direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: SD16×D15, C8⋊5D30, C40⋊15D6, Q8⋊2D30, C24⋊15D10, D4.2D30, D30.49D4, C120⋊10C22, C60.67C23, Dic15.23D4, D60.21C22, Dic30⋊13C22, (C5×Q8)⋊8D6, C3⋊5(D5×SD16), C5⋊5(S3×SD16), (C8×D15)⋊4C2, (C3×Q8)⋊5D10, (Q8×D15)⋊8C2, C24⋊D5⋊7C2, (C5×SD16)⋊3S3, (C3×SD16)⋊3D5, (C5×D4).14D6, (D4×D15).2C2, C2.18(D4×D15), C6.111(D4×D5), C15⋊19(C2×SD16), D4.D15⋊11C2, Q8⋊2D15⋊9C2, (C15×SD16)⋊3C2, (C3×D4).14D10, C30.318(C2×D4), C10.113(S3×D4), C4.4(C22×D15), C15⋊3C8⋊28C22, (Q8×C15)⋊12C22, C20.105(C22×S3), (D4×C15).21C22, (C4×D15).44C22, C12.105(C22×D5), SmallGroup(480,878)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for SD16×D15
G = < a,b,c,d | a8=b2=c15=d2=1, bab=a3, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Subgroups: 996 in 136 conjugacy classes, 43 normal (41 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, S3, C6, C6, C8, C8, C2×C4, D4, D4, Q8, Q8, C23, D5, C10, C10, Dic3, C12, C12, D6, C2×C6, C15, C2×C8, SD16, SD16, C2×D4, C2×Q8, Dic5, C20, C20, D10, C2×C10, C3⋊C8, C24, Dic6, C4×S3, D12, C3⋊D4, C3×D4, C3×Q8, C22×S3, D15, D15, C30, C30, C2×SD16, C5⋊2C8, C40, Dic10, C4×D5, D20, C5⋊D4, C5×D4, C5×Q8, C22×D5, S3×C8, C24⋊C2, D4.S3, Q8⋊2S3, C3×SD16, S3×D4, S3×Q8, Dic15, Dic15, C60, C60, D30, D30, C2×C30, C8×D5, C40⋊C2, D4.D5, Q8⋊D5, C5×SD16, D4×D5, Q8×D5, S3×SD16, C15⋊3C8, C120, Dic30, Dic30, C4×D15, C4×D15, D60, C15⋊7D4, D4×C15, Q8×C15, C22×D15, D5×SD16, C8×D15, C24⋊D5, D4.D15, Q8⋊2D15, C15×SD16, D4×D15, Q8×D15, SD16×D15
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, SD16, C2×D4, D10, C22×S3, D15, C2×SD16, C22×D5, S3×D4, D30, D4×D5, S3×SD16, C22×D15, D5×SD16, D4×D15, SD16×D15
►On 120 points
Generators in S120
(1 110 31 84 16 99 50 65)(2 111 32 85 17 100 51 66)(3 112 33 86 18 101 52 67)(4 113 34 87 19 102 53 68)(5 114 35 88 20 103 54 69)(6 115 36 89 21 104 55 70)(7 116 37 90 22 105 56 71)(8 117 38 76 23 91 57 72)(9 118 39 77 24 92 58 73)(10 119 40 78 25 93 59 74)(11 120 41 79 26 94 60 75)(12 106 42 80 27 95 46 61)(13 107 43 81 28 96 47 62)(14 108 44 82 29 97 48 63)(15 109 45 83 30 98 49 64)
(31 50)(32 51)(33 52)(34 53)(35 54)(36 55)(37 56)(38 57)(39 58)(40 59)(41 60)(42 46)(43 47)(44 48)(45 49)(61 95)(62 96)(63 97)(64 98)(65 99)(66 100)(67 101)(68 102)(69 103)(70 104)(71 105)(72 91)(73 92)(74 93)(75 94)(76 117)(77 118)(78 119)(79 120)(80 106)(81 107)(82 108)(83 109)(84 110)(85 111)(86 112)(87 113)(88 114)(89 115)(90 116)
(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 30)(2 29)(3 28)(4 27)(5 26)(6 25)(7 24)(8 23)(9 22)(10 21)(11 20)(12 19)(13 18)(14 17)(15 16)(31 49)(32 48)(33 47)(34 46)(35 60)(36 59)(37 58)(38 57)(39 56)(40 55)(41 54)(42 53)(43 52)(44 51)(45 50)(61 87)(62 86)(63 85)(64 84)(65 83)(66 82)(67 81)(68 80)(69 79)(70 78)(71 77)(72 76)(73 90)(74 89)(75 88)(91 117)(92 116)(93 115)(94 114)(95 113)(96 112)(97 111)(98 110)(99 109)(100 108)(101 107)(102 106)(103 120)(104 119)(105 118)
