metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: (C2×C30)⋊4D4, C15⋊4C22≀C2, (C6×D5)⋊16D4, C6.161(D4×D5), (C23×D5)⋊4S3, D10⋊7(C3⋊D4), C3⋊5(C23⋊D10), C5⋊2(C24⋊4S3), (C2×Dic3)⋊3D10, C30.243(C2×D4), (C22×S3)⋊2D10, C23.43(S3×D5), C22⋊3(C15⋊D4), (C22×D5).94D6, (C22×C6).93D10, (C22×C10).57D6, C30.38D4⋊27C2, D10⋊Dic3⋊33C2, (C2×C30).205C23, (C10×Dic3)⋊7C22, (C2×Dic15)⋊11C22, (C22×C30).67C22, (C2×C3⋊D4)⋊4D5, (D5×C22×C6)⋊1C2, (C10×C3⋊D4)⋊4C2, (S3×C2×C10)⋊2C22, C6.94(C2×C5⋊D4), C2.41(D5×C3⋊D4), (C2×C15⋊D4)⋊12C2, (C2×C10)⋊5(C3⋊D4), (C2×C6)⋊11(C5⋊D4), C10.65(C2×C3⋊D4), C2.24(C2×C15⋊D4), C22.234(C2×S3×D5), (D5×C2×C6).112C22, (C2×C6).217(C22×D5), (C2×C10).217(C22×S3), SmallGroup(480,639)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for (C2×C30)⋊D4
G = < a,b,c,d | a2=b30=c4=d2=1, ab=ba, cac-1=ab15, ad=da, cbc-1=b-1, dbd=b19, dcd=c-1 >
Subgroups: 1308 in 260 conjugacy classes, 60 normal (26 characteristic)
C1, C2, C2 [×2], C2 [×7], C3, C4 [×3], C22, C22 [×2], C22 [×21], C5, S3, C6, C6 [×2], C6 [×6], C2×C4 [×3], D4 [×6], C23, C23 [×9], D5 [×4], C10, C10 [×2], C10 [×3], Dic3 [×3], D6 [×3], C2×C6, C2×C6 [×2], C2×C6 [×18], C15, C22⋊C4 [×3], C2×D4 [×3], C24, Dic5 [×2], C20, D10 [×4], D10 [×12], C2×C10, C2×C10 [×2], C2×C10 [×5], C2×Dic3, C2×Dic3 [×2], C3⋊D4 [×6], C22×S3, C22×C6, C22×C6 [×8], C5×S3, C3×D5 [×4], C30, C30 [×2], C30 [×2], C22≀C2, C2×Dic5 [×2], C5⋊D4 [×4], C2×C20, C5×D4 [×2], C22×D5 [×2], C22×D5 [×6], C22×C10, C22×C10, C6.D4 [×3], C2×C3⋊D4, C2×C3⋊D4 [×2], C23×C6, C5×Dic3, Dic15 [×2], C6×D5 [×4], C6×D5 [×12], S3×C10 [×3], C2×C30, C2×C30 [×2], C2×C30 [×2], D10⋊C4 [×2], C23.D5, C2×C5⋊D4 [×2], D4×C10, C23×D5, C24⋊4S3, C15⋊D4 [×4], C10×Dic3, C5×C3⋊D4 [×2], C2×Dic15 [×2], D5×C2×C6 [×2], D5×C2×C6 [×6], S3×C2×C10, C22×C30, C23⋊D10, D10⋊Dic3 [×2], C30.38D4, C2×C15⋊D4 [×2], C10×C3⋊D4, D5×C22×C6, (C2×C30)⋊D4
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×6], C23, D5, D6 [×3], C2×D4 [×3], D10 [×3], C3⋊D4 [×6], C22×S3, C22≀C2, C5⋊D4 [×2], C22×D5, C2×C3⋊D4 [×3], S3×D5, D4×D5 [×2], C2×C5⋊D4, C24⋊4S3, C15⋊D4 [×2], C2×S3×D5, C23⋊D10, C2×C15⋊D4, D5×C3⋊D4 [×2], (C2×C30)⋊D4
►On 120 points
(1 45)(2 46)(3 47)(4 48)(5 49)(6 50)(7 51)(8 52)(9 53)(10 54)(11 55)(12 56)(13 57)(14 58)(15 59)(16 60)(17 31)(18 32)(19 33)(20 34)(21 35)(22 36)(23 37)(24 38)(25 39)(26 40)(27 41)(28 42)(29 43)(30 44)(61 106)(62 107)(63 108)(64 109)(65 110)(66 111)(67 112)(68 113)(69 114)(70 115)(71 116)(72 117)(73 118)(74 119)(75 120)(76 91)(77 92)(78 93)(79 94)(80 95)(81 96)(82 97)(83 98)(84 99)(85 100)(86 101)(87 102)(88 103)(89 104)(90 105)
