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, (C2×D60)⋊13C2, (C3×D4)⋊19D10, C5⋊5(D4⋊D6), (C3×Q8)⋊17D10, (C2×C4).21D30, C3⋊5(D4⋊D10), C15⋊37(C8⋊C22), (C2×C20).158D6, C30.391(C2×D4), 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), C20.103(C3⋊D4), C12.103(C5⋊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, C6.118(C2×C5⋊D4), C2.23(C2×C15⋊7D4), (C2×C6).20(C5⋊D4), C10.118(C2×C3⋊D4), (C2×C10).19(C3⋊D4), SmallGroup(480,914)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4⋊D30
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, C3, C4, C4, C22, C22, C5, S3, C6, C6, C8, C2×C4, C2×C4, D4, D4, Q8, C23, D5, C10, C10, C12, C12, D6, C2×C6, C2×C6, C15, M4(2), D8, SD16, C2×D4, C4○D4, C20, C20, D10, C2×C10, C2×C10, C3⋊C8, D12, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C22×S3, D15, C30, C30, C8⋊C22, C5⋊2C8, D20, C2×C20, C2×C20, C5×D4, C5×D4, C5×Q8, C22×D5, C4.Dic3, D4⋊S3, Q8⋊2S3, C2×D12, C3×C4○D4, C60, C60, D30, C2×C30, C2×C30, C4.Dic5, D4⋊D5, Q8⋊D5, C2×D20, C5×C4○D4, D4⋊D6, C15⋊3C8, D60, D60, C2×C60, C2×C60, D4×C15, D4×C15, Q8×C15, C22×D15, D4⋊D10, C60.7C4, D4⋊D15, Q8⋊2D15, C2×D60, C15×C4○D4, D4⋊D30
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, D10, C3⋊D4, C22×S3, D15, C8⋊C22, C5⋊D4, C22×D5, C2×C3⋊D4, D30, C2×C5⋊D4, D4⋊D6, C15⋊7D4, C22×D15, D4⋊D10, C2×C15⋊7D4, D4⋊D30
►On 120 points
Generators in S120
(1 51 21 36)(2 52 22 37)(3 53 23 38)(4 54 24 39)(5 55 25 40)(6 56 26 41)(7 57 27 42)(8 58 28 43)(9 59 29 44)(10 60 30 45)(11 46 16 31)(12 47 17 32)(13 48 18 33)(14 49 19 34)(15 50 20 35)(61 91 76 106)(62 92 77 107)(63 93 78 108)(64 94 79 109)(65 95 80 110)(66 96 81 111)(67 97 82 112)(68 98 83 113)(69 99 84 114)(70 100 85 115)(71 101 86 116)(72 102 87 117)(73 103 88 118)(74 104 89 119)(75 105 90 120)
(1 61 21 76)(2 77 22 62)(3 63 23 78)(4 79 24 64)(5 65 25 80)(6 81 26 66)(7 67 27 82)(8 83 28 68)(9 69 29 84)(10 85 30 70)(11 71 16 86)(12 87 17 72)(13 73 18 88)(14 89 19 74)(15 75 20 90)(31 101 46 116)(32 117 47 102)(33 103 48 118)(34 119 49 104)(35 105 50 120)(36 91 51 106)(37 107 52 92)(38 93 53 108)(39 109 54 94)(40 95 55 110)(41 111 56 96)(42 97 57 112)(43 113 58 98)(44 99 59 114)(45 115 60 100)
(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 30)(17 29)(18 28)(19 27)(20 26)(21 25)(22 24)(31 60)(32 59)(33 58)(34 57)(35 56)(36 55)(37 54)(38 53)(39 52)(40 51)(41 50)(42 49)(43 48)(44 47)(45 46)(61 110)(62 109)(63 108)(64 107)(65 106)(66 105)(67 104)(68 103)(69 102)(70 101)(71 100)(72 99)(73 98)(74 97)(75 96)(76 95)(77 94)(78 93)(79 92)(80 91)(81 120)(82 119)(83 118)(84 117)(85 116)(86 115)(87 114)(88 113)(89 112)(90 111)
G:=sub<Sym(120)| (1,51,21,36)(2,52,22,37)(3,53,23,38)(4,54,24,39)(5,55,25,40)(6,56,26,41)(7,57,27,42)(8,58,28,43)(9,59,29,44)(10,60,30,45)(11,46,16,31)(12,47,17,32)(13,48,18,33)(14,49,19,34)(15,50,20,35)(61,91,76,106)(62,92,77,107)(63,93,78,108)(64,94,79,109)(65,95,80,110)(66,96,81,111)(67,97,82,112)(68,98,83,113)(69,99,84,114)(70,100,85,115)(71,101,86,116)(72,102,87,117)(73,103,88,118)(74,104,89,119)(75,105,90,120), (1,61,21,76)(2,77,22,62)(3,63,23,78)(4,79,24,64)(5,65,25,80)(6,81,26,66)(7,67,27,82)(8,83,28,68)(9,69,29,84)(10,85,30,70)(11,71,16,86)(12,87,17,72)(13,73,18,88)(14,89,19,74)(15,75,20,90)(31,101,46,116)(32,117,47,102)(33,103,48,118)(34,119,49,104)(35,105,50,120)(36,91,51,106)(37,107,52,92)(38,93,53,108)(39,109,54,94)(40,95,55,110)(41,111,56,96)(42,97,57,112)(43,113,58,98)(44,99,59,114)(45,115,60,100), (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)(16,30)(17,29)(18,28)(19,27)(20,26)(21,25)(22,24)(31,60)(32,59)(33,58)(34,57)(35,56)(36,55)(37,54)(38,53)(39,52)(40,51)(41,50)(42,49)(43,48)(44,47)(45,46)(61,110)(62,109)(63,108)(64,107)(65,106)(66,105)(67,104)(68,103)(69,102)(70,101)(71,100)(72,99)(73,98)(74,97)(75,96)(76,95)(77,94)(78,93)(79,92)(80,91)(81,120)(82,119)(83,118)(84,117)(85,116)(86,115)(87,114)(88,113)(89,112)(90,111)>;
G:=Group( (1,51,21,36)(2,52,22,37)(3,53,23,38)(4,54,24,39)(5,55,25,40)(6,56,26,41)(7,57,27,42)(8,58,28,43)(9,59,29,44)(10,60,30,45)(11,46,16,31)(12,47,17,32)(13,48,18,33)(14,49,19,34)(15,50,20,35)(61,91,76,106)(62,92,77,107)(63,93,78,108)(64,94,79,109)(65,95,80,110)(66,96,81,111)(67,97,82,112)(68,98,83,113)(69,99,84,114)(70,100,85,115)(71,101,86,116)(72,102,87,117)(73,103,88,118)(74,104,89,119)(75,105,90,120), (1,61,21,76)(2,77,22,62)(3,63,23,78)(4,79,24,64)(5,65,25,80)(6,81,26,66)(7,67,27,82)(8,83,28,68)(9,69,29,84)(10,85,30,70)(11,71,16,86)(12,87,17,72)(13,73,18,88)(14,89,19,74)(15,75,20,90)(31,101,46,116)(32,117,47,102)(33,103,48,118)(34,119,49,104)(35,105,50,120)(36,91,51,106)(37,107,52,92)(38,93,53,108)(39,109,54,94)(40,95,55,110)(41,111,56,96)(42,97,57,112)(43,113,58,98)(44,99,59,114)(45,115,60,100), (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)(16,30)(17,29)(18,28)(19,27)(20,26)(21,25)(22,24)(31,60)(32,59)(33,58)(34,57)(35,56)(36,55)(37,54)(38,53)(39,52)(40,51)(41,50)(42,49)(43,48)(44,47)(45,46)(61,110)(62,109)(63,108)(64,107)(65,106)(66,105)(67,104)(68,103)(69,102)(70,101)(71,100)(72,99)(73,98)(74,97)(75,96)(76,95)(77,94)(78,93)(79,92)(80,91)(81,120)(82,119)(83,118)(84,117)(85,116)(86,115)(87,114)(88,113)(89,112)(90,111) );
G=PermutationGroup([[(1,51,21,36),(2,52,22,37),(3,53,23,38),(4,54,24,39),(5,55,25,40),(6,56,26,41),(7,57,27,42),(8,58,28,43),(9,59,29,44),(10,60,30,45),(11,46,16,31),(12,47,17,32),(13,48,18,33),(14,49,19,34),(15,50,20,35),(61,91,76,106),(62,92,77,107),(63,93,78,108),(64,94,79,109),(65,95,80,110),(66,96,81,111),(67,97,82,112),(68,98,83,113),(69,99,84,114),(70,100,85,115),(71,101,86,116),(72,102,87,117),(73,103,88,118),(74,104,89,119),(75,105,90,120)], [(1,61,21,76),(2,77,22,62),(3,63,23,78),(4,79,24,64),(5,65,25,80),(6,81,26,66),(7,67,27,82),(8,83,28,68),(9,69,29,84),(10,85,30,70),(11,71,16,86),(12,87,17,72),(13,73,18,88),(14,89,19,74),(15,75,20,90),(31,101,46,116),(32,117,47,102),(33,103,48,118),(34,119,49,104),(35,105,50,120),(36,91,51,106),(37,107,52,92),(38,93,53,108),(39,109,54,94),(40,95,55,110),(41,111,56,96),(42,97,57,112),(43,113,58,98),(44,99,59,114),(45,115,60,100)], [(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),(16,30),(17,29),(18,28),(19,27),(20,26),(21,25),(22,24),(31,60),(32,59),(33,58),(34,57),(35,56),(36,55),(37,54),(38,53),(39,52),(40,51),(41,50),(42,49),(43,48),(44,47),(45,46),(61,110),(62,109),(63,108),(64,107),(65,106),(66,105),(67,104),(68,103),(69,102),(70,101),(71,100),(72,99),(73,98),(74,97),(75,96),(76,95),(77,94),(78,93),(79,92),(80,91),(81,120),(82,119),(83,118),(84,117),(85,116),(86,115),(87,114),(88,113),(89,112),(90,111)]])
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 | D4⋊D30 |
| kernel | D4⋊D30 | 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 D4⋊D30 ►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] >;
D4⋊D30 in GAP, Magma, Sage, TeX
D_4\rtimes D_{30} % in TeX
G:=Group("D4:D30"); // 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