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, C2, C3, C4, C22, C22, C22, C5, S3, C6, C6, C6, C2×C4, D4, C23, C23, D5, C10, C10, C10, Dic3, D6, C2×C6, C2×C6, C2×C6, C15, C22⋊C4, C2×D4, C24, Dic5, C20, D10, D10, C2×C10, C2×C10, C2×C10, C2×Dic3, C2×Dic3, C3⋊D4, C22×S3, C22×C6, C22×C6, C5×S3, C3×D5, C30, C30, C30, C22≀C2, C2×Dic5, C5⋊D4, C2×C20, C5×D4, C22×D5, C22×D5, C22×C10, C22×C10, C6.D4, C2×C3⋊D4, C2×C3⋊D4, C23×C6, C5×Dic3, Dic15, C6×D5, C6×D5, S3×C10, C2×C30, C2×C30, C2×C30, D10⋊C4, C23.D5, C2×C5⋊D4, D4×C10, C23×D5, C24⋊4S3, C15⋊D4, C10×Dic3, C5×C3⋊D4, C2×Dic15, D5×C2×C6, D5×C2×C6, S3×C2×C10, C22×C30, C23⋊D10, D10⋊Dic3, C30.38D4, C2×C15⋊D4, C10×C3⋊D4, D5×C22×C6, (C2×C30)⋊D4
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, D10, C3⋊D4, C22×S3, C22≀C2, C5⋊D4, C22×D5, C2×C3⋊D4, S3×D5, D4×D5, C2×C5⋊D4, C24⋊4S3, C15⋊D4, C2×S3×D5, C23⋊D10, C2×C15⋊D4, D5×C3⋊D4, (C2×C30)⋊D4
►On 120 points
Generators in S120
(1 52)(2 53)(3 54)(4 55)(5 56)(6 57)(7 58)(8 59)(9 60)(10 31)(11 32)(12 33)(13 34)(14 35)(15 36)(16 37)(17 38)(18 39)(19 40)(20 41)(21 42)(22 43)(23 44)(24 45)(25 46)(26 47)(27 48)(28 49)(29 50)(30 51)(61 112)(62 113)(63 114)(64 115)(65 116)(66 117)(67 118)(68 119)(69 120)(70 91)(71 92)(72 93)(73 94)(74 95)(75 96)(76 97)(77 98)(78 99)(79 100)(80 101)(81 102)(82 103)(83 104)(84 105)(85 106)(86 107)(87 108)(88 109)(89 110)(90 111)
(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 68 52 104)(2 67 53 103)(3 66 54 102)(4 65 55 101)(5 64 56 100)(6 63 57 99)(7 62 58 98)(8 61 59 97)(9 90 60 96)(10 89 31 95)(11 88 32 94)(12 87 33 93)(13 86 34 92)(14 85 35 91)(15 84 36 120)(16 83 37 119)(17 82 38 118)(18 81 39 117)(19 80 40 116)(20 79 41 115)(21 78 42 114)(22 77 43 113)(23 76 44 112)(24 75 45 111)(25 74 46 110)(26 73 47 109)(27 72 48 108)(28 71 49 107)(29 70 50 106)(30 69 51 105)
(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 58)(32 47)(33 36)(34 55)(35 44)(37 52)(38 41)(39 60)(40 49)(42 57)(43 46)(45 54)(48 51)(50 59)(53 56)(61 106)(62 95)(63 114)(64 103)(65 92)(66 111)(67 100)(68 119)(69 108)(70 97)(71 116)(72 105)(73 94)(74 113)(75 102)(76 91)(77 110)(78 99)(79 118)(80 107)(81 96)(82 115)(83 104)(84 93)(85 112)(86 101)(87 120)(88 109)(89 98)(90 117)
G:=sub<Sym(120)| (1,52)(2,53)(3,54)(4,55)(5,56)(6,57)(7,58)(8,59)(9,60)(10,31)(11,32)(12,33)(13,34)(14,35)(15,36)(16,37)(17,38)(18,39)(19,40)(20,41)(21,42)(22,43)(23,44)(24,45)(25,46)(26,47)(27,48)(28,49)(29,50)(30,51)(61,112)(62,113)(63,114)(64,115)(65,116)(66,117)(67,118)(68,119)(69,120)(70,91)(71,92)(72,93)(73,94)(74,95)(75,96)(76,97)(77,98)(78,99)(79,100)(80,101)(81,102)(82,103)(83,104)(84,105)(85,106)(86,107)(87,108)(88,109)(89,110)(90,111), (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), (1,68,52,104)(2,67,53,103)(3,66,54,102)(4,65,55,101)(5,64,56,100)(6,63,57,99)(7,62,58,98)(8,61,59,97)(9,90,60,96)(10,89,31,95)(11,88,32,94)(12,87,33,93)(13,86,34,92)(14,85,35,91)(15,84,36,120)(16,83,37,119)(17,82,38,118)(18,81,39,117)(19,80,40,116)(20,79,41,115)(21,78,42,114)(22,77,43,113)(23,76,44,112)(24,75,45,111)(25,74,46,110)(26,73,47,109)(27,72,48,108)(28,71,49,107)(29,70,50,106)(30,69,51,105), (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)(31,58)(32,47)(33,36)(34,55)(35,44)(37,52)(38,41)(39,60)(40,49)(42,57)(43,46)(45,54)(48,51)(50,59)(53,56)(61,106)(62,95)(63,114)(64,103)(65,92)(66,111)(67,100)(68,119)(69,108)(70,97)(71,116)(72,105)(73,94)(74,113)(75,102)(76,91)(77,110)(78,99)(79,118)(80,107)(81,96)(82,115)(83,104)(84,93)(85,112)(86,101)(87,120)(88,109)(89,98)(90,117)>;
G:=Group( (1,52)(2,53)(3,54)(4,55)(5,56)(6,57)(7,58)(8,59)(9,60)(10,31)(11,32)(12,33)(13,34)(14,35)(15,36)(16,37)(17,38)(18,39)(19,40)(20,41)(21,42)(22,43)(23,44)(24,45)(25,46)(26,47)(27,48)(28,49)(29,50)(30,51)(61,112)(62,113)(63,114)(64,115)(65,116)(66,117)(67,118)(68,119)(69,120)(70,91)(71,92)(72,93)(73,94)(74,95)(75,96)(76,97)(77,98)(78,99)(79,100)(80,101)(81,102)(82,103)(83,104)(84,105)(85,106)(86,107)(87,108)(88,109)(89,110)(90,111), (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), (1,68,52,104)(2,67,53,103)(3,66,54,102)(4,65,55,101)(5,64,56,100)(6,63,57,99)(7,62,58,98)(8,61,59,97)(9,90,60,96)(10,89,31,95)(11,88,32,94)(12,87,33,93)(13,86,34,92)(14,85,35,91)(15,84,36,120)(16,83,37,119)(17,82,38,118)(18,81,39,117)(19,80,40,116)(20,79,41,115)(21,78,42,114)(22,77,43,113)(23,76,44,112)(24,75,45,111)(25,74,46,110)(26,73,47,109)(27,72,48,108)(28,71,49,107)(29,70,50,106)(30,69,51,105), (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)(31,58)(32,47)(33,36)(34,55)(35,44)(37,52)(38,41)(39,60)(40,49)(42,57)(43,46)(45,54)(48,51)(50,59)(53,56)(61,106)(62,95)(63,114)(64,103)(65,92)(66,111)(67,100)(68,119)(69,108)(70,97)(71,116)(72,105)(73,94)(74,113)(75,102)(76,91)(77,110)(78,99)(79,118)(80,107)(81,96)(82,115)(83,104)(84,93)(85,112)(86,101)(87,120)(88,109)(89,98)(90,117) );
G=PermutationGroup([[(1,52),(2,53),(3,54),(4,55),(5,56),(6,57),(7,58),(8,59),(9,60),(10,31),(11,32),(12,33),(13,34),(14,35),(15,36),(16,37),(17,38),(18,39),(19,40),(20,41),(21,42),(22,43),(23,44),(24,45),(25,46),(26,47),(27,48),(28,49),(29,50),(30,51),(61,112),(62,113),(63,114),(64,115),(65,116),(66,117),(67,118),(68,119),(69,120),(70,91),(71,92),(72,93),(73,94),(74,95),(75,96),(76,97),(77,98),(78,99),(79,100),(80,101),(81,102),(82,103),(83,104),(84,105),(85,106),(86,107),(87,108),(88,109),(89,110),(90,111)], [(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)], [(1,68,52,104),(2,67,53,103),(3,66,54,102),(4,65,55,101),(5,64,56,100),(6,63,57,99),(7,62,58,98),(8,61,59,97),(9,90,60,96),(10,89,31,95),(11,88,32,94),(12,87,33,93),(13,86,34,92),(14,85,35,91),(15,84,36,120),(16,83,37,119),(17,82,38,118),(18,81,39,117),(19,80,40,116),(20,79,41,115),(21,78,42,114),(22,77,43,113),(23,76,44,112),(24,75,45,111),(25,74,46,110),(26,73,47,109),(27,72,48,108),(28,71,49,107),(29,70,50,106),(30,69,51,105)], [(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),(31,58),(32,47),(33,36),(34,55),(35,44),(37,52),(38,41),(39,60),(40,49),(42,57),(43,46),(45,54),(48,51),(50,59),(53,56),(61,106),(62,95),(63,114),(64,103),(65,92),(66,111),(67,100),(68,119),(69,108),(70,97),(71,116),(72,105),(73,94),(74,113),(75,102),(76,91),(77,110),(78,99),(79,118),(80,107),(81,96),(82,115),(83,104),(84,93),(85,112),(86,101),(87,120),(88,109),(89,98),(90,117)]])
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