(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 74 45 104)(2 73 46 103)(3 72 47 102)(4 71 48 101)(5 70 49 100)(6 69 50 99)(7 68 51 98)(8 67 52 97)(9 66 53 96)(10 65 54 95)(11 64 55 94)(12 63 56 93)(13 62 57 92)(14 61 58 91)(15 90 59 120)(16 89 60 119)(17 88 31 118)(18 87 32 117)(19 86 33 116)(20 85 34 115)(21 84 35 114)(22 83 36 113)(23 82 37 112)(24 81 38 111)(25 80 39 110)(26 79 40 109)(27 78 41 108)(28 77 42 107)(29 76 43 106)(30 75 44 105)
(1 16)(2 5)(3 24)(4 13)(6 21)(7 10)(8 29)(9 18)(11 26)(12 15)(14 23)(17 20)(19 28)(22 25)(27 30)(31 34)(32 53)(33 42)(35 50)(36 39)(37 58)(38 47)(40 55)(41 44)(43 52)(45 60)(46 49)(48 57)(51 54)(56 59)(61 112)(62 101)(63 120)(64 109)(65 98)(66 117)(67 106)(68 95)(69 114)(70 103)(71 92)(72 111)(73 100)(74 119)(75 108)(76 97)(77 116)(78 105)(79 94)(80 113)(81 102)(82 91)(83 110)(84 99)(85 118)(86 107)(87 96)(88 115)(89 104)(90 93)
G:=sub<Sym(120)| (1,45)(2,46)(3,47)(4,48)(5,49)(6,50)(7,51)(8,52)(9,53)(10,54)(11,55)(12,56)(13,57)(14,58)(15,59)(16,60)(17,31)(18,32)(19,33)(20,34)(21,35)(22,36)(23,37)(24,38)(25,39)(26,40)(27,41)(28,42)(29,43)(30,44)(61,106)(62,107)(63,108)(64,109)(65,110)(66,111)(67,112)(68,113)(69,114)(70,115)(71,116)(72,117)(73,118)(74,119)(75,120)(76,91)(77,92)(78,93)(79,94)(80,95)(81,96)(82,97)(83,98)(84,99)(85,100)(86,101)(87,102)(88,103)(89,104)(90,105), (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), (1,74,45,104)(2,73,46,103)(3,72,47,102)(4,71,48,101)(5,70,49,100)(6,69,50,99)(7,68,51,98)(8,67,52,97)(9,66,53,96)(10,65,54,95)(11,64,55,94)(12,63,56,93)(13,62,57,92)(14,61,58,91)(15,90,59,120)(16,89,60,119)(17,88,31,118)(18,87,32,117)(19,86,33,116)(20,85,34,115)(21,84,35,114)(22,83,36,113)(23,82,37,112)(24,81,38,111)(25,80,39,110)(26,79,40,109)(27,78,41,108)(28,77,42,107)(29,76,43,106)(30,75,44,105), (1,16)(2,5)(3,24)(4,13)(6,21)(7,10)(8,29)(9,18)(11,26)(12,15)(14,23)(17,20)(19,28)(22,25)(27,30)(31,34)(32,53)(33,42)(35,50)(36,39)(37,58)(38,47)(40,55)(41,44)(43,52)(45,60)(46,49)(48,57)(51,54)(56,59)(61,112)(62,101)(63,120)(64,109)(65,98)(66,117)(67,106)(68,95)(69,114)(70,103)(71,92)(72,111)(73,100)(74,119)(75,108)(76,97)(77,116)(78,105)(79,94)(80,113)(81,102)(82,91)(83,110)(84,99)(85,118)(86,107)(87,96)(88,115)(89,104)(90,93)>;
