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 [×3], C3, C4 [×3], C4, C22 [×3], C5, S3 [×3], C6, C2×C4 [×3], D4 [×3], Q8, D5 [×3], C10, Dic3, C12 [×3], D6 [×3], C15, C4○D4, Dic5, C20 [×3], D10 [×3], C4×S3 [×3], D12 [×3], C3×Q8, D15 [×3], C30, C4×D5 [×3], D20 [×3], C5×Q8, Q8⋊3S3, Dic15, C60 [×3], D30 [×3], Q8⋊2D5, C4×D15 [×3], D60 [×3], Q8×C15, Q8⋊3D15
Quotients: C1, C2 [×7], C22 [×7], S3, C23, D5, D6 [×3], C4○D4, D10 [×3], C22×S3, D15, C22×D5, Q8⋊3S3, D30 [×3], Q8⋊2D5, C22×D15, Q8⋊3D15
►On 120 points
Generators in S120
(1 58 16 32)(2 59 17 33)(3 60 18 34)(4 46 19 35)(5 47 20 36)(6 48 21 37)(7 49 22 38)(8 50 23 39)(9 51 24 40)(10 52 25 41)(11 53 26 42)(12 54 27 43)(13 55 28 44)(14 56 29 45)(15 57 30 31)(61 98 87 120)(62 99 88 106)(63 100 89 107)(64 101 90 108)(65 102 76 109)(66 103 77 110)(67 104 78 111)(68 105 79 112)(69 91 80 113)(70 92 81 114)(71 93 82 115)(72 94 83 116)(73 95 84 117)(74 96 85 118)(75 97 86 119)
(1 88 16 62)(2 89 17 63)(3 90 18 64)(4 76 19 65)(5 77 20 66)(6 78 21 67)(7 79 22 68)(8 80 23 69)(9 81 24 70)(10 82 25 71)(11 83 26 72)(12 84 27 73)(13 85 28 74)(14 86 29 75)(15 87 30 61)(31 120 57 98)(32 106 58 99)(33 107 59 100)(34 108 60 101)(35 109 46 102)(36 110 47 103)(37 111 48 104)(38 112 49 105)(39 113 50 91)(40 114 51 92)(41 115 52 93)(42 116 53 94)(43 117 54 95)(44 118 55 96)(45 119 56 97)
(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 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 57)(17 56)(18 55)(19 54)(20 53)(21 52)(22 51)(23 50)(24 49)(25 48)(26 47)(27 46)(28 60)(29 59)(30 58)(61 99)(62 98)(63 97)(64 96)(65 95)(66 94)(67 93)(68 92)(69 91)(70 105)(71 104)(72 103)(73 102)(74 101)(75 100)(76 117)(77 116)(78 115)(79 114)(80 113)(81 112)(82 111)(83 110)(84 109)(85 108)(86 107)(87 106)(88 120)(89 119)(90 118)
G:=sub<Sym(120)| (1,58,16,32)(2,59,17,33)(3,60,18,34)(4,46,19,35)(5,47,20,36)(6,48,21,37)(7,49,22,38)(8,50,23,39)(9,51,24,40)(10,52,25,41)(11,53,26,42)(12,54,27,43)(13,55,28,44)(14,56,29,45)(15,57,30,31)(61,98,87,120)(62,99,88,106)(63,100,89,107)(64,101,90,108)(65,102,76,109)(66,103,77,110)(67,104,78,111)(68,105,79,112)(69,91,80,113)(70,92,81,114)(71,93,82,115)(72,94,83,116)(73,95,84,117)(74,96,85,118)(75,97,86,119), (1,88,16,62)(2,89,17,63)(3,90,18,64)(4,76,19,65)(5,77,20,66)(6,78,21,67)(7,79,22,68)(8,80,23,69)(9,81,24,70)(10,82,25,71)(11,83,26,72)(12,84,27,73)(13,85,28,74)(14,86,29,75)(15,87,30,61)(31,120,57,98)(32,106,58,99)(33,107,59,100)(34,108,60,101)(35,109,46,102)(36,110,47,103)(37,111,48,104)(38,112,49,105)(39,113,50,91)(40,114,51,92)(41,115,52,93)(42,116,53,94)(43,117,54,95)(44,118,55,96)(45,119,56,97), (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), (1,31)(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)(16,57)(17,56)(18,55)(19,54)(20,53)(21,52)(22,51)(23,50)(24,49)(25,48)(26,47)(27,46)(28,60)(29,59)(30,58)(61,99)(62,98)(63,97)(64,96)(65,95)(66,94)(67,93)(68,92)(69,91)(70,105)(71,104)(72,103)(73,102)(74,101)(75,100)(76,117)(77,116)(78,115)(79,114)(80,113)(81,112)(82,111)(83,110)(84,109)(85,108)(86,107)(87,106)(88,120)(89,119)(90,118)>;