G:=sub<Sym(120)| (1,110,31,84,16,99,50,65)(2,111,32,85,17,100,51,66)(3,112,33,86,18,101,52,67)(4,113,34,87,19,102,53,68)(5,114,35,88,20,103,54,69)(6,115,36,89,21,104,55,70)(7,116,37,90,22,105,56,71)(8,117,38,76,23,91,57,72)(9,118,39,77,24,92,58,73)(10,119,40,78,25,93,59,74)(11,120,41,79,26,94,60,75)(12,106,42,80,27,95,46,61)(13,107,43,81,28,96,47,62)(14,108,44,82,29,97,48,63)(15,109,45,83,30,98,49,64), (31,50)(32,51)(33,52)(34,53)(35,54)(36,55)(37,56)(38,57)(39,58)(40,59)(41,60)(42,46)(43,47)(44,48)(45,49)(61,95)(62,96)(63,97)(64,98)(65,99)(66,100)(67,101)(68,102)(69,103)(70,104)(71,105)(72,91)(73,92)(74,93)(75,94)(76,117)(77,118)(78,119)(79,120)(80,106)(81,107)(82,108)(83,109)(84,110)(85,111)(86,112)(87,113)(88,114)(89,115)(90,116), (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), (1,30)(2,29)(3,28)(4,27)(5,26)(6,25)(7,24)(8,23)(9,22)(10,21)(11,20)(12,19)(13,18)(14,17)(15,16)(31,49)(32,48)(33,47)(34,46)(35,60)(36,59)(37,58)(38,57)(39,56)(40,55)(41,54)(42,53)(43,52)(44,51)(45,50)(61,87)(62,86)(63,85)(64,84)(65,83)(66,82)(67,81)(68,80)(69,79)(70,78)(71,77)(72,76)(73,90)(74,89)(75,88)(91,117)(92,116)(93,115)(94,114)(95,113)(96,112)(97,111)(98,110)(99,109)(100,108)(101,107)(102,106)(103,120)(104,119)(105,118)>;
G:=Group( (1,110,31,84,16,99,50,65)(2,111,32,85,17,100,51,66)(3,112,33,86,18,101,52,67)(4,113,34,87,19,102,53,68)(5,114,35,88,20,103,54,69)(6,115,36,89,21,104,55,70)(7,116,37,90,22,105,56,71)(8,117,38,76,23,91,57,72)(9,118,39,77,24,92,58,73)(10,119,40,78,25,93,59,74)(11,120,41,79,26,94,60,75)(12,106,42,80,27,95,46,61)(13,107,43,81,28,96,47,62)(14,108,44,82,29,97,48,63)(15,109,45,83,30,98,49,64), (31,50)(32,51)(33,52)(34,53)(35,54)(36,55)(37,56)(38,57)(39,58)(40,59)(41,60)(42,46)(43,47)(44,48)(45,49)(61,95)(62,96)(63,97)(64,98)(65,99)(66,100)(67,101)(68,102)(69,103)(70,104)(71,105)(72,91)(73,92)(74,93)(75,94)(76,117)(77,118)(78,119)(79,120)(80,106)(81,107)(82,108)(83,109)(84,110)(85,111)(86,112)(87,113)(88,114)(89,115)(90,116), (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), (1,30)(2,29)(3,28)(4,27)(5,26)(6,25)(7,24)(8,23)(9,22)(10,21)(11,20)(12,19)(13,18)(14,17)(15,16)(31,49)(32,48)(33,47)(34,46)(35,60)(36,59)(37,58)(38,57)(39,56)(40,55)(41,54)(42,53)(43,52)(44,51)(45,50)(61,87)(62,86)(63,85)(64,84)(65,83)(66,82)(67,81)(68,80)(69,79)(70,78)(71,77)(72,76)(73,90)(74,89)(75,88)(91,117)(92,116)(93,115)(94,114)(95,113)(96,112)(97,111)(98,110)(99,109)(100,108)(101,107)(102,106)(103,120)(104,119)(105,118) );
G=PermutationGroup([[(1,110,31,84,16,99,50,65),(2,111,32,85,17,100,51,66),(3,112,33,86,18,101,52,67),(4,113,34,87,19,102,53,68),(5,114,35,88,20,103,54,69),(6,115,36,89,21,104,55,70),(7,116,37,90,22,105,56,71),(8,117,38,76,23,91,57,72),(9,118,39,77,24,92,58,73),(10,119,40,78,25,93,59,74),(11,120,41,79,26,94,60,75),(12,106,42,80,27,95,46,61),(13,107,43,81,28,96,47,62),(14,108,44,82,29,97,48,63),(15,109,45,83,30,98,49,64)], [(31,50),(32,51),(33,52),(34,53),(35,54),(36,55),(37,56),(38,57),(39,58),(40,59),(41,60),(42,46),(43,47),(44,48),(45,49),(61,95),(62,96),(63,97),(64,98),(65,99),(66,100),(67,101),(68,102),(69,103),(70,104),(71,105),(72,91),(73,92),(74,93),(75,94),(76,117),(77,118),(78,119),(79,120),(80,106),(81,107),(82,108),(83,109),(84,110),(85,111),(86,112),(87,113),(88,114),(89,115),(90,116)], [(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