G:=Group( (1,45)(2,46)(3,47)(4,48)(5,49)(6,50)(7,51)(8,52)(9,53)(10,54)(11,55)(12,56)(13,57)(14,58)(15,59)(16,60)(17,31)(18,32)(19,33)(20,34)(21,35)(22,36)(23,37)(24,38)(25,39)(26,40)(27,41)(28,42)(29,43)(30,44)(61,106)(62,107)(63,108)(64,109)(65,110)(66,111)(67,112)(68,113)(69,114)(70,115)(71,116)(72,117)(73,118)(74,119)(75,120)(76,91)(77,92)(78,93)(79,94)(80,95)(81,96)(82,97)(83,98)(84,99)(85,100)(86,101)(87,102)(88,103)(89,104)(90,105), (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), (1,74,45,104)(2,73,46,103)(3,72,47,102)(4,71,48,101)(5,70,49,100)(6,69,50,99)(7,68,51,98)(8,67,52,97)(9,66,53,96)(10,65,54,95)(11,64,55,94)(12,63,56,93)(13,62,57,92)(14,61,58,91)(15,90,59,120)(16,89,60,119)(17,88,31,118)(18,87,32,117)(19,86,33,116)(20,85,34,115)(21,84,35,114)(22,83,36,113)(23,82,37,112)(24,81,38,111)(25,80,39,110)(26,79,40,109)(27,78,41,108)(28,77,42,107)(29,76,43,106)(30,75,44,105), (1,16)(2,5)(3,24)(4,13)(6,21)(7,10)(8,29)(9,18)(11,26)(12,15)(14,23)(17,20)(19,28)(22,25)(27,30)(31,34)(32,53)(33,42)(35,50)(36,39)(37,58)(38,47)(40,55)(41,44)(43,52)(45,60)(46,49)(48,57)(51,54)(56,59)(61,112)(62,101)(63,120)(64,109)(65,98)(66,117)(67,106)(68,95)(69,114)(70,103)(71,92)(72,111)(73,100)(74,119)(75,108)(76,97)(77,116)(78,105)(79,94)(80,113)(81,102)(82,91)(83,110)(84,99)(85,118)(86,107)(87,96)(88,115)(89,104)(90,93) );
G=PermutationGroup([(1,45),(2,46),(3,47),(4,48),(5,49),(6,50),(7,51),(8,52),(9,53),(10,54),(11,55),(12,56),(13,57),(14,58),(15,59),(16,60),(17,31),(18,32),(19,33),(20,34),(21,35),(22,36),(23,37),(24,38),(25,39),(26,40),(27,41),(28,42),(29,43),(30,44),(61,106),(62,107),(63,108),(64,109),(65,110),(66,111),(67,112),(68,113),(69,114),(70,115),(71,116),(72,117),(73,118),(74,119),(75,120),(76,91),(77,92),(78,93),(79,94),(80,95),(81,96),(82,97),(83,98),(84,99),(85,100),(86,101),(87,102),(88,103),(89,104),(90,105)], [(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)], [(1,74,45,104),(2,73,46,103),(3,72,47,102),(4,71,48,101),(5,70,49,100),(6,69,50,99),(7,68,51,98),(8,67,52,97),(9,66,53,96),(10,65,54,95),(11,64,55,94),(12,63,56,93),(13,62,57,92),(14,61,58,91),(15,90,59,120),(16,89,60,119),(17,88,31,118),(18,87,32,117),(19,86,33,116),(20,85,34,115),(21,84,35,114),(22,83,36,113),(23,82,37,112),(24,81,38,111),(25,80,39,110),(26,79,40,109),(27,78,41,108),(28,77,42,107),(29,76,43,106),(30,75,44,105)], [(1,16),(2,5),(3,24),(4,13),(6,21),(7,10),(8,29),(9,18),(11,26),(12,15),(14,23),(17,20),(19,28),(22,25),(27,30),(31,34),(32,53),(33,42),(35,50),(36,39),(37,58),(38,47),(40,55),(41,44),(43,52),(45,60),(46,49),(48,57),(51,54),(56,59),(61,112),(62,101),(63,120),(64,109),(65,98),(66,117),(67,106),(68,95),(69,114),(70,103),(71,92),(72,111),(73,100),(74,119),(75,108),(76,97),(77,116),(78,105),(79,94),(80,113),(81,102),(82,91),(83,110),(84,99),(85,118),(86,107),(87,96),(88,115),(89,104),(90,93)])
66 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 3 | 4A | 4B | 4C | 5A | 5B | 6A | ··· | 6G | 6H | ··· | 6O | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 10K | 10L | 10M | 10N | 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 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 15 | 15 | 20 | 20 | 20 | 20 | 30 | ··· | 30 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 10 | 10 | 10 | 10 | 12 | 2 | 12 | 60 | 60 | 2 | 2 | 2 | ··· | 2 | 10 | ··· | 10 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 12 | 12 | 12 | 12 | 4 | 4 | 12 | 12 | 12 | 12 | 4 | ··· | 4 |
66 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D5 | D6 | D6 | D10 | D10 | D10 | C3⋊D4 | C3⋊D4 | C5⋊D4 | S3×D5 | D4×D5 | C15⋊D4 | C2×S3×D5 | D5×C3⋊D4 |
| kernel | (C2×C30)⋊D4 | D10⋊Dic3 | C30.38D4 | C2×C15⋊D4 | C10×C3⋊D4 | D5×C22×C6 | C23×D5 | C6×D5 | C2×C30 | C2×C3⋊D4 | C22×D5 | C22×C10 | C2×Dic3 | C22×S3 | C22×C6 | D10 | C2×C10 | C2×C6 | C23 | C6 | C22 | C22 | C2 |
| # reps | 1 | 2 | 1 | 2 | 1 | 1 | 1 | 4 | 2 | 2 | 2 | 1 | 2 | 2 | 2 | 8 | 4 | 8 | 2 | 4 | 4 | 2 | 8 |
Matrix representation of (C2×C30)⋊D4 ►in GL4(𝔽61) generated by
| 60 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 |
| 0 | 0 | 60 | 0 |
| 0 | 0 | 17 | 1 |
| 44 | 1 | 0 | 0 |
| 16 | 60 | 0 | 0 |
| 0 | 0 | 14 | 0 |
| 0 | 0 | 45 | 48 |
| 39 | 8 | 0 | 0 |
| 8 | 22 | 0 | 0 |
| 0 | 0 | 48 | 20 |
| 0 | 0 | 22 | 13 |
| 17 | 18 | 0 | 0 |
| 45 | 44 | 0 | 0 |
| 0 | 0 | 60 | 0 |
| 0 | 0 | 17 | 1 |
G:=sub<GL(4,GF(61))| [60,0,0,0,0,60,0,0,0,0,60,17,0,0,0,1],[44,16,0,0,1,60,0,0,0,0,14,45,0,0,0,48],[39,8,0,0,8,22,0,0,0,0,48,22,0,0,20,13],[17,45,0,0,18,44,0,0,0,0,60,17,0,0,0,1] >;
(C2×C30)⋊D4 in GAP, Magma, Sage, TeX
(C_2\times C_{30})\rtimes D_4 % in TeX
G:=Group("(C2xC30):D4"); // GroupNames label
G:=SmallGroup(480,639);
// by ID
G=gap.SmallGroup(480,639);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,141,219,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^30=c^4=d^2=1,a*b=b*a,c*a*c^-1=a*b^15,a*d=d*a,c*b*c^-1=b^-1,d*b*d=b^19,d*c*d=c^-1>;
// generators/relations