metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D30⋊8D4, (C6×D5)⋊8D4, (S3×C10)⋊8D4, C23⋊2(S3×D5), C15⋊10C22≀C2, C5⋊3(C23⋊2D6), D6⋊6(C5⋊D4), (C2×Dic5)⋊7D6, C6.167(D4×D5), (C22×C6)⋊3D10, (C22×C10)⋊6D6, D10⋊6(C3⋊D4), C3⋊3(C23⋊D10), D6⋊Dic5⋊38C2, (C2×Dic3)⋊7D10, C10.167(S3×D4), C30.257(C2×D4), D30⋊4C4⋊38C2, (C22×C30)⋊6C22, (C22×D5).66D6, D10⋊Dic3⋊38C2, (C2×C30).219C23, (C6×Dic5)⋊11C22, (C22×S3).57D10, C2.47(D10⋊D6), (C2×Dic15)⋊15C22, (C10×Dic3)⋊11C22, (C22×D15).71C22, (C22×S3×D5)⋊4C2, (C2×C5⋊D4)⋊10S3, (C2×C3⋊D4)⋊10D5, (C6×C5⋊D4)⋊10C2, C2.47(D5×C3⋊D4), C6.70(C2×C5⋊D4), C2.47(S3×C5⋊D4), (C10×C3⋊D4)⋊10C2, (C2×C15⋊7D4)⋊21C2, C10.71(C2×C3⋊D4), (D5×C2×C6).57C22, C22.248(C2×S3×D5), (S3×C2×C10).57C22, (C2×C6).231(C22×D5), (C2×C10).231(C22×S3), SmallGroup(480,653)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D30⋊8D4
G = < a,b,c,d | a30=b2=c4=d2=1, bab=a-1, cac-1=dad=a11, cbc-1=a25b, dbd=a10b, dcd=c-1 >
Subgroups: 1692 in 260 conjugacy classes, 54 normal (44 characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, S3, C6, C6, C2×C4, D4, C23, C23, D5, C10, C10, Dic3, C12, D6, D6, C2×C6, C2×C6, C15, C22⋊C4, C2×D4, C24, Dic5, C20, D10, D10, C2×C10, C2×C10, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C22×S3, C22×S3, C22×C6, C22×C6, C5×S3, C3×D5, D15, C30, C30, C22≀C2, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C5×D4, C22×D5, C22×D5, C22×C10, C22×C10, D6⋊C4, C6.D4, C2×C3⋊D4, C2×C3⋊D4, C6×D4, S3×C23, C5×Dic3, C3×Dic5, Dic15, S3×D5, C6×D5, C6×D5, S3×C10, S3×C10, D30, D30, C2×C30, C2×C30, D10⋊C4, C23.D5, C2×C5⋊D4, C2×C5⋊D4, D4×C10, C23×D5, C23⋊2D6, C6×Dic5, C3×C5⋊D4, C10×Dic3, C5×C3⋊D4, C2×Dic15, C15⋊7D4, C2×S3×D5, D5×C2×C6, S3×C2×C10, C22×D15, C22×C30, C23⋊D10, D10⋊Dic3, D6⋊Dic5, D30⋊4C4, C6×C5⋊D4, C10×C3⋊D4, C2×C15⋊7D4, C22×S3×D5, D30⋊8D4
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, D10, C3⋊D4, C22×S3, C22≀C2, C5⋊D4, C22×D5, S3×D4, C2×C3⋊D4, S3×D5, D4×D5, C2×C5⋊D4, C23⋊2D6, C2×S3×D5, C23⋊D10, D5×C3⋊D4, S3×C5⋊D4, D10⋊D6, D30⋊8D4
►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 53)(32 52)(33 51)(34 50)(35 49)(36 48)(37 47)(38 46)(39 45)(40 44)(41 43)(54 60)(55 59)(56 58)(61 78)(62 77)(63 76)(64 75)(65 74)(66 73)(67 72)(68 71)(69 70)(79 90)(80 89)(81 88)(82 87)(83 86)(84 85)(91 111)(92 110)(93 109)(94 108)(95 107)(96 106)(97 105)(98 104)(99 103)(100 102)(112 120)(113 119)(114 118)(115 117)
(1 35 85 109)(2 46 86 120)(3 57 87 101)(4 38 88 112)(5 49 89 93)(6 60 90 104)(7 41 61 115)(8 52 62 96)(9 33 63 107)(10 44 64 118)(11 55 65 99)(12 36 66 110)(13 47 67 91)(14 58 68 102)(15 39 69 113)(16 50 70 94)(17 31 71 105)(18 42 72 116)(19 53 73 97)(20 34 74 108)(21 45 75 119)(22 56 76 100)(23 37 77 111)(24 48 78 92)(25 59 79 103)(26 40 80 114)(27 51 81 95)(28 32 82 106)(29 43 83 117)(30 54 84 98)
(2 12)(3 23)(5 15)(6 26)(8 18)(9 29)(11 21)(14 24)(17 27)(20 30)(31 95)(32 106)(33 117)(34 98)(35 109)(36 120)(37 101)(38 112)(39 93)(40 104)(41 115)(42 96)(43 107)(44 118)(45 99)(46 110)(47 91)(48 102)(49 113)(50 94)(51 105)(52 116)(53 97)(54 108)(55 119)(56 100)(57 111)(58 92)(59 103)(60 114)(62 72)(63 83)(65 75)(66 86)(68 78)(69 89)(71 81)(74 84)(77 87)(80 90)
