metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4⋊4D30, Q8⋊5D30, C60.206D4, C60.80C23, D60.40C22, C4○D4⋊3D15, (C5×D4)⋊19D6, (C5×Q8)⋊20D6, (C2×C30).8D4, D4⋊D15⋊14C2, (C3×D4)⋊19D10, (C2×D60)⋊13C2, C5⋊5(D4⋊D6), (C3×Q8)⋊17D10, (C2×C4).21D30, C3⋊5(D4⋊D10), C15⋊37(C8⋊C22), C30.391(C2×D4), (C2×C20).158D6, C60.7C4⋊21C2, C15⋊3C8⋊18C22, Q8⋊2D15⋊14C2, (C2×C12).156D10, (D4×C15)⋊21C22, C4.24(C15⋊7D4), (C2×C60).83C22, (Q8×C15)⋊19C22, C4.17(C22×D15), C12.103(C5⋊D4), C20.103(C3⋊D4), C20.118(C22×S3), C12.118(C22×D5), C22.5(C15⋊7D4), (C5×C4○D4)⋊5S3, (C3×C4○D4)⋊1D5, (C15×C4○D4)⋊1C2, C2.23(C2×C15⋊7D4), C6.118(C2×C5⋊D4), C10.118(C2×C3⋊D4), (C2×C6).20(C5⋊D4), (C2×C10).19(C3⋊D4), SmallGroup(480,914)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q8⋊5D30
G = < a,b,c,d | a4=c30=d2=1, b2=a2, bab-1=dad=a-1, ac=ca, cbc-1=a2b, dbd=ab, dcd=c-1 >
Subgroups: 932 in 136 conjugacy classes, 47 normal (39 characteristic)
C1, C2, C2 [×4], C3, C4 [×2], C4, C22, C22 [×5], C5, S3 [×2], C6, C6 [×2], C8 [×2], C2×C4, C2×C4, D4, D4 [×4], Q8, C23, D5 [×2], C10, C10 [×2], C12 [×2], C12, D6 [×4], C2×C6, C2×C6, C15, M4(2), D8 [×2], SD16 [×2], C2×D4, C4○D4, C20 [×2], C20, D10 [×4], C2×C10, C2×C10, C3⋊C8 [×2], D12 [×3], C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C22×S3, D15 [×2], C30, C30 [×2], C8⋊C22, C5⋊2C8 [×2], D20 [×3], C2×C20, C2×C20, C5×D4, C5×D4, C5×Q8, C22×D5, C4.Dic3, D4⋊S3 [×2], Q8⋊2S3 [×2], C2×D12, C3×C4○D4, C60 [×2], C60, D30 [×4], C2×C30, C2×C30, C4.Dic5, D4⋊D5 [×2], Q8⋊D5 [×2], C2×D20, C5×C4○D4, D4⋊D6, C15⋊3C8 [×2], D60 [×2], D60, C2×C60, C2×C60, D4×C15, D4×C15, Q8×C15, C22×D15, D4⋊D10, C60.7C4, D4⋊D15 [×2], Q8⋊2D15 [×2], C2×D60, C15×C4○D4, Q8⋊5D30
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], C2×D4, D10 [×3], C3⋊D4 [×2], C22×S3, D15, C8⋊C22, C5⋊D4 [×2], C22×D5, C2×C3⋊D4, D30 [×3], C2×C5⋊D4, D4⋊D6, C15⋊7D4 [×2], C22×D15, D4⋊D10, C2×C15⋊7D4, Q8⋊5D30
►On 120 points
Generators in S120
(1 56 16 41)(2 57 17 42)(3 58 18 43)(4 59 19 44)(5 60 20 45)(6 46 21 31)(7 47 22 32)(8 48 23 33)(9 49 24 34)(10 50 25 35)(11 51 26 36)(12 52 27 37)(13 53 28 38)(14 54 29 39)(15 55 30 40)(61 96 76 111)(62 97 77 112)(63 98 78 113)(64 99 79 114)(65 100 80 115)(66 101 81 116)(67 102 82 117)(68 103 83 118)(69 104 84 119)(70 105 85 120)(71 106 86 91)(72 107 87 92)(73 108 88 93)(74 109 89 94)(75 110 90 95)
