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 [×2], C2 [×7], C3, C4 [×3], C22, C22 [×2], C22 [×21], C5, S3 [×4], C6, C6 [×2], C6 [×3], C2×C4, C2×C4 [×2], D4 [×6], C23 [×2], C23 [×8], D5 [×4], C10, C10 [×2], C10 [×3], Dic3 [×2], C12, D6 [×16], C2×C6, C2×C6 [×2], C2×C6 [×5], C15, C22⋊C4 [×3], C2×D4, C2×D4 [×2], C24, Dic5 [×2], C20, D10 [×16], C2×C10, C2×C10 [×2], C2×C10 [×5], C2×Dic3 [×2], C3⋊D4 [×4], C2×C12, C3×D4 [×2], C22×S3 [×8], C22×C6 [×2], D15 [×4], C30, C30 [×2], C30 [×3], C22≀C2, C2×Dic5 [×2], C5⋊D4 [×4], C2×C20, C5×D4 [×2], C22×D5 [×8], C22×C10 [×2], D6⋊C4 [×2], C6.D4, C2×C3⋊D4 [×2], C6×D4, S3×C23, Dic15 [×2], C60, D30 [×4], D30 [×12], C2×C30, C2×C30 [×2], C2×C30 [×5], D10⋊C4 [×2], C23.D5, C2×C5⋊D4 [×2], D4×C10, C23×D5, C23⋊2D6, C2×Dic15 [×2], C15⋊7D4 [×4], C2×C60, D4×C15 [×2], C22×D15 [×2], C22×D15 [×6], C22×C30 [×2], C23⋊D10, D30⋊3C4 [×2], C30.38D4, C2×C15⋊7D4 [×2], D4×C30, C23×D15, D30⋊17D4
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×6], C23, D5, D6 [×3], C2×D4 [×3], D10 [×3], C3⋊D4 [×2], C22×S3, D15, C22≀C2, C5⋊D4 [×2], C22×D5, S3×D4 [×2], C2×C3⋊D4, D30 [×3], D4×D5 [×2], C2×C5⋊D4, C23⋊2D6, C15⋊7D4 [×2], C22×D15, C23⋊D10, D4×D15 [×2], 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 59)(32 58)(33 57)(34 56)(35 55)(36 54)(37 53)(38 52)(39 51)(40 50)(41 49)(42 48)(43 47)(44 46)(61 71)(62 70)(63 69)(64 68)(65 67)(72 90)(73 89)(74 88)(75 87)(76 86)(77 85)(78 84)(79 83)(80 82)(91 104)(92 103)(93 102)(94 101)(95 100)(96 99)(97 98)(105 120)(106 119)(107 118)(108 117)(109 116)(110 115)(111 114)(112 113)
(1 89 113 53)(2 90 114 54)(3 61 115 55)(4 62 116 56)(5 63 117 57)(6 64 118 58)(7 65 119 59)(8 66 120 60)(9 67 91 31)(10 68 92 32)(11 69 93 33)(12 70 94 34)(13 71 95 35)(14 72 96 36)(15 73 97 37)(16 74 98 38)(17 75 99 39)(18 76 100 40)(19 77 101 41)(20 78 102 42)(21 79 103 43)(22 80 104 44)(23 81 105 45)(24 82 106 46)(25 83 107 47)(26 84 108 48)(27 85 109 49)(28 86 110 50)(29 87 111 51)(30 88 112 52)
(31 67)(32 68)(33 69)(34 70)(35 71)(36 72)(37 73)(38 74)(39 75)(40 76)(41 77)(42 78)(43 79)(44 80)(45 81)(46 82)(47 83)(48 84)(49 85)(50 86)(51 87)(52 88)(53 89)(54 90)(55 61)(56 62)(57 63)(58 64)(59 65)(60 66)
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)(31,59)(32,58)(33,57)(34,56)(35,55)(36,54)(37,53)(38,52)(39,51)(40,50)(41,49)(42,48)(43,47)(44,46)(61,71)(62,70)(63,69)(64,68)(65,67)(72,90)(73,89)(74,88)(75,87)(76,86)(77,85)(78,84)(79,83)(80,82)(91,104)(92,103)(93,102)(94,101)(95,100)(96,99)(97,98)(105,120)(106,119)(107,118)(108,117)(109,116)(110,115)(111,114)(112,113), (1,89,113,53)(2,90,114,54)(3,61,115,55)(4,62,116,56)(5,63,117,57)(6,64,118,58)(7,65,119,59)(8,66,120,60)(9,67