direct product, metacyclic, nilpotent (class 3), monomial, 2-elementary
Aliases: C15×D8, D4⋊C30, C8⋊1C30, C40⋊5C6, C24⋊3C10, C120⋊11C2, 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
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 43 59 69 94 25 81 119)(2 44 60 70 95 26 82 120)(3 45 46 71 96 27 83 106)(4 31 47 72 97 28 84 107)(5 32 48 73 98 29 85 108)(6 33 49 74 99 30 86 109)(7 34 50 75 100 16 87 110)(8 35 51 61 101 17 88 111)(9 36 52 62 102 18 89 112)(10 37 53 63 103 19 90 113)(11 38 54 64 104 20 76 114)(12 39 55 65 105 21 77 115)(13 40 56 66 91 22 78 116)(14 41 57 67 92 23 79 117)(15 42 58 68 93 24 80 118)
(1 119)(2 120)(3 106)(4 107)(5 108)(6 109)(7 110)(8 111)(9 112)(10 113)(11 114)(12 115)(13 116)(14 117)(15 118)(16 50)(17 51)(18 52)(19 53)(20 54)(21 55)(22 56)(23 57)(24 58)(25 59)(26 60)(27 46)(28 47)(29 48)(30 49)(31 84)(32 85)(33 86)(34 87)(35 88)(36 89)(37 90)(38 76)(39 77)(40 78)(41 79)(42 80)(43 81)(44 82)(45 83)(61 101)(62 102)(63 103)(64 104)(65 105)(66 91)(67 92)(68 93)(69 94)(70 95)(71 96)(72 97)(73 98)(74 99)(75 100)
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), (1,43,59,69,94,25,81,119)(2,44,60,70,95,26,82,120)(3,45,46,71,96,27,83,106)(4,31,47,72,97,28,84,107)(5,32,48,73,98,29,85,108)(6,33,49,74,99,30,86,109)(7,34,50,75,100,16,87,110)(8,35,51,61,101,17,88,111)(9,36,52,62,102,18,89,112)(10,37,53,63,103,19,90,113)(11,38,54,64,104,20,76,114)(12,39,55,65,105,21,77,115)(13,40,56,66,91,22,78,116)(14,41,57,67,92,23,79,117)(15,42,58,68,93,24,80,118), (1,119)(2,120)(3,106)(4,107)(5,108)(6,109)(7,110)(8,111)(9,112)(10,113)(11,114)(12,115)(13,116)(14,117)(15,118)(16,50)(17,51)(18,52)(19,53)(20,54)(21,55)(22,56)(23,57)(24,58)(25,59)(26,60)(27,46)(28,47)(29,48)(30,49)(31,84)(32,85)(33,86)(34,87)(35,88)(36,89)(37,90)(38,76)(39,77)(40,78)(41,79)(42,80)(43,81)(44,82)(45,83)(61,101)(62,102)(63,103)(64,104)(65,105)(66,91)(67,92)(68,93)(69,94)(70,95)(71,96)(72,97)(73,98)(74,99)(75,100)>;
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), (1,43,59,69,94,25,81,119)(2,44,60,70,95,26,82,120)(3,45,46,71,96,27,83,106)(4,31,47,72,97,28,84,107)(5,32,48,73,98,29,85,108)(6,33,49,74,99,30,86,109)(7,34,50,75,100,16,87,110)(8,35,51,61,101,17,88,111)(9,36,52,62,102,18,89,112)(10,37,53,63,103,19,90,113)(11,38,54,64,104,20,76,114)(12,39,55,65,105,21,77,115)(13,40,56,66,91,22,78,116)(14,41,57,67,92,23,79,117)(15,42,58,68,93,24,80,118), (1,119)(2,120)(3,106)(4,107)(5,108)(6,109)(7,110)(8,111)(9,112)(10,113)(11,114)(12,115)(13,116)(14,117)(15,118)(16,50)(17,51)(18,52)(19,53)(20,54)(21,55)(22,56)(23,57)(24,58)(25,59)(26,60)(27,46)(28,47)(29,48)(30,49)(31,84)(32,85)(33,86)(34,87)(35,88)(36,89)(37,90)(38,76)(39,77)(40,78)(41,79)(42,80)(43,81)(44,82)(45,83)(61,101)(62,102)(63,103)(64,104)(65,105)(66,91)(67,92)(68,93)(69,94)(70,95)(71,96)(72,97)(73,98)(74,99)(75,100) );
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)], [(1,43,59,69,94,25,81,119),(2,44,60,70,95,26,82,120),(3,45,46,71,96,27,83,106),(4,31,47,72,97,28,84,107),(5,32,48,73,98,29,85,108),(6,33,49,74,99,30,86,109),(7,34,50,75,100,16,87,110),(8,35,51,61,101,17,88,111),(9,36,52,62,102,18,89,112),(10,37,53,63,103,19,90,113),(11,38,54,64,104,20,76,114),(12,39,55,65,105,21,77,115),(13,40,56,66,91,22,78,116),(14,41,57,67,92,23,79,117),(15,42,58,68,93,24,80,118)], [(1,119),(2,120),(3,106),(4,107),(5,108),(6,109),(7,110),(8,111),(9,112),(10,113),(11,114),(12,115),(13,116),(14,117),(15,118),(16,50),(17,51),(18,52),(19,53),(20,54),(21,55),(22,56),(23,57),(24,58),(25,59),(26,60),(27,46),(28,47),(29,48),(30,49),(31,84),(32,85),(33,86),(34,87),(35,88),(36,89),(37,90),(38,76),(39,77),(40,78),(41,79),(42,80),(43,81),(44,82),(45,83),(61,101),(62,102),(63,103),(64,104),(65,105),(66,91),(67,92),(68,93),(69,94),(70,95),(71,96),(72,97),(73,98),(74,99),(75,100)]])
C15×D8 is a maximal subgroup of
C15⋊7D16 D8.D15 D8⋊D15 D8⋊3D15
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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