metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Q8⋊2D15, C4.3D30, D60.2C2, C20.11D6, C30.36D4, C15⋊13SD16, C12.11D10, C60.3C22, C3⋊3(Q8⋊D5), (C5×Q8)⋊3S3, (C3×Q8)⋊1D5, C15⋊3C8⋊3C2, (Q8×C15)⋊1C2, C5⋊3(Q8⋊2S3), C6.18(C5⋊D4), C2.6(C15⋊7D4), C10.18(C3⋊D4), SmallGroup(240,78)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q8⋊2D15
G = < a,b,c,d | a4=c15=d2=1, b2=a2, bab-1=dad=a-1, ac=ca, bc=cb, dbd=a-1b, dcd=c-1 >
Smallest permutation representation of Q8⋊2D15
►On 120 points
Generators in S120
(1 58 24 31)(2 59 25 32)(3 60 26 33)(4 46 27 34)(5 47 28 35)(6 48 29 36)(7 49 30 37)(8 50 16 38)(9 51 17 39)(10 52 18 40)(11 53 19 41)(12 54 20 42)(13 55 21 43)(14 56 22 44)(15 57 23 45)(61 91 87 106)(62 92 88 107)(63 93 89 108)(64 94 90 109)(65 95 76 110)(66 96 77 111)(67 97 78 112)(68 98 79 113)(69 99 80 114)(70 100 81 115)(71 101 82 116)(72 102 83 117)(73 103 84 118)(74 104 85 119)(75 105 86 120)
(1 76 24 65)(2 77 25 66)(3 78 26 67)(4 79 27 68)(5 80 28 69)(6 81 29 70)(7 82 30 71)(8 83 16 72)(9 84 17 73)(10 85 18 74)(11 86 19 75)(12 87 20 61)(13 88 21 62)(14 89 22 63)(15 90 23 64)(31 110 58 95)(32 111 59 96)(33 112 60 97)(34 113 46 98)(35 114 47 99)(36 115 48 100)(37 116 49 101)(38 117 50 102)(39 118 51 103)(40 119 52 104)(41 120 53 105)(42 106 54 91)(43 107 55 92)(44 108 56 93)(45 109 57 94)
(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 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(17 30)(18 29)(19 28)(20 27)(21 26)(22 25)(23 24)(31 57)(32 56)(33 55)(34 54)(35 53)(36 52)(37 51)(38 50)(39 49)(40 48)(41 47)(42 46)(43 60)(44 59)(45 58)(61 98)(62 97)(63 96)(64 95)(65 94)(66 93)(67 92)(68 91)(69 105)(70 104)(71 103)(72 102)(73 101)(74 100)(75 99)(76 109)(77 108)(78 107)(79 106)(80 120)(81 119)(82 118)(83 117)(84 116)(85 115)(86 114)(87 113)(88 112)(89 111)(90 110)
G:=sub<Sym(120)| (1,58,24,31)(2,59,25,32)(3,60,26,33)(4,46,27,34)(5,47,28,35)(6,48,29,36)(7,49,30,37)(8,50,16,38)(9,51,17,39)(10,52,18,40)(11,53,19,41)(12,54,20,42)(13,55,21,43)(14,56,22,44)(15,57,23,45)(61,91,87,106)(62,92,88,107)(63,93,89,108)(64,94,90,109)(65,95,76,110)(66,96,77,111)(67,97,78,112)(68,98,79,113)(69,99,80,114)(70,100,81,115)(71,101,82,116)(72,102,83,117)(73,103,84,118)(74,104,85,119)(75,105,86,120), (1,76,24,65)(2,77,25,66)(3,78,26,67)(4,79,27,68)(5,80,28,69)(6,81,29,70)(7,82,30,71)(8,83,16,72)(9,84,17,73)(10,85,18,74)(11,86,19,75)(12,87,20,61)(13,88,21,62)(14,89,22,63)(15,90,23,64)(31,110,58,95)(32,111,59,96)(33,112,60,97)(34,113,46,98)(35,114,47,99)(36,115,48,100)(37,116,49,101)(38,117,50,102)(39,118,51,103)(40,119,52,104)(41,120,53,105)(42,106,54,91)(43,107,55,92)(44,108,56,93)(45,109,57,94), (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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(17,30)(18,29)(19,28)(20,27)(21,26)(22,25)(23,24)(31,57)(32,56)(33,55)(34,54)(35,53)(36,52)(37,51)(38,50)(39,49)(40,48)(41,47)(42,46)(43,60)(44,59)(45,58)(61,98)(62,97)(63,96)(64,95)(65,94)(66,93)(67,92)(68,91)(69,105)(70,104)(71,103)(72,102)(73,101)(74,100)(75,99)(76,109)(77,108)(78,107)(79,106)(80,120)(81,119)(82,118)(83,117)(84,116)(85,115)(86,114)(87,113)(88,112)(89,111)(90,110)>;
