metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D60⋊4C2, Q8⋊3D15, C4.7D30, C20.21D6, C12.21D10, C60.7C22, C30.35C23, D30.7C22, Dic15.17C22, (C5×Q8)⋊5S3, (C3×Q8)⋊3D5, (C4×D15)⋊3C2, (Q8×C15)⋊3C2, C15⋊17(C4○D4), C5⋊3(Q8⋊3S3), C3⋊3(Q8⋊2D5), C6.35(C22×D5), C2.9(C22×D15), C10.35(C22×S3), SmallGroup(240,182)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q8⋊3D15
G = < a,b,c,d | a4=c15=d2=1, b2=a2, bab-1=dad=a-1, ac=ca, bc=cb, bd=db, dcd=c-1 >
Subgroups: 432 in 80 conjugacy classes, 35 normal (14 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, S3, C6, C2×C4, D4, Q8, D5, C10, Dic3, C12, D6, C15, C4○D4, Dic5, C20, D10, C4×S3, D12, C3×Q8, D15, C30, C4×D5, D20, C5×Q8, Q8⋊3S3, Dic15, C60, D30, Q8⋊2D5, C4×D15, D60, Q8×C15, Q8⋊3D15
Quotients: C1, C2, C22, S3, C23, D5, D6, C4○D4, D10, C22×S3, D15, C22×D5, Q8⋊3S3, D30, Q8⋊2D5, C22×D15, Q8⋊3D15
►On 120 points
Generators in S120
(1 58 28 39)(2 59 29 40)(3 60 30 41)(4 46 16 42)(5 47 17 43)(6 48 18 44)(7 49 19 45)(8 50 20 31)(9 51 21 32)(10 52 22 33)(11 53 23 34)(12 54 24 35)(13 55 25 36)(14 56 26 37)(15 57 27 38)(61 95 80 118)(62 96 81 119)(63 97 82 120)(64 98 83 106)(65 99 84 107)(66 100 85 108)(67 101 86 109)(68 102 87 110)(69 103 88 111)(70 104 89 112)(71 105 90 113)(72 91 76 114)(73 92 77 115)(74 93 78 116)(75 94 79 117)
(1 84 28 65)(2 85 29 66)(3 86 30 67)(4 87 16 68)(5 88 17 69)(6 89 18 70)(7 90 19 71)(8 76 20 72)(9 77 21 73)(10 78 22 74)(11 79 23 75)(12 80 24 61)(13 81 25 62)(14 82 26 63)(15 83 27 64)(31 114 50 91)(32 115 51 92)(33 116 52 93)(34 117 53 94)(35 118 54 95)(36 119 55 96)(37 120 56 97)(38 106 57 98)(39 107 58 99)(40 108 59 100)(41 109 60 101)(42 110 46 102)(43 111 47 103)(44 112 48 104)(45 113 49 105)
(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 38)(2 37)(3 36)(4 35)(5 34)(6 33)(7 32)(8 31)(9 45)(10 44)(11 43)(12 42)(13 41)(14 40)(15 39)(16 54)(17 53)(18 52)(19 51)(20 50)(21 49)(22 48)(23 47)(24 46)(25 60)(26 59)(27 58)(28 57)(29 56)(30 55)(61 102)(62 101)(63 100)(64 99)(65 98)(66 97)(67 96)(68 95)(69 94)(70 93)(71 92)(72 91)(73 105)(74 104)(75 103)(76 114)(77 113)(78 112)(79 111)(80 110)(81 109)(82 108)(83 107)(84 106)(85 120)(86 119)(87 118)(88 117)(89 116)(90 115)
G:=sub<Sym(120)| (1,58,28,39)(2,59,29,40)(3,60,30,41)(4,46,16,42)(5,47,17,43)(6,48,18,44)(7,49,19,45)(8,50,20,31)(9,51,21,32)(10,52,22,33)(11,53,23,34)(12,54,24,35)(13,55,25,36)(14,56,26,37)(15,57,27,38)(61,95,80,118)(62,96,81,119)(63,97,82,120)(64,98,83,106)(65,99,84,107)(66,100,85,108)(67,101,86,109)(68,102,87,110)(69,103,88,111)(70,104,89,112)(71,105,90,113)(72,91,76,114)(73,92,77,115)(74,93,78,116)(75,94,79,117), (1,84,28,65)(2,85,29,66)(3,86,30,67)(4,87,16,68)(5,88,17,69)(6,89,18,70)(7,90,19,71)(8,76,20,72)(9,77,21,73)(10,78,22,74)(11,79,23,75