metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C8⋊3D30, C40⋊9D6, C24⋊9D10, Q8⋊3D30, D120⋊9C2, D4.3D30, C120⋊4C22, SD16⋊1D15, D30.31D4, D60⋊15C22, C60.68C23, Dic15.36D4, (C5×Q8)⋊9D6, C5⋊5(Q8⋊3D6), (C3×Q8)⋊6D10, C40⋊S3⋊1C2, D4⋊D15⋊11C2, (D4×D15)⋊10C2, C3⋊5(D40⋊C2), (C3×SD16)⋊1D5, (C5×SD16)⋊1S3, (C5×D4).15D6, C2.19(D4×D15), C6.112(D4×D5), Q8⋊3D15⋊8C2, C15⋊31(C8⋊C22), (C15×SD16)⋊1C2, (C3×D4).15D10, C30.319(C2×D4), C10.114(S3×D4), C4.5(C22×D15), C15⋊3C8⋊16C22, Q8⋊2D15⋊10C2, (Q8×C15)⋊13C22, C20.106(C22×S3), (D4×C15).22C22, (C4×D15).25C22, C12.106(C22×D5), SmallGroup(480,879)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q8⋊3D30
G = < a,b,c,d | a4=c30=d2=1, b2=a2, bab-1=cac-1=dad=a-1, cbc-1=a-1b, dbd=ab, dcd=c-1 >
Subgroups: 1092 in 136 conjugacy classes, 41 normal (all characteristic)
C1, C2, C2 [×4], C3, C4, C4 [×2], C22 [×6], C5, S3 [×3], C6, C6, C8, C8, C2×C4 [×2], D4, D4 [×4], Q8, C23, D5 [×3], C10, C10, Dic3, C12, C12, D6 [×5], C2×C6, C15, M4(2), D8 [×2], SD16, SD16, C2×D4, C4○D4, Dic5, C20, C20, D10 [×5], C2×C10, C3⋊C8, C24, C4×S3 [×2], D12 [×3], C3⋊D4, C3×D4, C3×Q8, C22×S3, D15 [×3], C30, C30, C8⋊C22, C5⋊2C8, C40, C4×D5 [×2], D20 [×3], C5⋊D4, C5×D4, C5×Q8, C22×D5, C8⋊S3, D24, D4⋊S3, Q8⋊2S3, C3×SD16, S3×D4, Q8⋊3S3, Dic15, C60, C60, D30, D30 [×4], C2×C30, C8⋊D5, D40, D4⋊D5, Q8⋊D5, C5×SD16, D4×D5, Q8⋊2D5, Q8⋊3D6, C15⋊3C8, C120, C4×D15, C4×D15, D60 [×2], D60, C15⋊7D4, D4×C15, Q8×C15, C22×D15, D40⋊C2, C40⋊S3, D120, D4⋊D15, Q8⋊2D15, C15×SD16, D4×D15, Q8⋊3D15, Q8⋊3D30
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], C2×D4, D10 [×3], C22×S3, D15, C8⋊C22, C22×D5, S3×D4, D30 [×3], D4×D5, Q8⋊3D6, C22×D15, D40⋊C2, D4×D15, Q8⋊3D30
►On 120 points
Generators in S120
(1 34 30 49)(2 50 16 35)(3 36 17 51)(4 52 18 37)(5 38 19 53)(6 54 20 39)(7 40 21 55)(8 56 22 41)(9 42 23 57)(10 58 24 43)(11 44 25 59)(12 60 26 45)(13 46 27 31)(14 32 28 47)(15 48 29 33)(61 76 119 104)(62 105 120 77)(63 78 91 106)(64 107 92 79)(65 80 93 108)(66 109 94 81)(67 82 95 110)(68 111 96 83)(69 84 97 112)(70 113 98 85)(71 86 99 114)(72 115 100 87)(73 88 101 116)(74 117 102 89)(75 90 103 118)
(1 97 30 69)(2 113 16 85)(3 99 17 71)(4 115 18 87)(5 101 19 73)(6 117 20 89)(7 103 21 75)(8 119 22 61)(9 105 23 77)(10 91 24 63)(11 107 25 79)(12 93 26 65)(13 109 27 81)(14 95 28 67)(15 111 29 83)(31 94 46 66)(32 82 47 110)(33 96 48 68)(34 84 49 112)(35 98 50 70)(36 86 51 114)(37 100 52 72)(38 88 53 116)(39 102 54 74)(40 90 55 118)(41 104 56 76)(42 62 57 120)(43 106 58 78)(44 64 59 92)(45 108 60 80)
(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 24)(2 23)(3 22)(4 21)(5 20)(6 19)(7 18)(8 17)(9 16)(10 30)(11 29)(12 28)(13 27)(14 26)(15 25)(32 60)(33 59)(34 58)(35 57)(36 56)(37 55)(38 54)(39 53)(40 52)(41 51)(42 50)(43 49)(44 48)(45 47)(61 86)(62 85)(63 84)(64 83)(65 82)(66 81)(67 80)(68 79)(69 78)(70 77)(71 76)(72 75)(73 74)(87 90)(88 89)(91 112)(92 111)(93 110)(94 109)(95 108)(96 107)(97 106)(98 105)(99 104)(100 103)(101 102)(113 120)(114 119)(115 118)(116 117)
G:=sub<Sym(120)| (1,34,30,49)(2,50,16,35)(3,36,17,51)(4,52,18,37)(5,38,19,53)(6,54,20,39)(7,40,21,55)(8,56,22,41)(9,42,23,57)(10,58,24,43)(11,44,25,59)(12,60,26,45)(13,46,27,31)(14,32,28,47)(15,48,29,33)(61,76,119,104)(62,105,120,77)(63,78,91,106)(64,107,92,79)(65,80,93,108)(66,109,94,81)(67,82,95,110)(68,111,96,83)(69,84,97,112)(70,113,98,85)(71,86,99,114)(72,115,100,87)(73,88,101,116)(74,117,102,89)(75,90,103,118), (1,97,30,69)(2,113,16,85)(3,99,17,71)(4,115,18,87)(5,101,19,73)(6,117,20,89)(7,103,21,75)(8,119,22,61)(9,105,23,77)(10,91,24,63)(11,107,25,79)(12,93,26,65)(13,109,27,81)(14,95,28,67)(15,111,29,83)(31,94,46,66)(32,82,47,110)(33,96,48,68)(34,84,49,112)(35,98,50,70)(36,86,51,114)(37,100,52,72)(38,88,53,116)(39,102,54,74)(40,90,55,118)(41,104,56,76)(42,62,57,120)(43,106,58,78)(44,64,59,92)(45,108,60,80), (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), (1,24)(2,23)(3,22)(4,21)(5,20)(6,19)(7,18)(8,17)(9,16)(10,30)(11,29)(12,28)(13,27)(14,26)(15,25)(32,60)(33,59)(34,58)(35,57)(36,56)(37,55)(38,54)(39,53)(40,52)(41,51)(42,50)(43,49)(44,48)(45,47)(61,86)(62,85)(63,84)(64,83)(65,82)(66,81)(67,80)(68,79)(69,78)(70,77)(71,76)(72,75)(73,74)(87,90)(88,89)(91,112)(92,111)(93,110)(94,109)(95,108)(96,107)(97,106)(98,105)(99,104)(100,103)(101,102)(113,120)(114,119)(115,118)(116,117)>;
G:=Group( (1,34,30,49)(2,50,16,35)(3,36,17,51)(4,52,18,37)(5,38,19,53)(6,54,20,39)(7,40,21,55)(8,56,22,41)(9,42,23,57)(10,58,24,43)(11,44,25,59)(12,60,26,45)(13,46,27,31)(14,32,28,47)(15,48,29,33)(61,76,119,104)(62,105,120,77)(63,78,91,106)(64,107,92,79)(65,80,93,108)(66,109,94,81)(67,82,95,110)(68,111,96,83)(69,84,97,112)(70,113,98,85)(71,86,99,114)(72,115,100,87)(73,88,101,116)(74,117,102,89)(75,90,103,118), (1,97,30,69)(2,113,16,85)(3,99,17,71)(4,115,18,87)(5,101,19,73)(6,117,20,89)(7,103,21,75)(8,119,22,61)(9,105,23,77)(10,91,24,63)(11,107,25,79)(12,93,26,65)(13,109,27,81)(14,95,28,67)(15,111,29,83)(31,94,46,66)(32,82,47,110)(33,96,48,68)(34,84,49,112)(35,98,50,70)(36,86,51,114)(37,100,52,72)(38,88,53,116)(39,102,54,74)(40,90,55,118)(41,104,56,76)(42,62,57,120)(43,106,58,78)(44,64,59,92)(45,108,60,80), (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), (1,24)(2,23)(3,22)(4,21)(5,20)(6,19)(7,18)(8,17)(9,16)(10,30)(11,29)(12,28)(13,27)(14,26)(15,25)(32,60)(33,59)(34,58)(35,57)(36,56)(37,55)(38,54)(39,53)(40,52)(41,51)(42,50)(43,49)(44,48)(45,47)(61,86)(62,85)(63,84)(64,83)(65,82)(66,81)(67,80)(68,79)(69,78)(70,77)(71,76)(72,75)(73,74)(87,90)(88,89)(91,112)(92,111)(93,110)(94,109)(95,108)(96,107)(97,106)(98,105)(99,104)(100,103)(101,102)(113,120)(114,119)(115,118)(116,117) );
G=PermutationGroup([(1,34,30,49),(2,50,16,35),(3,36,17,51),(4,52,18,37),(5,38,19,53),(6,54,20,39),(7,40,21,55),(8,56,22,41),(9,42,23,57),(10,58,24,43),(11,44,25,59),(12,60,26,45),(13,46,27,31),(14,32,28,47),(15,48,29,33),(61,76,119,104),(62,105,120,77),(63,78,91,106),(64,107,92,79),(65,80,93,108),(66,109,94,81),(67,82,95,110),(68,111,96,83),(69,84,97,112),(70,113,98,85),(71,86,99,114),(72,115,100,87),(73,88,101,116),(74,117,102,89),(75,90,103,118)], [(1,97,30,69),(2,113,16,85),(3,99,17,71),(4,115,18,87),(5,101,19,73),(6,117,20,89),(7,103,21,75),(8,119,22,61),(9,105,23,77),(10,91,24,63),(11,107,25,79),(12,93,26,65),(13,109,27,81),(14,95,28,67),(15,111,29,83),(31,94,46,66),(32,82,47,110),(33,96,48,68),(34,84,49,112),(35,98,50,70),(36,86,51,114),(37,100,52,72),(38,88,53,116),(39,102,54,74),(40,90,55,118),(41,104,56,76),(42,62,57,120),(43,106,58,78),(44,64,59,92),(45,108,60,80)], [(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)], [(1,24),(2,23),(3,22),(4,21),(5,20),(6,19),(7,18),(8,17),(9,16),(10,30),(11,29),(12,28),(13,27),(14,26),(15,25),(32,60),(33,59),(34,58),(35,57),(36,56),(37,55),(38,54),(39,53),(40,52),(41,51),(42,50),(43,49),(44,48),(45,47),(61,86),(62,85),(63,84),(64,83),(65,82),(66,81),(67,80),(68,79),(69,78),(70,77),(71,76),(72,75),(73,74),(87,90),(88,89),(91,112),(92,111),(93,110),(94,109),(95,108),(96,107),(97,106),(98,105),(99,104),(100,103),(101,102),(113,120),(114,119),(115,118),(116,117)])
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 5A | 5B | 6A | 6B | 8A | 8B | 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 | 5 | 5 | 6 | 6 | 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 | 30 | 60 | 60 | 2 | 2 | 4 | 30 | 2 | 2 | 2 | 8 | 4 | 60 | 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 |
60 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 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D5 | D6 | D6 | D6 | D10 | D10 | D10 | D15 | D30 | D30 | D30 | C8⋊C22 | S3×D4 | D4×D5 | Q8⋊3D6 | D40⋊C2 | D4×D15 | Q8⋊3D30 |
| kernel | Q8⋊3D30 | C40⋊S3 | D120 | D4⋊D15 | Q8⋊2D15 | C15×SD16 | D4×D15 | Q8⋊3D15 | C5×SD16 | Dic15 | D30 | C3×SD16 | C40 | C5×D4 | C5×Q8 | C24 | C3×D4 | C3×Q8 | SD16 | C8 | D4 | Q8 | C15 | C10 | C6 | C5 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 1 | 1 | 2 | 2 | 4 | 4 | 8 |
Matrix representation of Q8⋊3D30 ►in GL4(𝔽241) generated by
| 1 | 0 | 0 | 103 |
| 0 | 1 | 138 | 138 |
| 234 | 234 | 240 | 0 |
| 7 | 0 | 0 | 240 |
| 0 | 0 | 25 | 206 |
| 0 | 0 | 181 | 60 |
| 217 | 227 | 0 | 0 |
| 217 | 231 | 0 | 0 |
| 64 | 148 | 180 | 24 |
| 109 | 173 | 15 | 156 |
| 0 | 0 | 161 | 93 |
| 0 | 0 | 148 | 84 |
| 30 | 110 | 238 | 195 |
| 143 | 211 | 198 | 226 |
| 0 | 0 | 161 | 93 |
| 0 | 0 | 110 | 80 |
G:=sub<GL(4,GF(241))| [1,0,234,7,0,1,234,0,0,138,240,0,103,138,0,240],[0,0,217,217,0,0,227,231,25,181,0,0,206,60,0,0],[64,109,0,0,148,173,0,0,180,15,161,148,24,156,93,84],[30,143,0,0,110,211,0,0,238,198,161,110,195,226,93,80] >;
Q8⋊3D30 in GAP, Magma, Sage, TeX
Q_8\rtimes_3D_{30} % in TeX
G:=Group("Q8:3D30"); // GroupNames label
G:=SmallGroup(480,879);
// by ID
G=gap.SmallGroup(480,879);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,422,135,100,346,185,80,2693,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^4=c^30=d^2=1,b^2=a^2,b*a*b^-1=c*a*c^-1=d*a*d=a^-1,c*b*c^-1=a^-1*b,d*b*d=a*b,d*c*d=c^-1>;
// generators/relations