metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C6.8D20, C30.7D4, C15⋊4SD16, D60.3C2, C12.6D10, C20.24D6, Dic10⋊1S3, C60.10C22, C3⋊C8⋊3D5, C4.3(S3×D5), C3⋊2(C40⋊C2), C5⋊1(Q8⋊2S3), (C3×Dic10)⋊1C2, C10.3(C3⋊D4), C2.6(C3⋊D20), (C5×C3⋊C8)⋊3C2, SmallGroup(240,19)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C15⋊SD16
G = < a,b,c | a15=b8=c2=1, bab-1=a11, cac=a-1, cbc=b3 >
Smallest permutation representation of C15⋊SD16
►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 87 31 114 24 66 54 100)(2 83 32 110 25 62 55 96)(3 79 33 106 26 73 56 92)(4 90 34 117 27 69 57 103)(5 86 35 113 28 65 58 99)(6 82 36 109 29 61 59 95)(7 78 37 120 30 72 60 91)(8 89 38 116 16 68 46 102)(9 85 39 112 17 64 47 98)(10 81 40 108 18 75 48 94)(11 77 41 119 19 71 49 105)(12 88 42 115 20 67 50 101)(13 84 43 111 21 63 51 97)(14 80 44 107 22 74 52 93)(15 76 45 118 23 70 53 104)
(1 31)(2 45)(3 44)(4 43)(5 42)(6 41)(7 40)(8 39)(9 38)(10 37)(11 36)(12 35)(13 34)(14 33)(15 32)(16 47)(17 46)(18 60)(19 59)(20 58)(21 57)(22 56)(23 55)(24 54)(25 53)(26 52)(27 51)(28 50)(29 49)(30 48)(61 77)(62 76)(63 90)(64 89)(65 88)(66 87)(67 86)(68 85)(69 84)(70 83)(71 82)(72 81)(73 80)(74 79)(75 78)(91 94)(92 93)(95 105)(96 104)(97 103)(98 102)(99 101)(106 107)(108 120)(109 119)(110 118)(111 117)(112 116)(113 115)
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), (1,87,31,114,24,66,54,100)(2,83,32,110,25,62,55,96)(3,79,33,106,26,73,56,92)(4,90,34,117,27,69,57,103)(5,86,35,113,28,65,58,99)(6,82,36,109,29,61,59,95)(7,78,37,120,30,72,60,91)(8,89,38,116,16,68,46,102)(9,85,39,112,17,64,47,98)(10,81,40,108,18,75,48,94)(11,77,41,119,19,71,49,105)(12,88,42,115,20,67,50,101)(13,84,43,111,21,63,51,97)(14,80,44,107,22,74,52,93)(15,76,45,118,23,70,53,104), (1,31)(2,45)(3,44)(4,43)(5,42)(6,41)(7,40)(8,39)(9,38)(10,37)(11,36)(12,35)(13,34)(14,33)(15,32)(16,47)(17,46)(18,60)(19,59)(20,58)(21,57)(22,56)(23,55)(24,54)(25,53)(26,52)(27,51)(28,50)(29,49)(30,48)(61,77)(62,76)(63,90)(64,89)(65,88)(66,87)(67,86)(68,85)(69,84)(70,83)(71,82)(72,81)(73,80)(74,79)(75,78)(91,94)(92,93)(95,105)(96,104)(97,103)(98,102)(99,101)(106,107)(108,120)(109,119)(110,118)(111,117)(112,116)(113,115)>;
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), (1,87,31,114,24,66,54,100)(2,83,32,110,25,62,55,96)(3,79,33,106,26,73,56,92)(4,90,34,117,27,69,57,103)(5,86,35,113,28,65,58,99)(6,82,36,109,29,61,59,95)(7,78,37,120,30,72,60,91)(8,89,38,116,16,68,46,102)(9,85,39,112,17,64,47,98)(10,81,40,108,18,75,48,94)(11,77,41,119,19,71,49,105)(12,88,42,115,20,67,50,101)(13,84,43,111,21,63,51,97)(14,80,44,107,22,74,52,93)(15,76,45,118,23,70,53,104), (1,31)(2,45)(3,44)(4,43)(5,42)(6,41)(7,40)(8,39)(9,38)(10,37)(11,36)(12,35)(13,34)(14,33)(15,32)(16,47)(17,46)(18,60)(19,59)(20,58)(21,57)(22,56)(23,55)(24,54)(25,53)(26,52)(27,51)(28,50)(29,49)(30,48)(61,77)(62,76)(63,90)(64,89)(65,88)(66,87)(67,86)(68,85)(69,84)(70,83)(71,82)(72,81)(73,80)(74,79)(75,78)(91,94)(92,93)(95,105)(96,104)(97,103)(98,102)(99,101)(106,107)(108,120)(109,119)(110,118)(111,117)(112,116)(113,115) );
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)], [(1,87,31,114,24,66,54,100),(2,83,32,110,25,62,55,96),(3,79,33,106,26,73,56,92),(4,90,34,117,27,69,57,103),(5,86,35,113,28,65,58,99),(6,82,36,109,29,61,59,95),(7,78,37,120,30,72,60,91),(8,89,38,116,16,68,46,102),(9,85,39,112,17,64,47,98),(10,81,40,108,18,75,48,94),(11,77,41,119,19,71,49,105),(12,88,42,115,20,67,50,101),(13,84,43,111,21,63,51,97),(14,80,44,107,22,74,52,93),(15,76,45,118,23,70,53,104)], [(1,31),(2,45),(3,44),(4,43),(5,42),(6,41),(7,40),(8,39),(9,38),(10,37),(11,36),(12,35),(13,34),(14,33),(15,32),(16,47),(17,46),(18,60),(19,59),(20,58),(21,57),(22,56),(23,55),(24,54),(25,53),(26,52),(27,51),(28,50),(29,49),(30,48),(61,77),(62,76),(63,90),(64,89),(65,88),(66,87),(67,86),(68,85),(69,84),(70,83),(71,82),(72,81),(73,80),(74,79),(75,78),(91,94),(92,93),(95,105),(96,104),(97,103),(98,102),(99,101),(106,107),(108,120),(109,119),(110,118),(111,117),(112,116),(113,115)]])
C15⋊SD16 is a maximal subgroup of
S3×C40⋊C2 C40⋊1D6 Dic20⋊S3 D120⋊5C2 D20⋊19D6 D20.31D6 C12.D20 Dic10⋊3D6 Dic10⋊D6 C60.16C23 D12⋊5D10 D5×Q8⋊2S3 C60.C23 C60.39C23 D12.D10
C15⋊SD16 is a maximal quotient of
D60⋊12C4 C6.Dic20 C60.Q8
36 conjugacy classes
| class | 1 | 2A | 2B | 3 | 4A | 4B | 5A | 5B | 6 | 8A | 8B | 10A | 10B | 12A | 12B | 12C | 15A | 15B | 20A | 20B | 20C | 20D | 30A | 30B | 40A | ··· | 40H | 60A | 60B | 60C | 60D |
| order | 1 | 2 | 2 | 3 | 4 | 4 | 5 | 5 | 6 | 8 | 8 | 10 | 10 | 12 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 30 | 30 | 40 | ··· | 40 | 60 | 60 | 60 | 60 |
| size | 1 | 1 | 60 | 2 | 2 | 20 | 2 | 2 | 2 | 6 | 6 | 2 | 2 | 4 | 20 | 20 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 6 | ··· | 6 | 4 | 4 | 4 | 4 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | S3 | D4 | D5 | D6 | SD16 | D10 | C3⋊D4 | D20 | C40⋊C2 | Q8⋊2S3 | S3×D5 | C3⋊D20 | C15⋊SD16 |
| kernel | C15⋊SD16 | C5×C3⋊C8 | C3×Dic10 | D60 | Dic10 | C30 | C3⋊C8 | C20 | C15 | C12 | C10 | C6 | C3 | C5 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 2 | 2 | 2 | 4 | 8 | 1 | 2 | 2 | 4 |
Matrix representation of C15⋊SD16 ►in GL4(𝔽241) generated by
| 240 | 189 | 0 | 0 |
| 52 | 52 | 0 | 0 |
| 0 | 0 | 1 | 5 |
| 0 | 0 | 144 | 239 |
| 166 | 72 | 0 | 0 |
| 169 | 37 | 0 | 0 |
| 0 | 0 | 210 | 218 |
| 0 | 0 | 199 | 31 |
| 41 | 85 | 0 | 0 |
| 122 | 200 | 0 | 0 |
| 0 | 0 | 240 | 0 |
| 0 | 0 | 97 | 1 |
G:=sub<GL(4,GF(241))| [240,52,0,0,189,52,0,0,0,0,1,144,0,0,5,239],[166,169,0,0,72,37,0,0,0,0,210,199,0,0,218,31],[41,122,0,0,85,200,0,0,0,0,240,97,0,0,0,1] >;
C15⋊SD16 in GAP, Magma, Sage, TeX
C_{15}\rtimes {\rm SD}_{16} % in TeX
G:=Group("C15:SD16"); // GroupNames label
G:=SmallGroup(240,19);
// by ID
G=gap.SmallGroup(240,19);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-5,48,73,31,218,50,490,6917]);
// Polycyclic
G:=Group<a,b,c|a^15=b^8=c^2=1,b*a*b^-1=a^11,c*a*c=a^-1,c*b*c=b^3>;
// generators/relations
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