Copied to
clipboard

G = C15×C7⋊C3order 315 = 32·5·7

Direct product of C15 and C7⋊C3

direct product, metacyclic, supersoluble, monomial, A-group, 3-hyperelementary

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

Series: Derived Chief Lower central Upper central

C1C7 — C15×C7⋊C3
C1C7C35C5×C7⋊C3 — C15×C7⋊C3
C7 — C15×C7⋊C3
C1C15

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

7C3
7C3
7C3
7C32
7C15
7C15
7C15
7C3×C15

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 3A3B3C···3H5A5B5C5D7A7B15A···15H15I···15AF21A21B21C21D35A···35H105A···105P
order1333···355557715···1515···152121212135···35105···105
size1117···71111331···17···733333···33···3

75 irreducible representations

dim1111113333
type+
imageC1C3C3C5C15C15C7⋊C3C3×C7⋊C3C5×C7⋊C3C15×C7⋊C3
kernelC15×C7⋊C3C5×C7⋊C3C105C3×C7⋊C3C7⋊C3C21C15C5C3C1
# reps162424824816

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

14000
018800
001880
000188
,
1000
0210201
00201
0210211
,
14000
0211191
0100
011190
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

Export

Subgroup lattice of C15×C7⋊C3 in TeX

׿
×
𝔽