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G = D6010C4order 480 = 25·3·5

4th semidirect product of D60 and C4 acting via C4/C2=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D6010C4, C60.211D4, C22.3D60, Dic3010C4, M4(2)⋊4D15, C1518C4≀C2, (C2×C6).4D20, C4.3(C4×D15), (C2×C30).1D4, C20.50(C4×S3), C60.80(C2×C4), C12.18(C4×D5), (C2×C10).4D12, (C2×C20).71D6, (C2×C4).37D30, C54(D12⋊C4), C32(D207C4), (C4×Dic15)⋊1C2, (C2×C12).72D10, C10.36(D6⋊C4), C4.29(C157D4), (C5×M4(2))⋊10S3, (C3×M4(2))⋊10D5, (C2×C60).57C22, D6011C2.8C2, C12.108(C5⋊D4), C20.108(C3⋊D4), C30.78(C22⋊C4), (C15×M4(2))⋊20C2, C6.21(D10⋊C4), C2.11(D303C4), SmallGroup(480,185)

Series: Derived Chief Lower central Upper central

C1C60 — D6010C4
C1C5C15C30C60C2×C60D6011C2 — D6010C4
C15C30C60 — D6010C4
C1C4C2×C4M4(2)

Generators and relations for D6010C4
 G = < a,b,c | a60=b2=c4=1, bab=a-1, cac-1=a29, cbc-1=a43b >

Subgroups: 596 in 88 conjugacy classes, 33 normal (all characteristic)
C1, C2, C2 [×2], C3, C4 [×2], C4 [×3], C22, C22, C5, S3, C6, C6, C8, C2×C4, C2×C4 [×2], D4 [×2], Q8, D5, C10, C10, Dic3 [×3], C12 [×2], D6, C2×C6, C15, C42, M4(2), C4○D4, Dic5 [×3], C20 [×2], D10, C2×C10, C24, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, D15, C30, C30, C4≀C2, C40, Dic10, C4×D5, D20, C2×Dic5, C5⋊D4, C2×C20, C4×Dic3, C3×M4(2), C4○D12, Dic15 [×3], C60 [×2], D30, C2×C30, C4×Dic5, C5×M4(2), C4○D20, D12⋊C4, C120, Dic30, C4×D15, D60, C2×Dic15, C157D4, C2×C60, D207C4, C4×Dic15, C15×M4(2), D6011C2, D6010C4
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C2×C4, D4 [×2], D5, D6, C22⋊C4, D10, C4×S3, D12, C3⋊D4, D15, C4≀C2, C4×D5, D20, C5⋊D4, D6⋊C4, D30, D10⋊C4, D12⋊C4, C4×D15, D60, C157D4, D207C4, D303C4, D6010C4

Smallest permutation representation of D6010C4
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 94)(2 93)(3 92)(4 91)(5 90)(6 89)(7 88)(8 87)(9 86)(10 85)(11 84)(12 83)(13 82)(14 81)(15 80)(16 79)(17 78)(18 77)(19 76)(20 75)(21 74)(22 73)(23 72)(24 71)(25 70)(26 69)(27 68)(28 67)(29 66)(30 65)(31 64)(32 63)(33 62)(34 61)(35 120)(36 119)(37 118)(38 117)(39 116)(40 115)(41 114)(42 113)(43 112)(44 111)(45 110)(46 109)(47 108)(48 107)(49 106)(50 105)(51 104)(52 103)(53 102)(54 101)(55 100)(56 99)(57 98)(58 97)(59 96)(60 95)
(1 16 31 46)(2 45 32 15)(3 14 33 44)(4 43 34 13)(5 12 35 42)(6 41 36 11)(7 10 37 40)(8 39 38 9)(17 60 47 30)(18 29 48 59)(19 58 49 28)(20 27 50 57)(21 56 51 26)(22 25 52 55)(23 54 53 24)(61 69)(62 98)(63 67)(64 96)(66 94)(68 92)(70 90)(71 119)(72 88)(73 117)(74 86)(75 115)(76 84)(77 113)(78 82)(79 111)(81 109)(83 107)(85 105)(87 103)(89 101)(91 99)(93 97)(100 120)(102 118)(104 116)(106 114)(108 112)

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), (1,94)(2,93)(3,92)(4,91)(5,90)(6,89)(7,88)(8,87)(9,86)(10,85)(11,84)(12,83)(13,82)(14,81)(15,80)(16,79)(17,78)(18,77)(19,76)(20,75)(21,74)(22,73)(23,72)(24,71)(25,70)(26,69)(27,68)(28,67)(29,66)(30,65)(31,64)(32,63)(33,62)(34,61)(35,120)(36,119)(37,118)(38,117)(39,116)(40,115)(41,114)(42,113)(43,112)(44,111)(45,110)(46,109)(47,108)(48,107)(49,106)(50,105)(51,104)(52,103)(53,102)(54,101)(55,100)(56,99)(57,98)(58,97)(59,96)(60,95), (1,16,31,46)(2,45,32,15)(3,14,33,44)(4,43,34,13)(5,12,35,42)(6,41,36,11)(7,10,37,40)(8,39,38,9)(17,60,47,30)(18,29,48,59)(19,58,49,28)(20,27,50,57)(21,56,51,26)(22,25,52,55)(23,54,53,24)(61,69)(62,98)(63,67)(64,96)(66,94)(68,92)(70,90)(71,119)(72,88)(73,117)(74,86)(75,115)(76,84)(77,113)(78,82)(79,111)(81,109)(83,107)(85,105)(87,103)(89,101)(91,99)(93,97)(100,120)(102,118)(104,116)(106,114)(108,112)>;

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), (1,94)(2,93)(3,92)(4,91)(5,90)(6,89)(7,88)(8,87)(9,86)(10,85)(11,84)(12,83)(13,82)(14,81)(15,80)(16,79)(17,78)(18,77)(19,76)(20,75)(21,74)(22,73)(23,72)(24,71)(25,70)(26,69)(27,68)(28,67)(29,66)(30,65)(31,64)(32,63)(33,62)(34,61)(35,120)(36,119)(37,118)(38,117)(39,116)(40,115)(41,114)(42,113)(43,112)(44,111)(45,110)(46,109)(47,108)(48,107)(49,106)(50,105)(51,104)(52,103)(53,102)(54,101)(55,100)(56,99)(57,98)(58,97)(59,96)(60,95), (1,16,31,46)(2,45,32,15)(3,14,33,44)(4,43,34,13)(5,12,35,42)(6,41,36,11)(7,10,37,40)(8,39,38,9)(17,60,47,30)(18,29,48,59)(19,58,49,28)(20,27,50,57)(21,56,51,26)(22,25,52,55)(23,54,53,24)(61,69)(62,98)(63,67)(64,96)(66,94)(68,92)(70,90)(71,119)(72,88)(73,117)(74,86)(75,115)(76,84)(77,113)(78,82)(79,111)(81,109)(83,107)(85,105)(87,103)(89,101)(91,99)(93,97)(100,120)(102,118)(104,116)(106,114)(108,112) );

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)], [(1,94),(2,93),(3,92),(4,91),(5,90),(6,89),(7,88),(8,87),(9,86),(10,85),(11,84),(12,83),(13,82),(14,81),(15,80),(16,79),(17,78),(18,77),(19,76),(20,75),(21,74),(22,73),(23,72),(24,71),(25,70),(26,69),(27,68),(28,67),(29,66),(30,65),(31,64),(32,63),(33,62),(34,61),(35,120),(36,119),(37,118),(38,117),(39,116),(40,115),(41,114),(42,113),(43,112),(44,111),(45,110),(46,109),(47,108),(48,107),(49,106),(50,105),(51,104),(52,103),(53,102),(54,101),(55,100),(56,99),(57,98),(58,97),(59,96),(60,95)], [(1,16,31,46),(2,45,32,15),(3,14,33,44),(4,43,34,13),(5,12,35,42),(6,41,36,11),(7,10,37,40),(8,39,38,9),(17,60,47,30),(18,29,48,59),(19,58,49,28),(20,27,50,57),(21,56,51,26),(22,25,52,55),(23,54,53,24),(61,69),(62,98),(63,67),(64,96),(66,94),(68,92),(70,90),(71,119),(72,88),(73,117),(74,86),(75,115),(76,84),(77,113),(78,82),(79,111),(81,109),(83,107),(85,105),(87,103),(89,101),(91,99),(93,97),(100,120),(102,118),(104,116),(106,114),(108,112)])

84 conjugacy classes

class 1 2A2B2C 3 4A4B4C4D4E4F4G4H5A5B6A6B8A8B10A10B10C10D12A12B12C15A15B15C15D20A20B20C20D20E20F24A24B24C24D30A30B30C30D30E30F30G30H40A···40H60A···60H60I60J60K60L120A···120P
order1222344444444556688101010101212121515151520202020202024242424303030303030303040···4060···6060606060120···120
size1126021123030303060222444224422422222222444444222244444···42···244444···4

84 irreducible representations

dim111111222222222222222222444
type+++++++++++++++
imageC1C2C2C2C4C4S3D4D4D5D6D10C4×S3C3⋊D4D12D15C4≀C2C4×D5C5⋊D4D20D30C4×D15C157D4D60D12⋊C4D207C4D6010C4
kernelD6010C4C4×Dic15C15×M4(2)D6011C2Dic30D60C5×M4(2)C60C2×C30C3×M4(2)C2×C20C2×C12C20C20C2×C10M4(2)C15C12C12C2×C6C2×C4C4C4C22C5C3C1
# reps111122111212222444444888248

Matrix representation of D6010C4 in GL4(𝔽241) generated by

688000
9821100
00640
00168177
,
17714800
1146400
002223
0012119
,
5224000
5218900
001770
002311
G:=sub<GL(4,GF(241))| [68,98,0,0,80,211,0,0,0,0,64,168,0,0,0,177],[177,114,0,0,148,64,0,0,0,0,222,121,0,0,3,19],[52,52,0,0,240,189,0,0,0,0,177,231,0,0,0,1] >;

D6010C4 in GAP, Magma, Sage, TeX

D_{60}\rtimes_{10}C_4
% in TeX

G:=Group("D60:10C4");
// GroupNames label

G:=SmallGroup(480,185);
// by ID

G=gap.SmallGroup(480,185);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,141,36,100,675,346,80,2693,18822]);
// Polycyclic

G:=Group<a,b,c|a^60=b^2=c^4=1,b*a*b=a^-1,c*a*c^-1=a^29,c*b*c^-1=a^43*b>;
// generators/relations

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