(1 61 16 76)(2 77 17 62)(3 63 18 78)(4 79 19 64)(5 65 20 80)(6 81 21 66)(7 67 22 82)(8 83 23 68)(9 69 24 84)(10 85 25 70)(11 71 26 86)(12 87 27 72)(13 73 28 88)(14 89 29 74)(15 75 30 90)(31 116 46 101)(32 102 47 117)(33 118 48 103)(34 104 49 119)(35 120 50 105)(36 106 51 91)(37 92 52 107)(38 108 53 93)(39 94 54 109)(40 110 55 95)(41 96 56 111)(42 112 57 97)(43 98 58 113)(44 114 59 99)(45 100 60 115)
(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 5)(2 4)(6 15)(7 14)(8 13)(9 12)(10 11)(16 20)(17 19)(21 30)(22 29)(23 28)(24 27)(25 26)(31 55)(32 54)(33 53)(34 52)(35 51)(36 50)(37 49)(38 48)(39 47)(40 46)(41 60)(42 59)(43 58)(44 57)(45 56)(61 115)(62 114)(63 113)(64 112)(65 111)(66 110)(67 109)(68 108)(69 107)(70 106)(71 105)(72 104)(73 103)(74 102)(75 101)(76 100)(77 99)(78 98)(79 97)(80 96)(81 95)(82 94)(83 93)(84 92)(85 91)(86 120)(87 119)(88 118)(89 117)(90 116)
G:=sub<Sym(120)| (1,56,16,41)(2,57,17,42)(3,58,18,43)(4,59,19,44)(5,60,20,45)(6,46,21,31)(7,47,22,32)(8,48,23,33)(9,49,24,34)(10,50,25,35)(11,51,26,36)(12,52,27,37)(13,53,28,38)(14,54,29,39)(15,55,30,40)(61,96,76,111)(62,97,77,112)(63,98,78,113)(64,99,79,114)(65,100,80,115)(66,101,81,116)(67,102,82,117)(68,103,83,118)(69,104,84,119)(70,105,85,120)(71,106,86,91)(72,107,87,92)(73,108,88,93)(74,109,89,94)(75,110,90,95), (1,61,16,76)(2,77,17,62)(3,63,18,78)(4,79,19,64)(5,65,20,80)(6,81,21,66)(7,67,22,82)(8,83,23,68)(9,69,24,84)(10,85,25,70)(11,71,26,86)(12,87,27,72)(13,73,28,88)(14,89,29,74)(15,75,30,90)(31,116,46,101)(32,102,47,117)(33,118,48,103)(34,104,49,119)(35,120,50,105)(36,106,51,91)(37,92,52,107)(38,108,53,93)(39,94,54,109)(40,110,55,95)(41,96,56,111)(42,112,57,97)(43,98,58,113)(44,114,59,99)(45,100,60,115), (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), (1,5)(2,4)(6,15)(7,14)(8,13)(9,12)(10,11)(16,20)(17,19)(21,30)(22,29)(23,28)(24,27)(25,26)(31,55)(32,54)(33,53)(34,52)(35,51)(36,50)(37,49)(38,48)(39,47)(40,46)(41,60)(42,59)(43,58)(44,57)(45,56)(61,115)(62,114)(63,113)(64,112)(65,111)(66,110)(67,109)(68,108)(69,107)(70,106)(71,105)(72,104)(73,103)(74,102)(75,101)(76,100)(77,99)(78,98)(79,97)(80,96)(81,95)(82,94)(83,93)(84,92)(85,91)(86,120)(87,119)(88,118)(89,117)(90,116)>;
G:=Group( (1,56,16,41)(2,57,17,42)(3,58,18,43)(4,59,19,44)(5,60,20,45)(6,46,21,31)(7,47,22,32)(8,48,23,33)(9,49,24,34)(10,50,25,35)(11,51,26,36)(12,52,27,37)(13,53,28,38)(14,54,29,39)(15,55,30,40)(61,96,76,111)(62,97,77,112)(63,98,78,113)(64,99,79,114)(65,100,80,115)(66,101,81,116)(67,102,82,117)(68,103,83,118)(69,104,84,119)(70,105,85,120)(71,106,86,91)(72,107,87,92)(73,108,88,93)(74,109,89,94)(75,110,90,95), (1,61,16,76)(2,77,17,62)(3,63,18,78)(4,79,19,64)(5,65,20,80)(6,