,91,31)(10,68,92,32)(11,69,93,33)(12,70,94,34)(13,71,95,35)(14,72,96,36)(15,73,97,37)(16,74,98,38)(17,75,99,39)(18,76,100,40)(19,77,101,41)(20,78,102,42)(21,79,103,43)(22,80,104,44)(23,81,105,45)(24,82,106,46)(25,83,107,47)(26,84,108,48)(27,85,109,49)(28,86,110,50)(29,87,111,51)(30,88,112,52), (31,67)(32,68)(33,69)(34,70)(35,71)(36,72)(37,73)(38,74)(39,75)(40,76)(41,77)(42,78)(43,79)(44,80)(45,81)(46,82)(47,83)(48,84)(49,85)(50,86)(51,87)(52,88)(53,89)(54,90)(55,61)(56,62)(57,63)(58,64)(59,65)(60,66)>;
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)(31,59)(32,58)(33,57)(34,56)(35,55)(36,54)(37,53)(38,52)(39,51)(40,50)(41,49)(42,48)(43,47)(44,46)(61,71)(62,70)(63,69)(64,68)(65,67)(72,90)(73,89)(74,88)(75,87)(76,86)(77,85)(78,84)(79,83)(80,82)(91,104)(92,103)(93,102)(94,101)(95,100)(96,99)(97,98)(105,120)(106,119)(107,118)(108,117)(109,116)(110,115)(111,114)(112,113), (1,89,113,53)(2,90,114,54)(3,61,115,55)(4,62,116,56)(5,63,117,57)(6,64,118,58)(7,65,119,59)(8,66,120,60)(9,67,91,31)(10,68,92,32)(11,69,93,33)(12,70,94,34)(13,71,95,35)(14,72,96,36)(15,73,97,37)(16,74,98,38)(17,75,99,39)(18,76,100,40)(19,77,101,41)(20,78,102,42)(21,79,103,43)(22,80,104,44)(23,81,105,45)(24,82,106,46)(25,83,107,47)(26,84,108,48)(27,85,109,49)(28,86,110,50)(29,87,111,51)(30,88,112,52), (31,67)(32,68)(33,69)(34,70)(35,71)(36,72)(37,73)(38,74)(39,75)(40,76)(41,77)(42,78)(43,79)(44,80)(45,81)(46,82)(47,83)(48,84)(49,85)(50,86)(51,87)(52,88)(53,89)(54,90)(55,61)(56,62)(57,63)(58,64)(59,65)(60,66) );
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),(31,59),(32,58),(33,57),(34,56),(35,55),(36,54),(37,53),(38,52),(39,51),(40,50),(41,49),(42,48),(43,47),(44,46),(61,71),(62,70),(63,69),(64,68),(65,67),(72,90),(73,89),(74,88),(75,87),(76,86),(77,85),(78,84),(79,83),(80,82),(91,104),(92,103),(93,102),(94,101),(95,100),(96,99),(97,98),(105,120),(106,119),(107,118),(108,117),(109,116),(110,115),(111,114),(112,113)], [(1,89,113,53),(2,90,114,54),(3,61,115,55),(4,62,116,56),(5,63,117,57),(6,64,118,58),(7,65,119,59),(8,66,120,60),(9,67,91,31),(10,68,92,32),(11,69,93,33),(12,70,94,34),(13,71,95,35),(14,72,96,36),(15,73,97,37),(16,74,98,38),(17,75,99,39),(18,76,100,40),(19,77,101,41),(20,78,102,42),(21,79,103,43),(22,80,104,44),(23,81,105,45),(24,82,106,46),(25,83,107,47),(26,84,108,48),(27,85,109,49),(28,86,110,50),(29,87,111,51),(30,88,112,52)], [(31,67),(32,68),(33,69),(34,70),(35,71),(36,72),(37,73),(38,74),(39,75),(40,76),(41,77),(42,78),(43,79),(44,80),(45,81),(46,82),(47,83),(48,84),(49,85),(50,86),(51,87),(52,88),(53,89),(54,90),(55,61),(56,62),(57,63),(58,64),(59,65),(60,66)])
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