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)(31,53)(32,52)(33,51)(34,50)(35,49)(36,48)(37,47)(38,46)(39,45)(40,44)(41,43)(54,60)(55,59)(56,58)(61,78)(62,77)(63,76)(64,75)(65,74)(66,73)(67,72)(68,71)(69,70)(79,90)(80,89)(81,88)(82,87)(83,86)(84,85)(91,111)(92,110)(93,109)(94,108)(95,107)(96,106)(97,105)(98,104)(99,103)(100,102)(112,120)(113,119)(114,118)(115,117), (1,35,85,109)(2,46,86,120)(3,57,87,101)(4,38,88,112)(5,49,89,93)(6,60,90,104)(7,41,61,115)(8,52,62,96)(9,33,63,107)(10,44,64,118)(11,55,65,99)(12,36,66,110)(13,47,67,91)(14,58,68,102)(15,39,69,113)(16,50,70,94)(17,31,71,105)(18,42,72,116)(19,53,73,97)(20,34,74,108)(21,45,75,119)(22,56,76,100)(23,37,77,111)(24,48,78,92)(25,59,79,103)(26,40,80,114)(27,51,81,95)(28,32,82,106)(29,43,83,117)(30,54,84,98), (2,12)(3,23)(5,15)(6,26)(8,18)(9,29)(11,21)(14,24)(17,27)(20,30)(31,95)(32,106)(33,117)(34,98)(35,109)(36,120)(37,101)(38,112)(39,93)(40,104)(41,115)(42,96)(43,107)(44,118)(45,99)(46,110)(47,91)(48,102)(49,113)(50,94)(51,105)(52,116)(53,97)(54,108)(55,119)(56,100)(57,111)(58,92)(59,103)(60,114)(62,72)(63,83)(65,75)(66,86)(68,78)(69,89)(71,81)(74,84)(77,87)(80,90)>;
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)(31,53)(32,52)(33,51)(34,50)(35,49)(36,48)(37,47)(38,46)(39,45)(40,44)(41,43)(54,60)(55,59)(56,58)(61,78)(62,77)(63,76)(64,75)(65,74)(66,73)(67,72)(68,71)(69,70)(79,90)(80,89)(81,88)(82,87)(83,86)(84,85)(91,111)(92,110)(93,109)(94,108)(95,107)(96,106)(97,105)(98,104)(99,103)(100,102)(112,120)(113,119)(114,118)(115,117), (1,35,85,109)(2,46,86,120)(3,57,87,101)(4,38,88,112)(5,49,89,93)(6,60,90,104)(7,41,61,115)(8,52,62,96)(9,33,63,107)(10,44,64,118)(11,55,65,99)(12,36,66,110)(13,47,67,91)(14,58,68,102)(15,39,69,113)(16,50,70,94)(17,31,71,105)(18,42,72,116)(19,53,73,97)(20,34,74,108)(21,45,75,119)(22,56,76,100)(23,37,77,111)(24,48,78,92)(25,59,79,103)(26,40,80,114)(27,51,81,95)(28,32,82,106)(29,43,83,117)(30,54,84,98), (2,12)(3,23)(5,15)(6,26)(8,18)(9,29)(11,21)(14,24)(17,27)(20,30)(31,95)(32,106)(33,117)(34,98)(35,109)(36,120)(37,101)(38,112)(39,93)(40,104)(41,115)(42,96)(43,107)(44,118)(45,99)(46,110)(47,91)(48,102)(49,113)(50,94)(51,105)(52,116)(53,97)(54,108)(55,119)(56,100)(57,111)(58,92)(59,103)(60,114)(62,72)(63,83)(65,75)(66,86)(68,78)(69,89)(71,81)(74,84)(77,87)(80,90) );
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),(31,53),(32,52),(33,51),(34,50),(35,49),(36,48),(37,47),(38,46),(39,45),(40,44),(41,43),(54,60),(55,59),(56,58),(61,78),(62,77),(63,76),(64,75),(65,74),(66,73),(67,72),(68,71),(69,70),(79,90),(80,89),(81,88),(82,87),(83,86),(84,85),(91,111),(92,110),(93,109),(94,108),(95,107),(96,106),(97,105),(98,104),(99,103),(100,102),(112,120),(113,119),(114,118),(115,117)], [(1,35,85,109),(2,46,86,120),(3,57,87,101),(4,38,88,112),(5,49,89,93),(6,60,90,104),(7,41,61,115),(8,52,62,96),(9,33,63,107),(10,44,64,118),(11,55,65,99),(12,36,66,110),(13,47,67,91),(14,58,68,102),(15,39,69,113),(16,50,70,94),(17,31,71,105),(18,42,72,116),(19,53,73,97),(20,34,74,108),(21,45,75,119),(22,56,76,100),(23,37,77,111),(24,48,78,92),(25,59,79,103),(26,40,80,114),(27,51,81,95),(28,32,82,106),(29,43,83,117),(30,54,84,98)], [(2,12),(3,23),(5,15),(6,26),(8,18),(9,29),(11,21),(14,24),(17,27),(20,30),(31,95),(32,106),(33,117),(34,98),(35,109),(36,120),(37,101),(38,112),(39,93),(40,104),(41,115),(42,96),(43,107),(44,118),(45,99),(46,110),(47,91),(48,102),(49,113),(50,94),(51,105),(52,116),(53,97),(54,108),(55,119),(56,100),(57,111),(58,92),(59,103),(60,114),(62,72),(63,83),(65,75),(66,86),(68,78),(69,89),(71,81),(74,84),(77,87),(80,90)]])
60 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 | 10H | 10I | 10J | 10K | 10L | 10M | 10N | 12A | 12B | 15A | 15B | 20A | 20B | 20C | 20D | 30A | ··· | 30N |
| 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 | 10 | 10 | 10 | 10 | 10 | 10 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 30 | ··· | 30 |
| size | 1 | 1 | 1 | 1 | 4 | 6 | 6 | 10 | 10 | 30 | 30 | 2 | 12 | 20 | 60 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 20 | 20 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 12 | 12 | 12 | 12 | 20 | 20 | 4 | 4 | 12 | 12 | 12 | 12 | 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 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D4 | D5 | D6 | D6 | D6 | D10 | D10 | D10 | C3⋊D4 | C5⋊D4 | S3×D4 | S3×D5 | D4×D5 | C2×S3×D5 | D5×C3⋊D4 | S3×C5⋊D4 | D10⋊D6 |
| kernel | D30⋊8D4 | D10⋊Dic3 | D6⋊Dic5 | D30⋊4C4 | C6×C5⋊D4 | C10×C3⋊D4 | C2×C15⋊7D4 | C22×S3×D5 | C2×C5⋊D4 | C6×D5 | S3×C10 | D30 | C2×C3⋊D4 | C2×Dic5 | C22×D5 | C22×C10 | C2×Dic3 | C22×S3 | C22×C6 | D10 | D6 | C10 | C23 | C6 | C22 | C2 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 8 | 2 | 2 | 4 | 2 | 4 | 4 | 4 |
Matrix representation of D30⋊8D4 ►in GL6(𝔽61)
| 0 | 60 | 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 17 | 60 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 0 | 0 | 60 |
| 0 | 60 | 0 | 0 | 0 | 0 |
| 60 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 17 | 60 | 0 | 0 |
| 0 | 0 | 44 | 44 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 0 | 31 | 1 |
| 9 | 18 | 0 | 0 | 0 | 0 |
| 9 | 52 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 47 | 5 |
| 0 | 0 | 0 | 0 | 22 | 14 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 60 | 60 | 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 |
G:=sub<GL(6,GF(61))| [0,1,0,0,0,0,60,1,0,0,0,0,0,0,17,1,0,0,0,0,60,0,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[0,60,0,0,0,0,60,0,0,0,0,0,0,0,17,44,0,0,0,0,60,44,0,0,0,0,0,0,60,31,0,0,0,0,0,1],[9,9,0,0,0,0,18,52,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,47,22,0,0,0,0,5,14],[1,60,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;
D30⋊8D4 in GAP, Magma, Sage, TeX
D_{30}\rtimes_8D_4 % in TeX
G:=Group("D30:8D4"); // GroupNames label
G:=SmallGroup(480,653);
// by ID
G=gap.SmallGroup(480,653);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,254,219,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^30=b^2=c^4=d^2=1,b*a*b=a^-1,c*a*c^-1=d*a*d=a^11,c*b*c^-1=a^25*b,d*b*d=a^10*b,d*c*d=c^-1>;
// generators/relations