)], [(1,30),(2,29),(3,28),(4,27),(5,26),(6,25),(7,24),(8,23),(9,22),(10,21),(11,20),(12,19),(13,18),(14,17),(15,16),(31,49),(32,48),(33,47),(34,46),(35,60),(36,59),(37,58),(38,57),(39,56),(40,55),(41,54),(42,53),(43,52),(44,51),(45,50),(61,87),(62,86),(63,85),(64,84),(65,83),(66,82),(67,81),(68,80),(69,79),(70,78),(71,77),(72,76),(73,90),(74,89),(75,88),(91,117),(92,116),(93,115),(94,114),(95,113),(96,112),(97,111),(98,110),(99,109),(100,108),(101,107),(102,106),(103,120),(104,119),(105,118)]])
63 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 5A | 5B | 6A | 6B | 8A | 8B | 8C | 8D | 10A | 10B | 10C | 10D | 12A | 12B | 15A | 15B | 15C | 15D | 20A | 20B | 20C | 20D | 24A | 24B | 30A | 30B | 30C | 30D | 30E | 30F | 30G | 30H | 40A | 40B | 40C | 40D | 60A | 60B | 60C | 60D | 60E | 60F | 60G | 60H | 120A | ··· | 120H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 10 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | 20 | 20 | 20 | 24 | 24 | 30 | 30 | 30 | 30 | 30 | 30 | 30 | 30 | 40 | 40 | 40 | 40 | 60 | 60 | 60 | 60 | 60 | 60 | 60 | 60 | 120 | ··· | 120 |
| size | 1 | 1 | 4 | 15 | 15 | 60 | 2 | 2 | 4 | 30 | 60 | 2 | 2 | 2 | 8 | 2 | 2 | 30 | 30 | 2 | 2 | 8 | 8 | 4 | 8 | 2 | 2 | 2 | 2 | 4 | 4 | 8 | 8 | 4 | 4 | 2 | 2 | 2 | 2 | 8 | 8 | 8 | 8 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | ··· | 4 |
63 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D5 | D6 | D6 | D6 | SD16 | D10 | D10 | D10 | D15 | D30 | D30 | D30 | S3×D4 | D4×D5 | S3×SD16 | D5×SD16 | D4×D15 | SD16×D15 |
| kernel | SD16×D15 | C8×D15 | C24⋊D5 | D4.D15 | Q8⋊2D15 | C15×SD16 | D4×D15 | Q8×D15 | C5×SD16 | Dic15 | D30 | C3×SD16 | C40 | C5×D4 | C5×Q8 | D15 | C24 | C3×D4 | C3×Q8 | SD16 | C8 | D4 | Q8 | C10 | C6 | C5 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 4 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 1 | 2 | 2 | 4 | 4 | 8 |
Matrix representation of SD16×D15 ►in GL6(𝔽241)
| 240 | 0 | 0 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 240 | 0 | 0 | 0 |
| 0 | 0 | 0 | 240 | 0 | 0 |
| 0 | 0 | 0 | 0 | 38 | 39 |
| 0 | 0 | 0 | 0 | 68 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 67 | 240 |
| 190 | 51 | 0 | 0 | 0 | 0 |
| 190 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 192 | 0 | 0 |
| 0 | 0 | 64 | 239 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 192 | 0 | 0 |
| 0 | 0 | 0 | 240 | 0 | 0 |
| 0 | 0 | 0 | 0 | 240 | 0 |
| 0 | 0 | 0 | 0 | 0 | 240 |
G:=sub<GL(6,GF(241))| [240,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,38,68,0,0,0,0,39,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,67,0,0,0,0,0,240],[190,190,0,0,0,0,51,240,0,0,0,0,0,0,1,64,0,0,0,0,192,239,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,192,240,0,0,0,0,0,0,240,0,0,0,0,0,0,240] >;
SD16×D15 in GAP, Magma, Sage, TeX
{\rm SD}_{16}\times D_{15} % in TeX
G:=Group("SD16xD15"); // GroupNames label
G:=SmallGroup(480,878);
// by ID
G=gap.SmallGroup(480,878);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,135,100,346,185,80,2693,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=c^15=d^2=1,b*a*b=a^3,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations