metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D30⋊17D4, C23⋊2D30, (C2×C4)⋊2D30, (C2×C30)⋊2D4, (C6×D4)⋊14D5, (C2×D4)⋊3D15, (C2×C20)⋊21D6, C15⋊14C22≀C2, (D4×C30)⋊26C2, (D4×C10)⋊14S3, (C2×C12)⋊21D10, C5⋊4(C23⋊2D6), C2.25(D4×D15), C6.119(D4×D5), (C22×C6)⋊4D10, (C22×C10)⋊7D6, C3⋊4(C23⋊D10), (C2×C60)⋊37C22, C30.326(C2×D4), C10.121(S3×D4), (C23×D15)⋊2C2, D30⋊3C4⋊36C2, C22⋊3(C15⋊7D4), (C22×C30)⋊3C22, C30.38D4⋊10C2, (C2×C30).308C23, (C2×Dic15)⋊2C22, C22.59(C22×D15), (C22×D15).87C22, (C2×C15⋊7D4)⋊4C2, (C2×C6)⋊5(C5⋊D4), (C2×C10)⋊9(C3⋊D4), C6.108(C2×C5⋊D4), C2.13(C2×C15⋊7D4), C10.108(C2×C3⋊D4), (C2×C6).304(C22×D5), (C2×C10).303(C22×S3), SmallGroup(480,902)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D30⋊17D4
G = < a,b,c,d | a30=b2=c4=d2=1, bab=a-1, ac=ca, ad=da, cbc-1=a15b, bd=db, dcd=c-1 >
Subgroups: 1844 in 260 conjugacy classes, 59 normal (31 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, C5, S3, C6, C6, C6, C2×C4, C2×C4, D4, C23, C23, D5, C10, C10, C10, Dic3, C12, D6, C2×C6, C2×C6, C2×C6, C15, C22⋊C4, C2×D4, C2×D4, C24, Dic5, C20, D10, C2×C10, C2×C10, C2×C10, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C22×S3, C22×C6, D15, C30, C30, C30, C22≀C2, C2×Dic5, C5⋊D4, C2×C20, C5×D4, C22×D5, C22×C10, D6⋊C4, C6.D4, C2×C3⋊D4, C6×D4, S3×C23, Dic15, C60, D30, D30, C2×C30, C2×C30, C2×C30, D10⋊C4, C23.D5, C2×C5⋊D4, D4×C10, C23×D5, C23⋊2D6, C2×Dic15, C15⋊7D4, C2×C60, D4×C15, C22×D15, C22×D15, C22×C30, C23⋊D10, D30⋊3C4, C30.38D4, C2×C15⋊7D4, D4×C30, C23×D15, D30⋊17D4
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, D10, C3⋊D4, C22×S3, D15, C22≀C2, C5⋊D4, C22×D5, S3×D4, C2×C3⋊D4, D30, D4×D5, C2×C5⋊D4, C23⋊2D6, C15⋊7D4, C22×D15, C23⋊D10, D4×D15, C2×C15⋊7D4, D30⋊17D4
►On 120 points
(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 30)(2 29)(3 28)(4 27)(5 26)(6 25)(7 24)(8 23)(9 22)(10 21)(11 20)(12 19)(13 18)(14 17)(15 16)(31 41)(32 40)(33 39)(34 38)(35 37)(42 60)(43 59)(44 58)(45 57)(46 56)(47 55)(48 54)(49 53)(50 52)(61 74)(62 73)(63 72)(64 71)(65 70)(66 69)(67 68)(75 90)(76 89)(77 88)(78 87)(79 86)(80 85)(81 84)(82 83)(91 119)(92 118)(93 117)(94 116)(95 115)(96 114)(97 113)(98 112)(99 111)(100 110)(101 109)(102 108)(103 107)(104 106)
(1 59 68 113)(2 60 69 114)(3 31 70 115)(4 32 71 116)(5 33 72 117)(6 34 73 118)(7 35 74 119)(8 36 75 120)(9 37 76 91)(10 38 77 92)(11 39 78 93)(12 40 79 94)(13 41 80 95)(14 42 81 96)(15 43 82 97)(16 44 83 98)(17 45 84 99)(18 46 85 100)(19 47 86 101)(20 48 87 102)(21 49 88 103)(22 50 89 104)(23 51 90 105)(24 52 61 106)(25 53 62 107)(26 54 63 108)(27 55 64 109)(28 56 65 110)(29 57 66 111)(30 58 67 112)
(31 115)(32 116)(33 117)(34 118)(35 119)(36 120)(37 91)(38 92)(39 93)(40 94)(41 95)(42 96)(43 97)(44 98)(45 99)(46 100)(47 101)(48 102)(49 103)(50 104)(51 105)(52 106)(53 107)(54 108)(55 109)(56 110)(57 111)(58 112)(59 113)(60 114)
G:=sub<Sym(120)| (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), (1,30)(2,29)(3,28)(4,27)(5,26)(6,25)(7,24)(8,23)(9,22)(10,21)(11,20)(12,19)(13,18)(14,17)(15,16)(31,41)(32,40)(33,39)(34,38)(35,37)(42,60)(43,59)(44,58)(45,57)(46,56)(47,55)(48,54)(49,53)(50,52)(61,74)(62,73)(63,72)(64,71)(65,70)(66,69)(67,68)(75,90)(76,89)(77,88)(78,87)(79,86)(80,85)(81,84)(82,83)(91,119)(92,118)(93,117)(94,116)(95,115)(96,114)(97,113)(98,112)(99,111)(100,110)(101,109)(102,108)(103,107)(104,106), (1,59,68,113)(2,60,69,114)(3,31,70,115)(4,32,71,116)(5,33,72,117)(6,34,73,118)(7,35,74,119)(8,36,75,120)(9,37,76,91)(10,38,77,92)(11,39,78,93)(12,40,79,94)(13,41,80,95)(14,42,81,96)(15,43,82,97)(16,44,83,98)(17,45,84,99)(18,46,85,100)(19,47,86,101)(20,48,87,102)(21,49,88,103)(22,50,89,104)(23,51,90,105)(24,52,61,106)(25,53,62,107)(26,54,63,108)(27,55,64,109)(28,56,65,110)(29,57,66,111)(30,58,67,112), (31,115)(32,116)(33,117)(34,118)(35,119)(36,120)(37,91)(38,92)(39,93)(40,94)(41,95)(42,96)(43,97)(44,98)(45,99)(46,100)(47,101)(48,102)(49,103)(50,104)(51,105)(52,106)(53,107)(54,108)(55,109)(56,110)(57,111)(58,112)(59,113)(60,114)>;
G:=Group( (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), (1,30)(2,29)(3,28)(4,27)(5,26)(6,25)(7,24)(8,23)(9,22)(10,21)(11,20)(12,19)(13,18)(14,17)(15,16)(31,41)(32,40)(33,39)(34,38)(35,37)(42,60)(43,59)(44,58)(45,57)(46,56)(47,55)(48,54)(49,53)(50,52)(61,74)(62,73)(63,72)(64,71)(65,70)(66,69)(67,68)(75,90)(76,89)(77,88)(78,87)(79,86)(80,85)(81,84)(82,83)(91,119)(92,118)(93,117)(94,116)(95,115)(96,114)(97,113)(98,112)(99,111)(100,110)(101,109)(102,108)(103,107)(104,106), (1,59,68,113)(2,60,69,114)(3,31,70,115)(4,32,71,116)(5,33,72,117)(6,34,73,118)(7,35,74,119)(8,36,75,120)(9,37,76,91)(10,38,77,92)(11,39,78,93)(12,40,79,94)(13,41,80,95)(14,42,81,96)(15,43,82,97)(16,44,83,98)(17,45,84,99)(18,46,85,100)(19,47,86,101)(20,48,87,102)(21,49,88,103)(22,50,89,104)(23,51,90,105)(24,52,61,106)(25,53,62,107)(26,54,63,108)(27,55,64,109)(28,56,65,110)(29,57,66,111)(30,58,67,112), (31,115)(32,116)(33,117)(34,118)(35,119)(36,120)(37,91)(38,92)(39,93)(40,94)(41,95)(42,96)(43,97)(44,98)(45,99)(46,100)(47,101)(48,102)(49,103)(50,104)(51,105)(52,106)(53,107)(54,108)(55,109)(56,110)(57,111)(58,112)(59,113)(60,114) );
G=PermutationGroup([[(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)], [(1,30),(2,29),(3,28),(4,27),(5,26),(6,25),(7,24),(8,23),(9,22),(10,21),(11,20),(12,19),(13,18),(14,17),(15,16),(31,41),(32,40),(33,39),(34,38),(35,37),(42,60),(43,59),(44,58),(45,57),(46,56),(47,55),(48,54),(49,53),(50,52),(61,74),(62,73),(63,72),(64,71),(65,70),(66,69),(67,68),(75,90),(76,89),(77,88),(78,87),(79,86),(80,85),(81,84),(82,83),(91,119),(92,118),(93,117),(94,116),(95,115),(96,114),(97,113),(98,112),(99,111),(100,110),(101,109),(102,108),(103,107),(104,106)], [(1,59,68,113),(2,60,69,114),(3,31,70,115),(4,32,71,116),(5,33,72,117),(6,34