direct product, metacyclic, nilpotent (class 3), monomial, 2-elementary
Aliases: C15×SD16, C8⋊2C30, C40⋊6C6, D4.C30, C24⋊6C10, Q8⋊2C30, C120⋊14C2, C30.55D4, C60.78C22, (C5×Q8)⋊6C6, C4.2(C2×C30), (C3×Q8)⋊4C10, (C5×D4).2C6, C2.4(D4×C15), C6.15(C5×D4), C20.18(C2×C6), (Q8×C15)⋊10C2, (D4×C15).4C2, (C3×D4).2C10, C10.15(C3×D4), C12.18(C2×C10), SmallGroup(240,87)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C15×SD16
G = < a,b,c | a15=b8=c2=1, ab=ba, ac=ca, 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 99 50 79 33 27 108 71)(2 100 51 80 34 28 109 72)(3 101 52 81 35 29 110 73)(4 102 53 82 36 30 111 74)(5 103 54 83 37 16 112 75)(6 104 55 84 38 17 113 61)(7 105 56 85 39 18 114 62)(8 91 57 86 40 19 115 63)(9 92 58 87 41 20 116 64)(10 93 59 88 42 21 117 65)(11 94 60 89 43 22 118 66)(12 95 46 90 44 23 119 67)(13 96 47 76 45 24 120 68)(14 97 48 77 31 25 106 69)(15 98 49 78 32 26 107 70)
(16 75)(17 61)(18 62)(19 63)(20 64)(21 65)(22 66)(23 67)(24 68)(25 69)(26 70)(27 71)(28 72)(29 73)(30 74)(46 119)(47 120)(48 106)(49 107)(50 108)(51 109)(52 110)(53 111)(54 112)(55 113)(56 114)(57 115)(58 116)(59 117)(60 118)(76 96)(77 97)(78 98)(79 99)(80 100)(81 101)(82 102)(83 103)(84 104)(85 105)(86 91)(87 92)(88 93)(89 94)(90 95)
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), (1,99,50,79,33,27,108,71)(2,100,51,80,34,28,109,72)(3,101,52,81,35,29,110,73)(4,102,53,82,36,30,111,74)(5,103,54,83,37,16,112,75)(6,104,55,84,38,17,113,61)(7,105,56,85,39,18,114,62)(8,91,57,86,40,19,115,63)(9,92,58,87,41,20,116,64)(10,93,59,88,42,21,117,65)(11,94,60,89,43,22,118,66)(12,95,46,90,44,23,119,67)(13,96,47,76,45,24,120,68)(14,97,48,77,31,25,106,69)(15,98,49,78,32,26,107,70), (16,75)(17,61)(18,62)(19,63)(20,64)(21,65)(22,66)(23,67)(24,68)(25,69)(26,70)(27,71)(28,72)(29,73)(30,74)(46,119)(47,120)(48,106)(49,107)(50,108)(51,109)(52,110)(53,111)(54,112)(55,113)(56,114)(57,115)(58,116)(59,117)(60,118)(76,96)(77,97)(78,98)(79,99)(80,100)(81,101)(82,102)(83,103)(84,104)(85,105)(86,91)(87,92)(88,93)(89,94)(90,95)>;
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), (1,99,50,79,33,27,108,71)(2,100,51,80,34,28,109,72)(3,101,52,81,35,29,110,73)(4,102,53,82,36,30,111,74)(5,103,54,83,37,16,112,75)(6,104,55,84,38,17,113,61)(7,105,56,85,39,18,114,62)(8,91,57,86,40,19,115,63)(9,92,58,87,41,20,116,64)(10,93,59,88,42,21,117,65)(11,94,60,89,43,22,118,66)(12,95,46,90,44,23,119,67)(13,96,47,76,45,24,120,68)(14,97,48,77,31,25,106,69)(15,98,49,78,32,26,107,70), (16,75)(17,61)(18,62)(19,63)(20,64)(21,65)(22,66)(23,67)(24,68)(25,69)(26,70)(27,71)(28,72)(29,73)(30,74)(46,119)(47,120)(48,106)(49,107)(50,108)(51,109)(52,110)(53,111)(54,112)(55,113)(56,114)(57,115)(58,116)(59,117)(60,118)(76,96)(77,97)(78,98)(79,99)(80,100)(81,101)(82,102)(83,103)(84,104)(85,105)(86,91)(87,92)(88,93)(89,94)(90,95) );
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)], [(1,99,50,79,33,27,108,71),(2,100,51,80,34,28,109,72),(3,101,52,81,35,29,110,73),(4,102,53,82,36,30,111,74),(5,103,54,83,37,16,112,75),(6,104,55,84,38,17,113,61),(7,105,56,85,39,18,114,62),(8,91,57,86,40,19,115,63),(9,92,58,87,41,20,116,64),(10,93,59,88,42,21,117,65),(11,94,60,89,43,22,118,66),(12,95,46,90,44,23,119,67),(13,96,47,76,45,24,120,68),(14,97,48,77,31,25,106,69),(15,98,49,78,32,26,107,70)], [(16,75),(17,61),(18,62),(19,63),(20,64),(21,65),(22,66),(23,67),(24,68),(25,69),(26,70),(27,71),(28,72),(29,73),(30,74),(46,119),(47,120),(48,106),(49,107),(50,108),(51,109),(52,110),(53,111),(54,112),(55,113),(56,114),(57,115),(58,116),(59,117),(60,118),(76,96),(77,97),(78,98),(79,99),(80,100),(81,101),(82,102),(83,103),(84,104),(85,105),(86,91),(87,92),(88,93),(89,94),(90,95)]])
C15×SD16 is a maximal subgroup of
Q8⋊3D30 SD16⋊D15 D4.5D30
105 conjugacy classes
| class | 1 | 2A | 2B | 3A | 3B | 4A | 4B | 5A | 5B | 5C | 5D | 6A | 6B | 6C | 6D | 8A | 8B | 10A | 10B | 10C | 10D | 10E | 10F | 10G | 10H | 12A | 12B | 12C | 12D | 15A | ··· | 15H | 20A | 20B | 20C | 20D | 20E | 20F | 20G | 20H | 24A | 24B | 24C | 24D | 30A | ··· | 30H | 30I | ··· | 30P | 40A | ··· | 40H | 60A | ··· | 60H | 60I | ··· | 60P | 120A | ··· | 120P |
| order | 1 | 2 | 2 | 3 | 3 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | 6 | 6 | 6 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 12 | 12 | 12 | 12 | 15 | ··· | 15 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 24 | 24 | 24 | 24 | 30 | ··· | 30 | 30 | ··· | 30 | 40 | ··· | 40 | 60 | ··· | 60 | 60 | ··· | 60 | 120 | ··· | 120 |
| size | 1 | 1 | 4 | 1 | 1 | 2 | 4 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 4 | 2 | 2 | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 2 | 2 | 4 | 4 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 4 | ··· | 4 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 |
105 irreducible representations
| dim | 1 | 1 | 1 | 1 | 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 | C2 | C3 | C5 | C6 | C6 | C6 | C10 | C10 | C10 | C15 | C30 | C30 | C30 | D4 | SD16 | C3×D4 | C5×D4 | C3×SD16 | C5×SD16 | D4×C15 | C15×SD16 |
| kernel | C15×SD16 | C120 | D4×C15 | Q8×C15 | C5×SD16 | C3×SD16 | C40 | C5×D4 | C5×Q8 | C24 | C3×D4 | C3×Q8 | SD16 | C8 | D4 | Q8 | C30 | C15 | C10 | C6 | C5 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 4 | 2 | 2 | 2 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 1 | 2 | 2 | 4 | 4 | 8 | 8 | 16 |
Matrix representation of C15×SD16 ►in GL3(𝔽241) generated by
| 225 | 0 | 0 |
| 0 | 98 | 0 |
| 0 | 0 | 98 |
| 240 | 0 | 0 |
| 0 | 0 | 203 |
| 0 | 19 | 203 |
| 240 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 1 | 240 |
G:=sub<GL(3,GF(241))| [225,0,0,0,98,0,0,0,98],[240,0,0,0,0,19,0,203,203],[240,0,0,0,1,1,0,0,240] >;
C15×SD16 in GAP, Magma, Sage, TeX
C_{15}\times {\rm SD}_{16} % in TeX
G:=Group("C15xSD16"); // GroupNames label
G:=SmallGroup(240,87);
// by ID
G=gap.SmallGroup(240,87);
# by ID
G:=PCGroup([6,-2,-2,-3,-5,-2,-2,720,745,5404,2710,88]);
// 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^3>;
// generators/relations
Export