G:=Group( (1,58,24,31)(2,59,25,32)(3,60,26,33)(4,46,27,34)(5,47,28,35)(6,48,29,36)(7,49,30,37)(8,50,16,38)(9,51,17,39)(10,52,18,40)(11,53,19,41)(12,54,20,42)(13,55,21,43)(14,56,22,44)(15,57,23,45)(61,91,87,106)(62,92,88,107)(63,93,89,108)(64,94,90,109)(65,95,76,110)(66,96,77,111)(67,97,78,112)(68,98,79,113)(69,99,80,114)(70,100,81,115)(71,101,82,116)(72,102,83,117)(73,103,84,118)(74,104,85,119)(75,105,86,120), (1,76,24,65)(2,77,25,66)(3,78,26,67)(4,79,27,68)(5,80,28,69)(6,81,29,70)(7,82,30,71)(8,83,16,72)(9,84,17,73)(10,85,18,74)(11,86,19,75)(12,87,20,61)(13,88,21,62)(14,89,22,63)(15,90,23,64)(31,110,58,95)(32,111,59,96)(33,112,60,97)(34,113,46,98)(35,114,47,99)(36,115,48,100)(37,116,49,101)(38,117,50,102)(39,118,51,103)(40,119,52,104)(41,120,53,105)(42,106,54,91)(43,107,55,92)(44,108,56,93)(45,109,57,94), (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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(17,30)(18,29)(19,28)(20,27)(21,26)(22,25)(23,24)(31,57)(32,56)(33,55)(34,54)(35,53)(36,52)(37,51)(38,50)(39,49)(40,48)(41,47)(42,46)(43,60)(44,59)(45,58)(61,98)(62,97)(63,96)(64,95)(65,94)(66,93)(67,92)(68,91)(69,105)(70,104)(71,103)(72,102)(73,101)(74,100)(75,99)(76,109)(77,108)(78,107)(79,106)(80,120)(81,119)(82,118)(83,117)(84,116)(85,115)(86,114)(87,113)(88,112)(89,111)(90,110) );
G=PermutationGroup([[(1,58,24,31),(2,59,25,32),(3,60,26,33),(4,46,27,34),(5,47,28,35),(6,48,29,36),(7,49,30,37),(8,50,16,38),(9,51,17,39),(10,52,18,40),(11,53,19,41),(12,54,20,42),(13,55,21,43),(14,56,22,44),(15,57,23,45),(61,91,87,106),(62,92,88,107),(63,93,89,108),(64,94,90,109),(65,95,76,110),(66,96,77,111),(67,97,78,112),(68,98,79,113),(69,99,80,114),(70,100,81,115),(71,101,82,116),(72,102,83,117),(73,103,84,118),(74,104,85,119),(75,105,86,120)], [(1,76,24,65),(2,77,25,66),(3,78,26,67),(4,79,27,68),(5,80,28,69),(6,81,29,70),(7,82,30,71),(8,83,16,72),(9,84,17,73),(10,85,18,74),(11,86,19,75),(12,87,20,61),(13,88,21,62),(14,89,22,63),(15,90,23,64),(31,110,58,95),(32,111,59,96),(33,112,60,97),(34,113,46,98),(35,114,47,99),(36,115,48,100),(37,116,49,101),(38,117,50,102),(39,118,51,103),(40,119,52,104),(41,120,53,105),(42,106,54,91),(43,107,55,92),(44,108,56,93),(45,109,57,94)], [(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)], [(1,15),(2,14),(3,13),(4,12),(5,11),(6,10),(7,9),(17,30),(18,29),(19,28),(20,27),(21,26),(22,25),(23,24),(31,57),(32,56),(33,55),(34,54),(35,53),(36,52),(37,51),(38,50),(39,49),(40,48),(41,47),(42,46),(43,60),(44,59),(45,58),(61,98),(62,97),(63,96),(64,95),(65,94),(66,93),(67,92),(68,91),(69,105),(70,104),(71,103),(72,102),(73,101),(74,100),(75,99),(76,109),(77,108),(78,107),(79,106),(80,120),(81,119),(82,118),(83,117),(84,116),(85,115),(86,114),(87,113),(88,112),(89,111),(90,110)]])