G:=Group( (1,58,16,32)(2,59,17,33)(3,60,18,34)(4,46,19,35)(5,47,20,36)(6,48,21,37)(7,49,22,38)(8,50,23,39)(9,51,24,40)(10,52,25,41)(11,53,26,42)(12,54,27,43)(13,55,28,44)(14,56,29,45)(15,57,30,31)(61,98,87,120)(62,99,88,106)(63,100,89,107)(64,101,90,108)(65,102,76,109)(66,103,77,110)(67,104,78,111)(68,105,79,112)(69,91,80,113)(70,92,81,114)(71,93,82,115)(72,94,83,116)(73,95,84,117)(74,96,85,118)(75,97,86,119), (1,88,16,62)(2,89,17,63)(3,90,18,64)(4,76,19,65)(5,77,20,66)(6,78,21,67)(7,79,22,68)(8,80,23,69)(9,81,24,70)(10,82,25,71)(11,83,26,72)(12,84,27,73)(13,85,28,74)(14,86,29,75)(15,87,30,61)(31,120,57,98)(32,106,58,99)(33,107,59,100)(34,108,60,101)(35,109,46,102)(36,110,47,103)(37,111,48,104)(38,112,49,105)(39,113,50,91)(40,114,51,92)(41,115,52,93)(42,116,53,94)(43,117,54,95)(44,118,55,96)(45,119,56,97), (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), (1,31)(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)(16,57)(17,56)(18,55)(19,54)(20,53)(21,52)(22,51)(23,50)(24,49)(25,48)(26,47)(27,46)(28,60)(29,59)(30,58)(61,99)(62,98)(63,97)(64,96)(65,95)(66,94)(67,93)(68,92)(69,91)(70,105)(71,104)(72,103)(73,102)(74,101)(75,100)(76,117)(77,116)(78,115)(79,114)(80,113)(81,112)(82,111)(83,110)(84,109)(85,108)(86,107)(87,106)(88,120)(89,119)(90,118) );
G=PermutationGroup([(1,58,16,32),(2,59,17,33),(3,60,18,34),(4,46,19,35),(5,47,20,36),(6,48,21,37),(7,49,22,38),(8,50,23,39),(9,51,24,40),(10,52,25,41),(11,53,26,42),(12,54,27,43),(13,55,28,44),(14,56,29,45),(15,57,30,31),(61,98,87,120),(62,99,88,106),(63,100,89,107),(64,101,90,108),(65,102,76,109),(66,103,77,110),(67,104,78,111),(68,105,79,112),(69,91,80,113),(70,92,81,114),(71,93,82,115),(72,94,83,116),(73,95,84,117),(74,96,85,118),(75,97,86,119)], [(1,88,16,62),(2,89,17,63),(3,90,18,64),(4,76,19,65),(5,77,20,66),(6,78,21,67),(7,79,22,68),(8,80,23,69),(9,81,24,70),(10,82,25,71),(11,83,26,72),(12,84,27,73),(13,85,28,74),(14,86,29,75),(15,87,30,61),(31,120,57,98),(32,106,58,99),(33,107,59,100),(34,108,60,101),(35,109,46,102),(36,110,47,103),(37,111,48,104),(38,112,49,105),(39,113,50,91),(40,114,51,92),(41,115,52,93),(42,116,53,94),(43,117,54,95),(44,118,55,96),(45,119,56,97)], [(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)], [(1,31),(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),(16,57),(17,56),(18,55),(19,54),(20,53),(21,52),(22,51),(23,50),(24,49),(25,48),(26,47),(27,46),(28,60),(29,59),(30,58),(61,99),(62,98),(63,97),(64,96),(65,95),(66,94),(67,93),(68,92),(69,91),(70,105),(71,104),(72,103),(73,102),(74,101),(75,100),(76,117),(77,116),(78,115),(79,114),(80,113),(81,112),(82,111),(83,110),(84,109),(85,108),(86,107),(87,106),(88,120),(89,119),(90,118)])
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