)(12,80,24,61)(13,81,25,62)(14,82,26,63)(15,83,27,64)(31,114,50,91)(32,115,51,92)(33,116,52,93)(34,117,53,94)(35,118,54,95)(36,119,55,96)(37,120,56,97)(38,106,57,98)(39,107,58,99)(40,108,59,100)(41,109,60,101)(42,110,46,102)(43,111,47,103)(44,112,48,104)(45,113,49,105), (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), (1,38)(2,37)(3,36)(4,35)(5,34)(6,33)(7,32)(8,31)(9,45)(10,44)(11,43)(12,42)(13,41)(14,40)(15,39)(16,54)(17,53)(18,52)(19,51)(20,50)(21,49)(22,48)(23,47)(24,46)(25,60)(26,59)(27,58)(28,57)(29,56)(30,55)(61,102)(62,101)(63,100)(64,99)(65,98)(66,97)(67,96)(68,95)(69,94)(70,93)(71,92)(72,91)(73,105)(74,104)(75,103)(76,114)(77,113)(78,112)(79,111)(80,110)(81,109)(82,108)(83,107)(84,106)(85,120)(86,119)(87,118)(88,117)(89,116)(90,115)>;
G:=Group( (1,58,28,39)(2,59,29,40)(3,60,30,41)(4,46,16,42)(5,47,17,43)(6,48,18,44)(7,49,19,45)(8,50,20,31)(9,51,21,32)(10,52,22,33)(11,53,23,34)(12,54,24,35)(13,55,25,36)(14,56,26,37)(15,57,27,38)(61,95,80,118)(62,96,81,119)(63,97,82,120)(64,98,83,106)(65,99,84,107)(66,100,85,108)(67,101,86,109)(68,102,87,110)(69,103,88,111)(70,104,89,112)(71,105,90,113)(72,91,76,114)(73,92,77,115)(74,93,78,116)(75,94,79,117), (1,84,28,65)(2,85,29,66)(3,86,30,67)(4,87,16,68)(5,88,17,69)(6,89,18,70)(7,90,19,71)(8,76,20,72)(9,77,21,73)(10,78,22,74)(11,79,23,75)(12,80,24,61)(13,81,25,62)(14,82,26,63)(15,83,27,64)(31,114,50,91)(32,115,51,92)(33,116,52,93)(34,117,53,94)(35,118,54,95)(36,119,55,96)(37,120,56,97)(38,106,57,98)(39,107,58,99)(40,108,59,100)(41,109,60,101)(42,110,46,102)(43,111,47,103)(44,112,48,104)(45,113,49,105), (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), (1,38)(2,37)(3,36)(4,35)(5,34)(6,33)(7,32)(8,31)(9,45)(10,44)(11,43)(12,42)(13,41)(14,40)(15,39)(16,54)(17,53)(18,52)(19,51)(20,50)(21,49)(22,48)(23,47)(24,46)(25,60)(26,59)(27,58)(28,57)(29,56)(30,55)(61,102)(62,101)(63,100)(64,99)(65,98)(66,97)(67,96)(68,95)(69,94)(70,93)(71,92)(72,91)(73,105)(74,104)(75,103)(76,114)(77,113)(78,112)(79,111)(80,110)(81,109)(82,108)(83,107)(84,106)(85,120)(86,119)(87,118)(88,117)(89,116)(90,115) );
G=PermutationGroup([[(1,58,28,39),(2,59,29,40),(3,60,30,41),(4,46,16,42),(5,47,17,43),(6,48,18,44),(7,49,19,45),(8,50,20,31),(9,51,21,32),(10,52,22,33),(11,53,23,34),(12,54,24,35),(13,55,25,36),(14,56,26,37),(15,57,27,38),(61,95,80,118),(62,96,81,119),(63,97,82,120),(64,98,83,106),(65,99,84,107),(66,100,85,108),(67,101,86,109),(68,102,87,110),(69,103,88,111),(70,104,89,112),(71,105,90,113),(72,91,76,114),(73,92,77,115),(74,93,78,116),(75,94,79,117)], [(1,84,28,65),(2,85,29,66),(3,86,30,67),(4,87,16,68),(5,88,17,69),(6,89,18,70),(7,90,19,71),(8,76,20,72),(9,77,21,73),(10,78,22,74),(11,79,23,75),(12,80,24,61),(13,81,25,62),(14,82,26,63),(15,83,27,64),(31,114