metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D60⋊16C4, C60.101D4, C20.57D12, Dic30⋊16C4, C15⋊10C4≀C2, C20.41(C4×S3), (C4×Dic5)⋊2S3, (C2×C20).57D6, C12.33(C4×D5), (C2×C6).33D20, (C2×C30).28D4, C4.Dic3⋊1D5, C5⋊3(C42⋊4S3), C3⋊1(D20⋊7C4), C60.110(C2×C4), (C12×Dic5)⋊2C2, C10.24(D6⋊C4), (C2×C12).313D10, C4.29(C5⋊D12), C12.65(C5⋊D4), (C2×C60).44C22, C4.8(D30.C2), D60⋊11C2.6C2, C6.9(D10⋊C4), C30.64(C22⋊C4), C22.2(C3⋊D20), C2.10(D30⋊4C4), (C2×C4).138(S3×D5), (C5×C4.Dic3)⋊4C2, (C2×C10).3(C3⋊D4), SmallGroup(480,57)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D60⋊16C4
G = < a,b,c | a60=b2=c4=1, bab=a-1, cac-1=a49, cbc-1=a3b >
Subgroups: 524 in 88 conjugacy classes, 32 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, S3, C6, C6, C8, C2×C4, C2×C4, D4, Q8, D5, C10, C10, Dic3, C12, C12, D6, C2×C6, C15, C42, M4(2), C4○D4, Dic5, C20, D10, C2×C10, C3⋊C8, Dic6, C4×S3, D12, C3⋊D4, C2×C12, C2×C12, D15, C30, C30, C4≀C2, C40, Dic10, C4×D5, D20, C2×Dic5, C5⋊D4, C2×C20, C4.Dic3, C4×C12, C4○D12, C3×Dic5, Dic15, C60, D30, C2×C30, C4×Dic5, C5×M4(2), C4○D20, C42⋊4S3, C5×C3⋊C8, C6×Dic5, Dic30, C4×D15, D60, C15⋊7D4, C2×C60, D20⋊7C4, C12×Dic5, C5×C4.Dic3, D60⋊11C2, D60⋊16C4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, D5, D6, C22⋊C4, D10, C4×S3, D12, C3⋊D4, C4≀C2, C4×D5, D20, C5⋊D4, D6⋊C4, S3×D5, D10⋊C4, C42⋊4S3, D30.C2, C3⋊D20, C5⋊D12, D20⋊7C4, D30⋊4C4, D60⋊16C4
►On 120 points
Generators in S120
(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)
(1 120)(2 119)(3 118)(4 117)(5 116)(6 115)(7 114)(8 113)(9 112)(10 111)(11 110)(12 109)(13 108)(14 107)(15 106)(16 105)(17 104)(18 103)(19 102)(20 101)(21 100)(22 99)(23 98)(24 97)(25 96)(26 95)(27 94)(28 93)(29 92)(30 91)(31 90)(32 89)(33 88)(34 87)(35 86)(36 85)(37 84)(38 83)(39 82)(40 81)(41 80)(42 79)(43 78)(44 77)(45 76)(46 75)(47 74)(48 73)(49 72)(50 71)(51 70)(52 69)(53 68)(54 67)(55 66)(56 65)(57 64)(58 63)(59 62)(60 61)
(1 31)(2 20)(3 9)(4 58)(5 47)(6 36)(7 25)(8 14)(10 52)(11 41)(12 30)(13 19)(15 57)(16 46)(17 35)(18 24)(21 51)(22 40)(23 29)(26 56)(27 45)(28 34)(32 50)(33 39)(37 55)(38 44)(42 60)(43 49)(48 54)(53 59)(61 106 91 76)(62 95 92 65)(63 84 93 114)(64 73 94 103)(66 111 96 81)(67 100 97 70)(68 89 98 119)(69 78 99 108)(71 116 101 86)(72 105 102 75)(74 83 104 113)(77 110 107 80)(79 88 109 118)(82 115 112 85)(87 120 117 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), (1,120)(2,119)(3,118)(4,117)(5,116)(6,115)(7,114)(8,113)(9,112)(10,111)(11,110)(12,109)(13,108)(14,107)(15,106)(16,105)(17,104)(18,103)(19,102)(20,101)(21,100)(22,99)(23,98)(24,97)(25,96)(26,95)(27,94)(28,93)(29,92)(30,91)(31,90)(32,89)(33,88)(34,87)(35,86)(36,85)(37,84)(38,83)(39,82)(40,81)(41,80)(42,79)(43,78)(44,77)(45,76)(46,75)(47,74)(48,73)(49,72)(50,71)(51,70)(52,69)(53,68)(54,67)(55,66)(56,65)(57,64)(58,63)(59,62)(60,61), (1,31)(2,20)(3,9)(4,58)(5,47)(6,36)(7,25)(8,14)(10,52)(11,41)(12,30)(13,19)(15,57)(16,46)(17,35)(18,24)(21,51)(22,40)(23,29)(26,56)(27,45)(28,34)(32,50)(33,39)(37,55)(38,44)(42,60)(43,49)(48,54)(53,59)(61,106,91,76)(62,95,92,65)(63,84,93,114)(64,73,94,103)(66,111,96,81)(67,100,97,70)(68,89,98,119)(69,78,99,108)(71,116,101,86)(72,105,102,75)(74,83,104,113)(77,110,107,80)(79,88,109,118)(82,115,112,85)(87,120,117,90)>;
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), (1,120)(2,119)(3,118)(4,117)(5,116)(6,115)(7,114)(8,113)(9,112)(10,111)(11,110)(12,109)(13,108)(14,107)(15,106)(16,105)(17,104)(18,103)(19,102)(20,101)(21,100)(22,99)(23,98)(24,97)(25,96)(26,95)(27,94)(28,93)(29,92)(30,91)(31,90)(32,89)(33,88)(34,87)(35,86)(36,85)(37,84)(38,83)(39,82)(40,81)(41,80)(42,79)(43,78)(44,77)(45,76)(46,75)(47,74)(48,73)(49,72)(50,71)(51,70)(52,69)(53,68)(54,67)(55,66)(56,65)(57,64)(58,63)(59,62)(60,61), (1,31)(2,20)(3,9)(4,58)(5,47)(6,36)(7,25)(8,14)(10,52)(11,41)(12,30)(13,19)(15,57)(16,46)(17,35)(18,24)(21,51)(22,40)(23,29)(26,56)(27,45)(28,34)(32,50)(33,39)(37,55)(38,44)(42,60)(43,49)(48,54)(53,59)(61,106,91,76)(62,95,92,65)(63,84,93,114)(64,73,94,103)(66,111,96,81)(67,100,97,70)(68,89,98,119)(69,78,99,108)(71,116,101,86)(72,105,102,75)(74,83,104,113)(77,110,107,80)(79,88,109,118)(82,115,112,85)(87,120,117,90) );
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)], [(1,120),(2,119),(3,118),(4,117),(5,116),(6,115),(7,114),(8,113),(9,112),(10,111),(11,110),(12,109),(13,108),(14,107),(15,106),(16,105),(17,104),(18,103),(19,102),(20,101),(21,100),(22,99),(23,98),(24,97),(25,96),(26,95),(27,94),(28,93),(29,92),(30,91),(31,90),(32,89),(33,88),(34,87),(35,86),(36,85),(37,84),(38,83),(39,82),(40,81),(41,80),(42,79),(43,78),(44,77),(45,76),(46,75),(47,74),(48,73),(49,72),(50,71),(51,70),(52,69),(53,68),(54,67),(55,66),(56,65),(57,64),(58,63),(59,62),(60,61)], [(1,31),(2,20),(3,9),(4,58),(5,47),(6,36),(7,25),(8,14),(10,52),(11,41),(12,30),(13,19),(15,57),(16,46),(17,35),(18,24),(21,51),(22,40),(23,29),(26,56),(27,45),(28,34),(32,50),(33,39),(37,55),(38,44),(42,60),(43,49),(48,54),(53,59),(61,106,91,76),(62,95,92,65),(63,84,93,114),(64,73,94,103),(66,111,96,81),(67,100,97,70),(68,89,98,119),(69,78,99,108),(71,116,101,86),(72,105,102,75),(74,83,104,113),(77,110,107,80),(79,88,109,118),(82,115,112,85),(87,120,117,90)]])
66 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5A | 5B | 6A | 6B | 6C | 8A | 8B | 10A | 10B | 10C | 10D | 12A | 12B | 12C | 12D | 12E | ··· | 12L | 15A | 15B | 20A | 20B | 20C | 20D | 20E | 20F | 30A | ··· | 30F | 40A | ··· | 40H | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 8 | 8 | 10 | 10 | 10 | 10 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | 20 | 30 | ··· | 30 | 40 | ··· | 40 | 60 | ··· | 60 |
| size | 1 | 1 | 2 | 60 | 2 | 1 | 1 | 2 | 10 | 10 | 10 | 10 | 60 | 2 | 2 | 2 | 2 | 2 | 12 | 12 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 2 | 10 | ··· | 10 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | ··· | 4 | 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 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | S3 | D4 | D4 | D5 | D6 | D10 | C4×S3 | D12 | C3⋊D4 | C4≀C2 | C4×D5 | C5⋊D4 | D20 | C42⋊4S3 | S3×D5 | D30.C2 | C5⋊D12 | C3⋊D20 | D20⋊7C4 | D60⋊16C4 |
| kernel | D60⋊16C4 | C12×Dic5 | C5×C4.Dic3 | D60⋊11C2 | Dic30 | D60 | C4×Dic5 | C60 | C2×C30 | C4.Dic3 | C2×C20 | C2×C12 | C20 | C20 | C2×C10 | C15 | C12 | C12 | C2×C6 | C5 | C2×C4 | C4 | C4 | C22 | C3 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 2 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 2 | 2 | 2 | 2 | 4 | 8 |
Matrix representation of D60⋊16C4 ►in GL4(𝔽241) generated by
| 181 | 0 | 0 | 0 |
| 144 | 4 | 0 | 0 |
| 0 | 0 | 190 | 51 |
| 0 | 0 | 190 | 240 |
| 21 | 170 | 0 | 0 |
| 125 | 220 | 0 | 0 |
| 0 | 0 | 0 | 240 |
| 0 | 0 | 240 | 0 |
| 240 | 0 | 0 | 0 |
| 39 | 177 | 0 | 0 |
| 0 | 0 | 51 | 190 |
| 0 | 0 | 240 | 190 |
G:=sub<GL(4,GF(241))| [181,144,0,0,0,4,0,0,0,0,190,190,0,0,51,240],[21,125,0,0,170,220,0,0,0,0,0,240,0,0,240,0],[240,39,0,0,0,177,0,0,0,0,51,240,0,0,190,190] >;
D60⋊16C4 in GAP, Magma, Sage, TeX
D_{60}\rtimes_{16}C_4 % in TeX
G:=Group("D60:16C4"); // GroupNames label
G:=SmallGroup(480,57);
// by ID
G=gap.SmallGroup(480,57);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,141,92,219,346,80,1356,18822]);
// Polycyclic
G:=Group<a,b,c|a^60=b^2=c^4=1,b*a*b=a^-1,c*a*c^-1=a^49,c*b*c^-1=a^3*b>;
// generators/relations