,73,118),(7,35,74,119),(8,36,75,120),(9,37,76,91),(10,38,77,92),(11,39,78,93),(12,40,79,94),(13,41,80,95),(14,42,81,96),(15,43,82,97),(16,44,83,98),(17,45,84,99),(18,46,85,100),(19,47,86,101),(20,48,87,102),(21,49,88,103),(22,50,89,104),(23,51,90,105),(24,52,61,106),(25,53,62,107),(26,54,63,108),(27,55,64,109),(28,56,65,110),(29,57,66,111),(30,58,67,112)], [(31,115),(32,116),(33,117),(34,118),(35,119),(36,120),(37,91),(38,92),(39,93),(40,94),(41,95),(42,96),(43,97),(44,98),(45,99),(46,100),(47,101),(48,102),(49,103),(50,104),(51,105),(52,106),(53,107),(54,108),(55,109),(56,110),(57,111),(58,112),(59,113),(60,114)]])
84 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 3 | 4A | 4B | 4C | 5A | 5B | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 10A | ··· | 10F | 10G | ··· | 10N | 12A | 12B | 15A | 15B | 15C | 15D | 20A | 20B | 20C | 20D | 30A | ··· | 30L | 30M | ··· | 30AB | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 10 | ··· | 10 | 10 | ··· | 10 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | 20 | 20 | 20 | 30 | ··· | 30 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 30 | 30 | 30 | 30 | 2 | 4 | 60 | 60 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
84 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D5 | D6 | D6 | D10 | D10 | C3⋊D4 | D15 | C5⋊D4 | D30 | D30 | C15⋊7D4 | S3×D4 | D4×D5 | D4×D15 |
| kernel | D30⋊17D4 | D30⋊3C4 | C30.38D4 | C2×C15⋊7D4 | D4×C30 | C23×D15 | D4×C10 | D30 | C2×C30 | C6×D4 | C2×C20 | C22×C10 | C2×C12 | C22×C6 | C2×C10 | C2×D4 | C2×C6 | C2×C4 | C23 | C22 | C10 | C6 | C2 |
| # reps | 1 | 2 | 1 | 2 | 1 | 1 | 1 | 4 | 2 | 2 | 1 | 2 | 2 | 4 | 4 | 4 | 8 | 4 | 8 | 16 | 2 | 4 | 8 |
Matrix representation of D30⋊17D4 ►in GL6(𝔽61)
| 60 | 0 | 0 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 23 | 53 | 0 | 0 |
| 0 | 0 | 8 | 45 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 60 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 18 | 1 | 0 | 0 |
| 0 | 0 | 43 | 43 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 48 | 41 |
| 0 | 0 | 0 | 0 | 39 | 13 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 17 | 60 |
G:=sub<GL(6,GF(61))| [60,0,0,0,0,0,0,60,0,0,0,0,0,0,23,8,0,0,0,0,53,45,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[60,0,0,0,0,0,0,1,0,0,0,0,0,0,18,43,0,0,0,0,1,43,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,48,39,0,0,0,0,41,13],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,17,0,0,0,0,0,60] >;
D30⋊17D4 in GAP, Magma, Sage, TeX
D_{30}\rtimes_{17}D_4 % in TeX
G:=Group("D30:17D4"); // GroupNames label
G:=SmallGroup(480,902);
// by ID
G=gap.SmallGroup(480,902);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,254,219,2693,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^30=b^2=c^4=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=a^15*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations