Copied to
clipboard

G = C15×D8order 240 = 24·3·5

Direct product of C15 and D8

direct product, metacyclic, nilpotent (class 3), monomial, 2-elementary

Aliases: C15×D8, D4⋊C30, C81C30, C405C6, C243C10, C12011C2, C30.54D4, C60.77C22, (C5×D4)⋊4C6, (C3×D4)⋊4C10, C4.1(C2×C30), C6.14(C5×D4), C2.3(D4×C15), (D4×C15)⋊10C2, C20.17(C2×C6), C10.14(C3×D4), C12.17(C2×C10), SmallGroup(240,86)

Series: Derived Chief Lower central Upper central

C1C4 — C15×D8
C1C2C4C20C60D4×C15 — C15×D8
C1C2C4 — C15×D8
C1C30C60 — C15×D8

Generators and relations for C15×D8
 G = < a,b,c | a15=b8=c2=1, ab=ba, ac=ca, cbc=b-1 >

4C2
4C2
2C22
2C22
4C6
4C6
4C10
4C10
2C2×C6
2C2×C6
2C2×C10
2C2×C10
4C30
4C30
2C2×C30
2C2×C30

Smallest permutation representation of C15×D8
On 120 points
Generators in S120
(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)(106 107 108 109 110 111 112 113 114 115 116 117 118 119 120)
(1 85 92 29 59 38 108 65)(2 86 93 30 60 39 109 66)(3 87 94 16 46 40 110 67)(4 88 95 17 47 41 111 68)(5 89 96 18 48 42 112 69)(6 90 97 19 49 43 113 70)(7 76 98 20 50 44 114 71)(8 77 99 21 51 45 115 72)(9 78 100 22 52 31 116 73)(10 79 101 23 53 32 117 74)(11 80 102 24 54 33 118 75)(12 81 103 25 55 34 119 61)(13 82 104 26 56 35 120 62)(14 83 105 27 57 36 106 63)(15 84 91 28 58 37 107 64)
(1 65)(2 66)(3 67)(4 68)(5 69)(6 70)(7 71)(8 72)(9 73)(10 74)(11 75)(12 61)(13 62)(14 63)(15 64)(16 46)(17 47)(18 48)(19 49)(20 50)(21 51)(22 52)(23 53)(24 54)(25 55)(26 56)(27 57)(28 58)(29 59)(30 60)(31 100)(32 101)(33 102)(34 103)(35 104)(36 105)(37 91)(38 92)(39 93)(40 94)(41 95)(42 96)(43 97)(44 98)(45 99)(76 114)(77 115)(78 116)(79 117)(80 118)(81 119)(82 120)(83 106)(84 107)(85 108)(86 109)(87 110)(88 111)(89 112)(90 113)

G:=sub<Sym(120)| (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)(106,107,108,109,110,111,112,113,114,115,116,117,118,119,120), (1,85,92,29,59,38,108,65)(2,86,93,30,60,39,109,66)(3,87,94,16,46,40,110,67)(4,88,95,17,47,41,111,68)(5,89,96,18,48,42,112,69)(6,90,97,19,49,43,113,70)(7,76,98,20,50,44,114,71)(8,77,99,21,51,45,115,72)(9,78,100,22,52,31,116,73)(10,79,101,23,53,32,117,74)(11,80,102,24,54,33,118,75)(12,81,103,25,55,34,119,61)(13,82,104,26,56,35,120,62)(14,83,105,27,57,36,106,63)(15,84,91,28,58,37,107,64), (1,65)(2,66)(3,67)(4,68)(5,69)(6,70)(7,71)(8,72)(9,73)(10,74)(11,75)(12,61)(13,62)(14,63)(15,64)(16,46)(17,47)(18,48)(19,49)(20,50)(21,51)(22,52)(23,53)(24,54)(25,55)(26,56)(27,57)(28,58)(29,59)(30,60)(31,100)(32,101)(33,102)(34,103)(35,104)(36,105)(37,91)(38,92)(39,93)(40,94)(41,95)(42,96)(43,97)(44,98)(45,99)(76,114)(77,115)(78,116)(79,117)(80,118)(81,119)(82,120)(83,106)(84,107)(85,108)(86,109)(87,110)(88,111)(89,112)(90,113)>;

