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G = D6036C22order 480 = 25·3·5

17th semidirect product of D60 and C22 acting via C22/C2=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C60.81D4, C20.14D12, D12.31D10, D6036C22, C60.128C23, Dic3032C22, C52C83D6, (C2×D12)⋊9D5, C54(C8⋊D6), (C10×D12)⋊2C2, C5⋊D2413C2, (C2×C20).90D6, (C2×C30).47D4, C30.79(C2×D4), C4.Dic57S3, C1510(C8⋊C22), C10.49(C2×D12), (C2×C10).39D12, (C2×C12).90D10, D6011C22C2, D12.D513C2, C31(D4.D10), C12.27(C5⋊D4), C4.23(C5⋊D12), C20.90(C22×S3), (C2×C60).25C22, (C5×D12).36C22, C12.151(C22×D5), C22.4(C5⋊D12), C4.76(C2×S3×D5), C6.3(C2×C5⋊D4), (C2×C4).10(S3×D5), C2.7(C2×C5⋊D12), (C3×C4.Dic5)⋊2C2, (C3×C52C8)⋊17C22, (C2×C6).11(C5⋊D4), SmallGroup(480,380)

Series: Derived Chief Lower central Upper central

C1C60 — D6036C22
C1C5C15C30C60C3×C52C8C5⋊D24 — D6036C22
C15C30C60 — D6036C22
C1C2C2×C4

Generators and relations for D6036C22
 G = < a,b,c,d | a60=b2=c2=d2=1, bab=a-1, cac=a11, ad=da, cbc=a25b, dbd=a30b, cd=dc >

Subgroups: 764 in 136 conjugacy classes, 44 normal (32 characteristic)
C1, C2, C2 [×4], C3, C4 [×2], C4, C22, C22 [×5], C5, S3 [×3], C6, C6, C8 [×2], C2×C4, C2×C4, D4 [×5], Q8, C23, D5, C10, C10 [×3], Dic3, C12 [×2], D6 [×5], C2×C6, C15, M4(2), D8 [×2], SD16 [×2], C2×D4, C4○D4, Dic5, C20 [×2], D10, C2×C10, C2×C10 [×4], C24 [×2], Dic6, C4×S3, D12 [×2], D12 [×2], C3⋊D4, C2×C12, C22×S3, C5×S3 [×2], D15, C30, C30, C8⋊C22, C52C8 [×2], Dic10, C4×D5, D20, C5⋊D4, C2×C20, C5×D4 [×3], C22×C10, C24⋊C2 [×2], D24 [×2], C3×M4(2), C2×D12, C4○D12, Dic15, C60 [×2], S3×C10 [×4], D30, C2×C30, C4.Dic5, D4⋊D5 [×2], D4.D5 [×2], C4○D20, D4×C10, C8⋊D6, C3×C52C8 [×2], C5×D12 [×2], C5×D12, Dic30, C4×D15, D60, C157D4, C2×C60, S3×C2×C10, D4.D10, C5⋊D24 [×2], D12.D5 [×2], C3×C4.Dic5, C10×D12, D6011C2, D6036C22
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], C2×D4, D10 [×3], D12 [×2], C22×S3, C8⋊C22, C5⋊D4 [×2], C22×D5, C2×D12, S3×D5, C2×C5⋊D4, C8⋊D6, C5⋊D12 [×2], C2×S3×D5, D4.D10, C2×C5⋊D12, D6036C22