81,21,66)(7,67,22,82)(8,83,23,68)(9,69,24,84)(10,85,25,70)(11,71,26,86)(12,87,27,72)(13,73,28,88)(14,89,29,74)(15,75,30,90)(31,116,46,101)(32,102,47,117)(33,118,48,103)(34,104,49,119)(35,120,50,105)(36,106,51,91)(37,92,52,107)(38,108,53,93)(39,94,54,109)(40,110,55,95)(41,96,56,111)(42,112,57,97)(43,98,58,113)(44,114,59,99)(45,100,60,115), (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), (1,5)(2,4)(6,15)(7,14)(8,13)(9,12)(10,11)(16,20)(17,19)(21,30)(22,29)(23,28)(24,27)(25,26)(31,55)(32,54)(33,53)(34,52)(35,51)(36,50)(37,49)(38,48)(39,47)(40,46)(41,60)(42,59)(43,58)(44,57)(45,56)(61,115)(62,114)(63,113)(64,112)(65,111)(66,110)(67,109)(68,108)(69,107)(70,106)(71,105)(72,104)(73,103)(74,102)(75,101)(76,100)(77,99)(78,98)(79,97)(80,96)(81,95)(82,94)(83,93)(84,92)(85,91)(86,120)(87,119)(88,118)(89,117)(90,116) );
G=PermutationGroup([(1,56,16,41),(2,57,17,42),(3,58,18,43),(4,59,19,44),(5,60,20,45),(6,46,21,31),(7,47,22,32),(8,48,23,33),(9,49,24,34),(10,50,25,35),(11,51,26,36),(12,52,27,37),(13,53,28,38),(14,54,29,39),(15,55,30,40),(61,96,76,111),(62,97,77,112),(63,98,78,113),(64,99,79,114),(65,100,80,115),(66,101,81,116),(67,102,82,117),(68,103,83,118),(69,104,84,119),(70,105,85,120),(71,106,86,91),(72,107,87,92),(73,108,88,93),(74,109,89,94),(75,110,90,95)], [(1,61,16,76),(2,77,17,62),(3,63,18,78),(4,79,19,64),(5,65,20,80),(6,81,21,66),(7,67,22,82),(8,83,23,68),(9,69,24,84),(10,85,25,70),(11,71,26,86),(12,87,27,72),(13,73,28,88),(14,89,29,74),(15,75,30,90),(31,116,46,101),(32,102,47,117),(33,118,48,103),(34,104,49,119),(35,120,50,105),(36,106,51,91),(37,92,52,107),(38,108,53,93),(39,94,54,109),(40,110,55,95),(41,96,56,111),(42,112,57,97),(43,98,58,113),(44,114,59,99),(45,100,60,115)], [(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)], [(1,5),(2,4),(6,15),(7,14),(8,13),(9,12),(10,11),(16,20),(17,19),(21,30),(22,29),(23,28),(24,27),(25,26),(31,55),(32,54),(33,53),(34,52),(35,51),(36,50),(37,49),(38,48),(39,47),(40,46),(41,60),(42,59),(43,58),(44,57),(45,56),(61,115),(62,114),(63,113),(64,112),(65,111),(66,110),(67,109),(68,108),(69,107),(70,106),(71,105),(72,104),(73,103),(74,102),(75,101),(76,100),(77,99),(78,98),(79,97),(80,96),(81,95),(82,94),(83,93),(84,92),(85,91),(86,120),(87,119),(88,118),(89,117),(90,116)])
81 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 5A | 5B | 6A | 6B | 6C | 6D | 8A | 8B | 10A | 10B | 10C | ··· | 10H | 12A | 12B | 12C | 12D | 12E | 15A | 15B | 15C | 15D | 20A | 20B | 20C | 20D | 20E | ··· | 20J | 30A | 30B | 30C | 30D | 30E | ··· | 30P | 60A | ··· | 60H | 60I | ··· | 60T |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 8 | 8 | 10 | 10 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 30 | 30 | 30 | 30 | 30 | ··· | 30 | 60 | ··· | 60 | 60 | ··· | 60 |
| size | 1 | 1 | 2 | 4 | 60 | 60 | 2 | 2 | 2 | 4 | 2 | 2 | 2 | 4 | 4 | 4 | 60 | 60 | 2 | 2 | 4 | ··· | 4 | 2 | 2 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
81 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D5 | D6 | D6 | D6 | D10 | D10 | D10 | C3⋊D4 | C3⋊D4 | D15 | C5⋊D4 | C5⋊D4 | D30 | D30 | D30 | C15⋊7D4 | C15⋊7D4 | C8⋊C22 | D4⋊D6 | D4⋊D10 | Q8⋊5D30 |
| kernel | Q8⋊5D30 | C60.7C4 | D4⋊D15 | Q8⋊2D15 | C2×D60 | C15×C4○D4 | C5×C4○D4 | C60 | C2×C30 | C3×C4○D4 | C2×C20 | C5×D4 | C5×Q8 | C2×C12 | C3×D4 | C3×Q8 | C20 | C2×C10 | C4○D4 | C12 | C2×C6 | C2×C4 | D4 | Q8 | C4 | C22 | C15 | C5 | C3 | C1 |
| # reps | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 1 | 2 | 4 | 8 |
Matrix representation of Q8⋊5D30 ►in GL6(𝔽241)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 200 | 156 | 0 | 0 |
| 0 | 0 | 85 | 41 | 0 | 0 |
| 0 | 0 | 0 | 0 | 41 | 85 |
| 0 | 0 | 0 | 0 | 156 | 200 |
| 240 | 0 | 0 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 240 | 0 |
| 0 | 0 | 0 | 0 | 0 | 240 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 1 | 240 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 189 | 240 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 52 | 1 |
| 0 | 0 | 0 | 0 | 240 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 1 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 189 | 240 | 0 | 0 |
| 0 | 0 | 52 | 52 | 0 | 0 |
| 0 | 0 | 0 | 0 | 122 | 200 |
| 0 | 0 | 0 | 0 | 122 | 119 |
G:=sub<GL(6,GF(241))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,200,85,0,0,0,0,156,41,0,0,0,0,0,0,41,156,0,0,0,0,85,200],[240,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,240,0,0,0,0,0,0,240,0,0],[1,1,0,0,0,0,240,0,0,0,0,0,0,0,189,1,0,0,0,0,240,0,0,0,0,0,0,0,52,240,0,0,0,0,1,0],[1,1,0,0,0,0,0,240,0,0,0,0,0,0,189,52,0,0,0,0,240,52,0,0,0,0,0,0,122,122,0,0,0,0,200,119] >;
Q8⋊5D30 in GAP, Magma, Sage, TeX
Q_8\rtimes_5D_{30} % in TeX
G:=Group("Q8:5D30"); // GroupNames label
G:=SmallGroup(480,914);
// by ID
G=gap.SmallGroup(480,914);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,254,219,675,185,80,2693,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^4=c^30=d^2=1,b^2=a^2,b*a*b^-1=d*a*d=a^-1,a*c=c*a,c*b*c^-1=a^2*b,d*b*d=a*b,d*c*d=c^-1>;
// generators/relations