71),(40,81,72),(41,82,73),(42,83,74),(43,84,75),(44,85,61),(45,86,62)])

75 conjugacy classes

 class 1 3A 3B 3C ··· 3H 5A 5B 5C 5D 7A 7B 15A ··· 15H 15I ··· 15AF 21A 21B 21C 21D 35A ··· 35H 105A ··· 105P order 1 3 3 3 ··· 3 5 5 5 5 7 7 15 ··· 15 15 ··· 15 21 21 21 21 35 ··· 35 105 ··· 105 size 1 1 1 7 ··· 7 1 1 1 1 3 3 1 ··· 1 7 ··· 7 3 3 3 3 3 ··· 3 3 ··· 3

75 irreducible representations

 dim 1 1 1 1 1 1 3 3 3 3 type + image C1 C3 C3 C5 C15 C15 C7⋊C3 C3×C7⋊C3 C5×C7⋊C3 C15×C7⋊C3 kernel C15×C7⋊C3 C5×C7⋊C3 C105 C3×C7⋊C3 C7⋊C3 C21 C15 C5 C3 C1 # reps 1 6 2 4 24 8 2 4 8 16

Matrix representation of C15×C7⋊C3 in GL4(𝔽211) generated by

 14 0 0 0 0 188 0 0 0 0 188 0 0 0 0 188
,
 1 0 0 0 0 210 20 1 0 0 20 1 0 210 21 1
,
 14 0 0 0 0 21 1 191 0 1 0 0 0 1 1 190
G:=sub<GL(4,GF(211))| [14,0,0,0,0,188,0,0,0,0,188,0,0,0,0,188],[1,0,0,0,0,210,0,210,0,20,20,21,0,1,1,1],[14,0,0,0,0,21,1,1,0,1,0,1,0,191,0,190] >;

C15×C7⋊C3 in GAP, Magma, Sage, TeX

C_{15}\times C_7\rtimes C_3
% in TeX

G:=Group("C15xC7:C3");
// GroupNames label

G:=SmallGroup(315,3);
// by ID

G=gap.SmallGroup(315,3);
# by ID

G:=PCGroup([4,-3,-3,-5,-7,1443]);
// Polycyclic

G:=Group<a,b,c|a^15=b^7=c^3=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^4>;
// generators/relations

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