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G = D6030C22order 480 = 25·3·5

11st semidirect product of D60 and C22 acting via C22/C2=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C60.84D4, D20.32D6, C12.15D20, D6030C22, C60.105C23, Dic3027C22, C3⋊C84D10, (C6×D20)⋊2C2, (C2×D20)⋊9S3, C3⋊D4014C2, C34(C8⋊D10), C6.50(C2×D20), (C2×C30).55D4, (C2×C6).40D20, C30.87(C2×D4), (C2×C20).96D6, C4.Dic38D5, C1512(C8⋊C22), (C2×C12).98D10, C51(D126C22), D6011C25C2, C6.D2013C2, C4.23(C3⋊D20), C20.29(C3⋊D4), (C2×C60).33C22, C12.96(C22×D5), C20.155(C22×S3), (C3×D20).37C22, C22.4(C3⋊D20), C4.104(C2×S3×D5), (C5×C3⋊C8)⋊18C22, (C2×C4).14(S3×D5), C2.9(C2×C3⋊D20), C10.5(C2×C3⋊D4), (C5×C4.Dic3)⋊2C2, (C2×C10).12(C3⋊D4), SmallGroup(480,388)

Series: Derived Chief Lower central Upper central

C1C60 — D6030C22
C1C5C15C30C60C3×D20C6.D20 — D6030C22
C15C30C60 — D6030C22
C1C2C2×C4

Generators and relations for D6030C22
 G = < a,b,c,d | a60=b2=c2=d2=1, bab=a-1, cac=a19, ad=da, cbc=a3b, dbd=a30b, cd=dc >

Subgroups: 860 in 136 conjugacy classes, 44 normal (32 characteristic)
C1, C2, C2 [×4], C3, C4 [×2], C4, C22, C22 [×5], C5, S3, C6, C6 [×3], C8 [×2], C2×C4, C2×C4, D4 [×5], Q8, C23, D5 [×3], C10, C10, Dic3, C12 [×2], D6, C2×C6, C2×C6 [×4], C15, M4(2), D8 [×2], SD16 [×2], C2×D4, C4○D4, Dic5, C20 [×2], D10 [×5], C2×C10, C3⋊C8 [×2], Dic6, C4×S3, D12, C3⋊D4, C2×C12, C3×D4 [×3], C22×C6, C3×D5 [×2], D15, C30, C30, C8⋊C22, C40 [×2], Dic10, C4×D5, D20 [×2], D20 [×2], C5⋊D4, C2×C20, C22×D5, C4.Dic3, D4⋊S3 [×2], D4.S3 [×2], C4○D12, C6×D4, Dic15, C60 [×2], C6×D5 [×4], D30, C2×C30, C40⋊C2 [×2], D40 [×2], C5×M4(2), C2×D20, C4○D20, D126C22, C5×C3⋊C8 [×2], C3×D20 [×2], C3×D20, Dic30, C4×D15, D60, C157D4, C2×C60, D5×C2×C6, C8⋊D10, C3⋊D40 [×2], C6.D20 [×2], C5×C4.Dic3, C6×D20, D6011C2, D6030C22
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], C2×D4, D10 [×3], C3⋊D4 [×2], C22×S3, C8⋊C22, D20 [×2], C22×D5, C2×C3⋊D4, S3×D5, C2×D20, D126C22, C3⋊D20 [×2], C2×S3×D5, C8⋊D10, C2×C3⋊D20, D6030C22

Smallest permutation representation of D6030C22
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)
(1 102)(2 101)(3 100)(4 99)(5 98)(6 97)(7 96)(8 95)(9 94)(10 93)(11 92)(12 91)(13 90)(14 89)(15 88)(16 87)(17 86)(18 85)(19 84)(20 83)(21 82)(22 81)(23 80)(24 79)(25 78)(26 77)(27 76)(28 75)(29 74)(30 73)(31 72)(32 71)(33 70)(34 69)(35 68)(36 67)(37 66)(38 65)(39 64)(40 63)(41 62)(42 61)(43 120)(44 119)(45 118)(46 117)(47 116)(48 115)(49 114)(50 113)(51 112)(52 111)(53 110)(54 109)(55 108)(56 107)(57 106)(58 105)(59 104)(60 103)
(2 20)(3 39)(4 58)(5 17)(6 36)(7 55)(8 14)(9 33)(10 52)(12 30)(13 49)(15 27)(16 46)(18 24)(19 43)(22 40)(23 59)(25 37)(26 56)(28 34)(29 53)(32 50)(35 47)(38 44)(42 60)(45 57)(48 54)(61 100)(62 119)(63 78)(64 97)(65 116)(66 75)(67 94)(68 113)(69 72)(70 91)(71 110)(73 88)(74 107)(76 85)(77 104)(79 82)(80 101)(81 120)(83 98)(84 117)(86 95)(87 114)(89 92)(90 111)(93 108)(96 105)(99 102)(103 118)(106 115)(109 112)
(61 91)(62 92)(63 93)(64 94)(65 95)(66 96)(67 97)(68 98)(69 99)(70 100)(71 101)(72 102)(73 103)(74 104)(75 105)(76 106)(77 107)(78 108)(79 109)(80 110)(81 111)(82 112)(83 113)(84 114)(85 115)(86 116)(87 117)(88 118)(89 119)(90 120)

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

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

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

57 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C5A5B6A6B6C6D6E6F6G8A8B10A10B10C10D12A12B15A15B20A20B20C20D20E20F30A···30F40A···40H60A···60H
order122222344455666666688101010101212151520202020202030···3040···4060···60
size1122020602226022222202020201212224444442222444···412···124···4

57 irreducible representations

dim11111122222222222244444444
type++++++++++++++++++++++
imageC1C2C2C2C2C2S3D4D4D5D6D6D10D10C3⋊D4C3⋊D4D20D20C8⋊C22S3×D5D126C22C3⋊D20C2×S3×D5C3⋊D20C8⋊D10D6030C22
kernelD6030C22C3⋊D40C6.D20C5×C4.Dic3C6×D20D6011C2C2×D20C60C2×C30C4.Dic3D20C2×C20C3⋊C8C2×C12C20C2×C10C12C2×C6C15C2×C4C5C4C4C22C3C1
# reps12211111122142224412222248

Matrix representation of D6030C22 in GL4(𝔽241) generated by

989800
14313300
002424
0021767
,
002250
0013216
15000
5722600
,
118900
024000
00200119
0015641
,
1000
0100
002400
000240
G:=sub<GL(4,GF(241))| [98,143,0,0,98,133,0,0,0,0,24,217,0,0,24,67],[0,0,15,57,0,0,0,226,225,132,0,0,0,16,0,0],[1,0,0,0,189,240,0,0,0,0,200,156,0,0,119,41],[1,0,0,0,0,1,0,0,0,0,240,0,0,0,0,240] >;

D6030C22 in GAP, Magma, Sage, TeX

D_{60}\rtimes_{30}C_2^2
% in TeX

G:=Group("D60:30C2^2");
// GroupNames label

G:=SmallGroup(480,388);
// by ID

G=gap.SmallGroup(480,388);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,141,64,219,675,80,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^60=b^2=c^2=d^2=1,b*a*b=a^-1,c*a*c=a^19,a*d=d*a,c*b*c=a^3*b,d*b*d=a^30*b,c*d=d*c>;
// generators/relations

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