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, C3, C4, C4, C22, C5, S3, C6, C6, C8, C8, C2×C4, D4, D4, Q8, C23, D5, C10, C10, Dic3, C12, C12, D6, C2×C6, C15, M4(2), D8, SD16, SD16, C2×D4, C4○D4, Dic5, C20, C20, D10, C2×C10, C3⋊C8, C24, C4×S3, D12, C3⋊D4, C3×D4, C3×Q8, C22×S3, D15, C30, C30, C8⋊C22, C5⋊2C8, C40, C4×D5, D20, 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, 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, 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, C22, S3, D4, C23, D5, D6, C2×D4, D10, C22×S3, D15, C8⋊C22, C22×D5, S3×D4, D30, D4×D5, Q8⋊3D6, C22×D15, D40⋊C2, D4×D15, Q8⋊3D30
►On 120 points
Generators in S120
(1 114 17 99)(2 100 18 115)(3 116 19 101)(4 102 20 117)(5 118 21 103)(6 104 22 119)(7 120 23 105)(8 106 24 91)(9 92 25 107)(10 108 26 93)(11 94 27 109)(12 110 28 95)(13 96 29 111)(14 112 30 97)(15 98 16 113)(31 78 63 46)(32 47 64 79)(33 80 65 48)(34 49 66 81)(35 82 67 50)(36 51 68 83)(37 84 69 52)(38 53 70 85)(39 86 71 54)(40 55 72 87)(41 88 73 56)(42 57 74 89)(43 90 75 58)(44 59 76 61)(45 62 77 60)
(1 68 17 36)(2 84 18 52)(3 70 19 38)(4 86 20 54)(5 72 21 40)(6 88 22 56)(7 74 23 42)(8 90 24 58)(9 76 25 44)(10 62 26 60)(11 78 27 46)(12 64 28 32)(13 80 29 48)(14 66 30 34)(15 82 16 50)(31 109 63 94)(33 111 65 96)(35 113 67 98)(37 115 69 100)(39 117 71 102)(41 119 73 104)(43 91 75 106)(45 93 77 108)(47 95 79 110)(49 97 81 112)(51 99 83 114)(53 101 85 116)(55 103 87 118)(57 105 89 120)(59 107 61 92)
(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 17)(2 16)(3 30)(4 29)(5 28)(6 27)(7 26)(8 25)(9 24)(10 23)(11 22)(12 21)(13 20)(14 19)(15 18)(31 56)(32 55)(33 54)(34 53)(35 52)(36 51)(37 50)(38 49)(39 48)(40 47)(41 46)(42 45)(43 44)(57 60)(58 59)(61 90)(62 89)(63 88)(64 87)(65 86)(66 85)(67 84)(68 83)(69 82)(70 81)(71 80)(72 79)(73 78)(74 77)(75 76)(91 107)(92 106)(93 105)(94 104)(95 103)(96 102)(97 101)(98 100)(108 120)(109 119)(110 118)(111 117)(112 116)(113 115)
G:=sub<Sym(120)| (1,114,17,99)(2,100,18,115)(3,116,19,101)(4,102,20,117)(5,118,21,103)(6,104,22,119)(7,120,23,105)(8,106,24,91)(9,92,25,107)(10,108,26,93)(11,94,27,109)(12,110,28,95)(13,96,29,111)(14,112,30,97)(15,98,16,113)(31,78,63,46)(32,47,64,79)(33,80,65,48)(34,49,66,81)(35,82,67,50)(36,51,68,83)(37,84,69,52)(38,53,70,85)(39,86,71,54)(40,55,72,87)(41,88,73,56)(42,57,74,89)(43,90,75,58)(44,59,76,61)(45,62,77,60), (1,68,17,36)(2,84,18,52)(3,70,19,38)(4,86,20,54)(5,72,21,40)(6,88,22,56)(7,74,23,42)(8,90,24,58)(9,76,25,44)(10,62,26,60)(11,78,27,46)(12,64,28,32)(13,80,29,48)(14,66,30,34)(15,82,16,50)(31,109,63,94)(33,111,65,96)(35,113,67,98)(37,115,69,100)(39,117,71,102)(41,119,73,104)(43,91,75,106)(45,93,77,108)(47,95,79,110)(49,97,81,112)(51,99,83,114)(53,101,85,116)(55,103,87,118)(57,105,89,120)(59,107,61,92), (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), (1,17)(2,16)(3,30)(4,29)(5,28)(6,27)(7,26)(8,25)(9,24)(10,23)(11,22)(12,21)(13,20)(14,19)(15,18)(31,56)(32,55)(33,54)(34,53)(35,52)(36,51)(37,50)(38,49)(39,48)(40,47)(41,46)(42,45)(43,44)(57,60)(58,59)(61,90)(62,89)(63,88)(64,87)(65,86)(66,85)(67,84)(68,83)(69,82)(70,81)(71,80)(72,79)(73,78)(74,77)(75,76)(91,107)(92,106)(93,105)(94,104)(95,103)(96,102)(97,101)(98,100)(108,120)(109,119)(110,118)(111,117)(112,116)(113,115)>;
G:=Group( (1,114,17,99)(2,100,18,115)(3,116,19,101)(4,102,20,117)(5,118,21,103)(6,104,22,119)(7,120,23,105)(8,106,24,91)(9,92,25,107)(10,108,26,93)(11,94,27,109)(12,110,28,95)(13,96,29,111)(14,112,30,97)(15,98,16,113)(31,78,63,46)(32,47,64,79)(33,80,65,48)(34,49,66,81)(35,82,67,50)(36,51,68,83)(37,84,69,52)(38,53,70,85)(39,86,71,54)(40,55,72,87)(41,88,73,56)(42,57,74,89)(43,90,75,58)(44,59,76,61)(45,62,77,60), (1,68,17,36)(2,84,18,52)(3,70,19,38)(4,86,20,54)(5,72,21,40)(6,88,22,56)(7,74,23,42)(8,90,24,58)(9,76,25,44)(10,62,26,60)(11,78,27,46)(12,64,28,32)(13,80,29,48)(14,66,30,34)(15,82,16,50)(31,109,63,94)(33,111,65,96)(35,113,67,98)(37,115,69,100)(39,117,71,102)(41,119,73,104)(43,91,75,106)(45,93,77,108)(47,95,79,110)(49,97,81,112)(51,99,83,114)(53,101,85,116)(55,103,87,118)(57,105,89,120)(59,107,61,92), (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), (1,17)(2,16)(3,30)(4,29)(5,28)(6,27)(7,26)(8,25)(9,24)(10,23)(11,22)(12,21)(13,20)(14,19)(15,18)(31,56)(32,55)(33,54)(34,53)(35,52)(36,51)(37,50)(38,49)(39,48)(40,47)(41,46)(42,45)(43,44)(57,60)(58,59)(61,90)(62,89)(63,88)(64,87)(65,86)(66,85)(67,84)(68,83)(69,82)(70,81)(71,80)(72,79)(73,78)(74,77)(75,76)(91,107)(92,106)(93,105)(94,104)(95,103)(96,102)(97,101)(98,100)(108,120)(109,119)(110,118)(111,117)(112,116)(113,115) );
G=PermutationGroup([[(1,114,17,99),(2,100,18,115),(3,116,19,101),(4,102,20,117),(5,118,21,103),(6,104,22,119),(7,120,23,105),(8,106,24,91),(9,92,25,107),(10,108,26,93),(11,94,27,109),(12,110,28,95),(13,96,29,111),(14,112,30,97),(15,98,16,113),(31,78,63,46),(32,47,64,79),(33,80,65,48),(34,49,66,81),(35,82,67,50),(36,51,68,83),(37,84,69,52),(38,53,70,85),(39,86,71,54),(40,55,72,87),(41,88,73,56),(42,57,74,89),(43,90,75,58),(44,59,76,61),(45,62,77,60)], [(1,68,17,36),(2,84,18,52),(3,70,19,38),(4,86,20,54),(5,72,21,40),(6,88,22,56),(7,74,23,42),(8,90,24,58),(9,76,25,44),(10,62,26,60),(11,78,27,46),(12,64,28,32),(13,80,29,48),(14,66,30,34),(15,82,16,50),(31,109,63,94),(33,111,65,96),(35,113,67,98),(37,115,69,100),(39,117,71,102),(41,119,73,104),(43,91,75,106),(45,93,77,108),(47,95,79,110),(49,97,81,112),(51,99,83,114),(53,101,85,116),(55,103,87,118),(57,105,89,120),(59,107,61,92)], [(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)], [(1,17),(2,16),(3,30),(4,29),(5,28),(6,27),(7,26),(8,25),(9,24),(10,23),(11,22),(12,21),(13,20),(14,19),(15,18),(31,56),(32,55),(33,54),(34,53),(35,52),(36,51),(37,50),(38,49),(39,48),(40,47),(41,46),(42,45),(43,44),(57,60),(58,59),(61,90),(62,89),(63,88),(64,87),(65,86),(66,85),(67,84),(68,83),(69,82),(70,81),(71,80),(72,79),(73,78),(74,77),(75,76),(91,107),(92,106),(93,105),(94,104),(95,103),(96,102),(97,101),(98,100),(108,120),(109,119),(110,118),(111,117),(112,116),(113,115)]])
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