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)(106,107,108,109,110,111,112,113,114,115,116,117,118,119,120), (1,85,92,29,59,38,108,65)(2,86,93,30,60,39,109,66)(3,87,94,16,46,40,110,67)(4,88,95,17,47,41,111,68)(5,89,96,18,48,42,112,69)(6,90,97,19,49,43,113,70)(7,76,98,20,50,44,114,71)(8,77,99,21,51,45,115,72)(9,78,100,22,52,31,116,73)(10,79,101,23,53,32,117,74)(11,80,102,24,54,33,118,75)(12,81,103,25,55,34,119,61)(13,82,104,26,56,35,120,62)(14,83,105,27,57,36,106,63)(15,84,91,28,58,37,107,64), (1,65)(2,66)(3,67)(4,68)(5,69)(6,70)(7,71)(8,72)(9,73)(10,74)(11,75)(12,61)(13,62)(14,63)(15,64)(16,46)(17,47)(18,48)(19,49)(20,50)(21,51)(22,52)(23,53)(24,54)(25,55)(26,56)(27,57)(28,58)(29,59)(30,60)(31,100)(32,101)(33,102)(34,103)(35,104)(36,105)(37,91)(38,92)(39,93)(40,94)(41,95)(42,96)(43,97)(44,98)(45,99)(76,114)(77,115)(78,116)(79,117)(80,118)(81,119)(82,120)(83,106)(84,107)(85,108)(86,109)(87,110)(88,111)(89,112)(90,113) );

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),(106,107,108,109,110,111,112,113,114,115,116,117,118,119,120)], [(1,85,92,29,59,38,108,65),(2,86,93,30,60,39,109,66),(3,87,94,16,46,40,110,67),(4,88,95,17,47,41,111,68),(5,89,96,18,48,42,112,69),(6,90,97,19,49,43,113,70),(7,76,98,20,50,44,114,71),(8,77,99,21,51,45,115,72),(9,78,100,22,52,31,116,73),(10,79,101,23,53,32,117,74),(11,80,102,24,54,33,118,75),(12,81,103,25,55,34,119,61),(13,82,104,26,56,35,120,62),(14,83,105,27,57,36,106,63),(15,84,91,28,58,37,107,64)], [(1,65),(2,66),(3,67),(4,68),(5,69),(6,70),(7,71),(8,72),(9,73),(10,74),(11,75),(12,61),(13,62),(14,63),(15,64),(16,46),(17,47),(18,48),(19,49),(20,50),(21,51),(22,52),(23,53),(24,54),(25,55),(26,56),(27,57),(28,58),(29,59),(30,60),(31,100),(32,101),(33,102),(34,103),(35,104),(36,105),(37,91),(38,92),(39,93),(40,94),(41,95),(42,96),(43,97),(44,98),(45,99),(76,114),(77,115),(78,116),(79,117),(80,118),(81,119),(82,120),(83,106),(84,107),(85,108),(86,109),(87,110),(88,111),(89,112),(90,113)])

C15×D8 is a maximal subgroup of   C157D16  D8.D15  D8⋊D15  D83D15

105 conjugacy classes

class 1 2A2B2C3A3B 4 5A5B5C5D6A6B6C6D6E6F8A8B10A10B10C10D10E···10L12A12B15A···15H20A20B20C20D24A24B24C24D30A···30H30I···30X40A···40H60A···60H120A···120P
order12223345555666666881010101010···10121215···15202020202424242430···3030···3040···4060···60120···120
size114411211111144442211114···4221···1222222221···14···42···22···22···2

105 irreducible representations

dim11111111111122222222
type+++++
imageC1C2C2C3C5C6C6C10C10C15C30C30D4D8C3×D4C5×D4C3×D8C5×D8D4×C15C15×D8
kernelC15×D8C120D4×C15C5×D8C3×D8C40C5×D4C24C3×D4D8C8D4C30C15C10C6C5C3C2C1
# reps1122424488816122448816

Matrix representation of C15×D8 in GL2(𝔽31) generated by

180
018
,
88
270
,
08
40
G:=sub<GL(2,GF(31))| [18,0,0,18],[8,27,8,0],[0,4,8,0] >;

C15×D8 in GAP, Magma, Sage, TeX

C_{15}\times D_8
% in TeX

G:=Group("C15xD8");
// GroupNames label

G:=SmallGroup(240,86);
// by ID

G=gap.SmallGroup(240,86);
# by ID

G:=PCGroup([6,-2,-2,-3,-5,-2,-2,745,5404,2710,88]);
// Polycyclic

G:=Group<a,b,c|a^15=b^8=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C15×D8 in TeX

׿
×
𝔽