Smallest permutation representation of D6036C22
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 73)(2 72)(3 71)(4 70)(5 69)(6 68)(7 67)(8 66)(9 65)(10 64)(11 63)(12 62)(13 61)(14 120)(15 119)(16 118)(17 117)(18 116)(19 115)(20 114)(21 113)(22 112)(23 111)(24 110)(25 109)(26 108)(27 107)(28 106)(29 105)(30 104)(31 103)(32 102)(33 101)(34 100)(35 99)(36 98)(37 97)(38 96)(39 95)(40 94)(41 93)(42 92)(43 91)(44 90)(45 89)(46 88)(47 87)(48 86)(49 85)(50 84)(51 83)(52 82)(53 81)(54 80)(55 79)(56 78)(57 77)(58 76)(59 75)(60 74)
(2 12)(3 23)(4 34)(5 45)(6 56)(8 18)(9 29)(10 40)(11 51)(14 24)(15 35)(16 46)(17 57)(20 30)(21 41)(22 52)(26 36)(27 47)(28 58)(32 42)(33 53)(38 48)(39 59)(44 54)(50 60)(61 96)(62 107)(63 118)(64 69)(65 80)(66 91)(67 102)(68 113)(70 75)(71 86)(72 97)(73 108)(74 119)(76 81)(77 92)(78 103)(79 114)(82 87)(83 98)(84 109)(85 120)(88 93)(89 104)(90 115)(94 99)(95 110)(100 105)(101 116)(106 111)(112 117)
(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), (1,73)(2,72)(3,71)(4,70)(5,69)(6,68)(7,67)(8,66)(9,65)(10,64)(11,63)(12,62)(13,61)(14,120)(15,119)(16,118)(17,117)(18,116)(19,115)(20,114)(21,113)(22,112)(23,111)(24,110)(25,109)(26,108)(27,107)(28,106)(29,105)(30,104)(31,103)(32,102)(33,101)(34,100)(35,99)(36,98)(37,97)(38,96)(39,95)(40,94)(41,93)(42,92)(43,91)(44,90)(45,89)(46,88)(47,87)(48,86)(49,85)(50,84)(51,83)(52,82)(53,81)(54,80)(55,79)(56,78)(57,77)(58,76)(59,75)(60,74), (2,12)(3,23)(4,34)(5,45)(6,56)(8,18)(9,29)(10,40)(11,51)(14,24)(15,35)(16,46)(17,57)(20,30)(21,41)(22,52)(26,36)(27,47)(28,58)(32,42)(33,53)(38,48)(39,59)(44,54)(50,60)(61,96)(62,107)(63,118)(64,69)(65,80)(66,91)(67,102)(68,113)(70,75)(71,86)(72,97)(73,108)(74,119)(76,81)(77,92)(78,103)(79,114)(82,87)(83,98)(84,109)(85,120)(88,93)(89,104)(90,115)(94,99)(95,110)(100,105)(101,116)(106,111)(112,117), (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), (1,73)(2,72)(3,71)(4,70)(5,69)(6,68)(7,67)(8,66)(9,65)(10,64)(11,63)(12,62)(13,61)(14,120)(15,119)(16,118)(17,117)(18,116)(19,115)(20,114)(21,113)(22,112)(23,111)(24,110)(25,109)(26,108)(27,107)(28,106)(29,105)(30,104)(31,103)(32,102)(33,101)(34,100)(35,99)(36,98)(37,97)(38,96)(39,95)(40,94)(41,93)(42,92)(43,91)(44,90)(45,89)(46,88)(47,87)(48,86)(49,85)(50,84)(51,83)(52,82)(53,81)(54,80)(55,79)(56,78)(57,77)(58,76)(59,75)(60,74), (2,12)(3,23)(4,34)(5,45)(6,56)(8,18)(9,29)(10,40)(11,51)(14,24)(15,35)(16,46)(17,57)(20,30)(21,41)(22,52)(26,36)(27,47)(28,58)(32,42)(33,53)(38,48)(39,59)(44,54)(50,60)(61,96)(62,107)(63,118)(64,69)(65,80)(66,91)(67,102)(68,113)(70,75)(71,86)(72,97)(73,108)(74,119)(76,81)(77,92)(78,103)(79,114)(82,87)(83,98)(84,109)(85,120)(88,93)(89,104)(90,115)(94,99)(95,110)(100,105)(101,116)(106,111)(112,117), (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)], [(1,73),(2,72),(3,71),(4,70),(5,69),(6,68),(7,67),(8,66),(9,65),(10,64),(11,63),(12,62),(13,61),(14,120),(15,119),(16,118),(17,117),(18,116),(19,115),(20,114),(21,113),(22,112),(23,111),(24,110),(25,109),(26,108),(27,107),(28,106),(29,105),(30,104),(31,103),(32,102),(33,101),(34,100),(35,99),(36,98),(37,97),(38,96),(39,95),(40,94),(41,93),(42,92),(43,91),(44,90),(45,89),(46,88),(47,87),(48,86),(49,85),(50,84),(51,83),(52,82),(53,81),(54,80),(55,79),(56,78),(57,77),(58,76),(59,75),(60,74)], [(2,12),(3,23),(4,34),(5,45),(6,56),(8,18),(9,29),(10,40),(11,51),(14,24),(15,35),(16,46),(17,57),(20,30),(21,41),(22,52),(26,36),(27,47),(28,58),(32,42),(33,53),(38,48),(39,59),(44,54),(50,60),(61,96),(62,107),(63,118),(64,69),(65,80),(66,91),(67,102),(68,113),(70,75),(71,86),(72,97),(73,108),(74,119),(76,81),(77,92),(78,103),(79,114),(82,87),(83,98),(84,109),(85,120),(88,93),(89,104),(90,115),(94,99),(95,110),(100,105),(101,116),(106,111),(112,117)], [(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 4A4B4C5A5B6A6B8A8B10A···10F10G···10N12A12B12C15A15B20A20B20C20D24A24B24C24D30A···30F60A···60H
order122222344455668810···1010···101212121515202020202424242430···3060···60
size11212126022260222420202···212···12224444444202020204···44···4

57 irreducible representations

dim11111122222222222244444444
type++++++++++++++++++++++
imageC1C2C2C2C2C2S3D4D4D5D6D6D10D10D12D12C5⋊D4C5⋊D4C8⋊C22S3×D5C8⋊D6C5⋊D12C2×S3×D5C5⋊D12D4.D10D6036C22
kernelD6036C22C5⋊D24D12.D5C3×C4.Dic5C10×D12D6011C2C4.Dic5C60C2×C30C2×D12C52C8C2×C20D12C2×C12C20C2×C10C12C2×C6C15C2×C4C5C4C4C22C3C1
# reps12211111122142224412222248

Matrix representation of D6036C22 in GL4(𝔽241) generated by

552700
162000
00206180
001160
,
000180
001250
02700
79000
,
11900
024000
0056194
00159185
,
1000
0100
002400
000240
G:=sub<GL(4,GF(241))| [55,162,0,0,27,0,0,0,0,0,206,116,0,0,180,0],[0,0,0,79,0,0,27,0,0,125,0,0,180,0,0,0],[1,0,0,0,19,240,0,0,0,0,56,159,0,0,194,185],[1,0,0,0,0,1,0,0,0,0,240,0,0,0,0,240] >;

D6036C22 in GAP, Magma, Sage, TeX

D_{60}\rtimes_{36}C_2^2
% in TeX

G:=Group("D60:36C2^2");
// GroupNames label

G:=SmallGroup(480,380);
// by ID

G=gap.SmallGroup(480,380);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,141,422,346,80,1356,18822]);
// 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^11,a*d=d*a,c*b*c=a^25*b,d*b*d=a^30*b,c*d=d*c>;
// generators/relations

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