,50,91),(32,115,51,92),(33,116,52,93),(34,117,53,94),(35,118,54,95),(36,119,55,96),(37,120,56,97),(38,106,57,98),(39,107,58,99),(40,108,59,100),(41,109,60,101),(42,110,46,102),(43,111,47,103),(44,112,48,104),(45,113,49,105)], [(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)], [(1,38),(2,37),(3,36),(4,35),(5,34),(6,33),(7,32),(8,31),(9,45),(10,44),(11,43),(12,42),(13,41),(14,40),(15,39),(16,54),(17,53),(18,52),(19,51),(20,50),(21,49),(22,48),(23,47),(24,46),(25,60),(26,59),(27,58),(28,57),(29,56),(30,55),(61,102),(62,101),(63,100),(64,99),(65,98),(66,97),(67,96),(68,95),(69,94),(70,93),(71,92),(72,91),(73,105),(74,104),(75,103),(76,114),(77,113),(78,112),(79,111),(80,110),(81,109),(82,108),(83,107),(84,106),(85,120),(86,119),(87,118),(88,117),(89,116),(90,115)]])
Q8⋊3D15 is a maximal subgroup of
D60⋊C22 C60.C23 D20.16D6 D12.D10 Q8⋊3D30 D4.5D30 Q16⋊D15 D120⋊8C2 C30.33C24 D5×Q8⋊3S3 S3×Q8⋊2D5 D20⋊17D6 Q8.15D30 C4○D4×D15 D4⋊8D30
Q8⋊3D15 is a maximal quotient of
C4.Dic30 C4⋊C4⋊7D15 D60⋊11C4 D30.29D4 C4⋊D60 C4⋊C4⋊D15 Q8×Dic15 D30⋊7Q8 C60.23D4
45 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 4C | 4D | 4E | 5A | 5B | 6 | 10A | 10B | 12A | 12B | 12C | 15A | 15B | 15C | 15D | 20A | ··· | 20F | 30A | 30B | 30C | 30D | 60A | ··· | 60L |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 10 | 10 | 12 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 30 | 30 | 30 | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 30 | 30 | 30 | 2 | 2 | 2 | 2 | 15 | 15 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
45 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | S3 | D5 | D6 | C4○D4 | D10 | D15 | D30 | Q8⋊3S3 | Q8⋊2D5 | Q8⋊3D15 |
| kernel | Q8⋊3D15 | C4×D15 | D60 | Q8×C15 | C5×Q8 | C3×Q8 | C20 | C15 | C12 | Q8 | C4 | C5 | C3 | C1 |
| # reps | 1 | 3 | 3 | 1 | 1 | 2 | 3 | 2 | 6 | 4 | 12 | 1 | 2 | 4 |
Matrix representation of Q8⋊3D15 ►in GL4(𝔽61) generated by
| 60 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 60 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 50 |
| 0 | 0 | 50 | 0 |
| 9 | 56 | 0 | 0 |
| 5 | 38 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 52 | 5 | 0 | 0 |
| 45 | 9 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(61))| [60,0,0,0,0,60,0,0,0,0,0,60,0,0,1,0],[1,0,0,0,0,1,0,0,0,0,0,50,0,0,50,0],[9,5,0,0,56,38,0,0,0,0,1,0,0,0,0,1],[52,45,0,0,5,9,0,0,0,0,0,1,0,0,1,0] >;
Q8⋊3D15 in GAP, Magma, Sage, TeX
Q_8\rtimes_3D_{15} % in TeX
G:=Group("Q8:3D15"); // GroupNames label
G:=SmallGroup(240,182);
// by ID
G=gap.SmallGroup(240,182);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-5,55,218,116,50,964,6917]);
// Polycyclic
G:=Group<a,b,c,d|a^4=c^15=d^2=1,b^2=a^2,b*a*b^-1=d*a*d=a^-1,a*c=c*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations