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## 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

 Derived series C1 — C4 — C15×D8
 Chief series C1 — C2 — C4 — C20 — C60 — D4×C15 — C15×D8
 Lower central C1 — C2 — C4 — C15×D8
 Upper central C1 — C30 — C60 — C15×D8

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

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 2A 2B 2C 3A 3B 4 5A 5B 5C 5D 6A 6B 6C 6D 6E 6F 8A 8B 10A 10B 10C 10D 10E ··· 10L 12A 12B 15A ··· 15H 20A 20B 20C 20D 24A 24B 24C 24D 30A ··· 30H 30I ··· 30X 40A ··· 40H 60A ··· 60H 120A ··· 120P order 1 2 2 2 3 3 4 5 5 5 5 6 6 6 6 6 6 8 8 10 10 10 10 10 ··· 10 12 12 15 ··· 15 20 20 20 20 24 24 24 24 30 ··· 30 30 ··· 30 40 ··· 40 60 ··· 60 120 ··· 120 size 1 1 4 4 1 1 2 1 1 1 1 1 1 4 4 4 4 2 2 1 1 1 1 4 ··· 4 2 2 1 ··· 1 2 2 2 2 2 2 2 2 1 ··· 1 4 ··· 4 2 ··· 2 2 ··· 2 2 ··· 2

105 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + + + image C1 C2 C2 C3 C5 C6 C6 C10 C10 C15 C30 C30 D4 D8 C3×D4 C5×D4 C3×D8 C5×D8 D4×C15 C15×D8 kernel C15×D8 C120 D4×C15 C5×D8 C3×D8 C40 C5×D4 C24 C3×D4 D8 C8 D4 C30 C15 C10 C6 C5 C3 C2 C1 # reps 1 1 2 2 4 2 4 4 8 8 8 16 1 2 2 4 4 8 8 16

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

 18 0 0 18
,
 8 8 27 0
,
 0 8 4 0
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

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