Q8⋊2D15 is a maximal subgroup of
D5×Q8⋊2S3 D20⋊D6 S3×Q8⋊D5 D12⋊D10 C60.39C23 D20.D6 Dic10.27D6 C60.44C23 SD16×D15 Q8⋊3D30 Q16⋊D15 D120⋊8C2 Q8.11D30 D4⋊D30 D4.8D30
Q8⋊2D15 is a maximal quotient of
C60.2Q8 D60⋊9C4 Q8⋊2Dic15
42 conjugacy classes
| class | 1 | 2A | 2B | 3 | 4A | 4B | 5A | 5B | 6 | 8A | 8B | 10A | 10B | 12A | 12B | 12C | 15A | 15B | 15C | 15D | 20A | ··· | 20F | 30A | 30B | 30C | 30D | 60A | ··· | 60L |
| order | 1 | 2 | 2 | 3 | 4 | 4 | 5 | 5 | 6 | 8 | 8 | 10 | 10 | 12 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 30 | 30 | 30 | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 60 | 2 | 2 | 4 | 2 | 2 | 2 | 30 | 30 | 2 | 2 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | S3 | D4 | D5 | D6 | SD16 | D10 | C3⋊D4 | D15 | C5⋊D4 | D30 | C15⋊7D4 | Q8⋊2S3 | Q8⋊D5 | Q8⋊2D15 |
| kernel | Q8⋊2D15 | C15⋊3C8 | D60 | Q8×C15 | C5×Q8 | C30 | C3×Q8 | C20 | C15 | C12 | C10 | Q8 | C6 | C4 | C2 | C5 | C3 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 8 | 1 | 2 | 4 |
Matrix representation of Q8⋊2D15 ►in GL6(𝔽241)
| 240 | 0 | 0 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 240 | 192 | 0 | 0 |
| 0 | 0 | 123 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 240 | 0 |
| 0 | 0 | 0 | 0 | 0 | 240 |
| 76 | 192 | 0 | 0 | 0 | 0 |
| 49 | 165 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 33 | 0 | 0 |
| 0 | 0 | 73 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 70 | 101 |
| 0 | 0 | 0 | 0 | 140 | 171 |
| 190 | 51 | 0 | 0 | 0 | 0 |
| 190 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 240 | 1 |
| 0 | 0 | 0 | 0 | 240 | 0 |
| 240 | 0 | 0 | 0 | 0 | 0 |
| 51 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 118 | 240 | 0 | 0 |
| 0 | 0 | 0 | 0 | 240 | 0 |
| 0 | 0 | 0 | 0 | 240 | 1 |
G:=sub<GL(6,GF(241))| [240,0,0,0,0,0,0,240,0,0,0,0,0,0,240,123,0,0,0,0,192,1,0,0,0,0,0,0,240,0,0,0,0,0,0,240],[76,49,0,0,0,0,192,165,0,0,0,0,0,0,0,73,0,0,0,0,33,0,0,0,0,0,0,0,70,140,0,0,0,0,101,171],[190,190,0,0,0,0,51,240,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,240,240,0,0,0,0,1,0],[240,51,0,0,0,0,0,1,0,0,0,0,0,0,1,118,0,0,0,0,0,240,0,0,0,0,0,0,240,240,0,0,0,0,0,1] >;
Q8⋊2D15 in GAP, Magma, Sage, TeX
Q_8\rtimes_2D_{15} % in TeX
G:=Group("Q8:2D15"); // GroupNames label
G:=SmallGroup(240,78);
// by ID
G=gap.SmallGroup(240,78);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-5,73,55,218,116,50,964,6917]);
// 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,d*b*d=a^-1*b,d*c*d=c^-1>;
// generators/relations
Export