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G = D60.C22order 480 = 25·3·5

2nd non-split extension by D60 of C22 acting faithfully

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Dic63D10, D20.23D6, C60.4C23, D60.2C22, D4⋊D51S3, (C5×D4)⋊8D6, D42(S3×D5), C52C85D6, D4⋊D154C2, (C3×D4)⋊2D10, (S3×D20)⋊2C2, C57(D8⋊S3), D42S31D5, (C4×S3).5D10, (S3×C10).8D4, C32(D4⋊D10), C1514(C8⋊C22), C153C84C22, C30.166(C2×D4), C30.D41C2, C10.140(S3×D4), Dic6⋊D51C2, D6.6(C5⋊D4), (D4×C15)⋊4C22, D6.Dic51C2, C20.4(C22×S3), C12.4(C22×D5), (S3×C20).2C22, (C5×Dic3).34D4, (C5×Dic6)⋊1C22, (C3×D20).1C22, Dic3.15(C5⋊D4), C4.4(C2×S3×D5), (C3×D4⋊D5)⋊2C2, C6.43(C2×C5⋊D4), C2.21(S3×C5⋊D4), (C5×D42S3)⋊1C2, (C3×C52C8)⋊2C22, SmallGroup(480,556)

Series: Derived Chief Lower central Upper central

C1C60 — D60.C22
C1C5C15C30C60C3×D20S3×D20 — D60.C22
C15C30C60 — D60.C22
C1C2C4D4

Generators and relations for D60.C22
 G = < a,b,c,d | a60=b2=c2=d2=1, bab=a-1, cac=a19, dad=a31, cbc=a18b, dbd=a45b, dcd=a45c >

Subgroups: 876 in 136 conjugacy classes, 40 normal (all characteristic)
C1, C2, C2 [×4], C3, C4, C4 [×2], C22 [×6], C5, S3 [×2], C6, C6 [×2], C8 [×2], C2×C4 [×2], D4, D4 [×4], Q8, C23, D5 [×2], C10, C10 [×2], Dic3, Dic3, C12, D6, D6 [×3], C2×C6 [×2], C15, M4(2), D8 [×2], SD16 [×2], C2×D4, C4○D4, C20, C20 [×2], D10 [×4], C2×C10 [×2], C3⋊C8, C24, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4 [×2], C3×D4, C3×D4, C22×S3, C5×S3, C3×D5, D15, C30, C30, C8⋊C22, C52C8, C52C8, D20, D20 [×2], C2×C20 [×2], C5×D4, C5×D4, C5×Q8, C22×D5, C8⋊S3, C24⋊C2, D4⋊S3, D4.S3, C3×D8, S3×D4, D42S3, C5×Dic3, C5×Dic3, C60, S3×D5 [×2], C6×D5, S3×C10, D30, C2×C30, C4.Dic5, D4⋊D5, D4⋊D5, Q8⋊D5 [×2], C2×D20, C5×C4○D4, D8⋊S3, C3×C52C8, C153C8, C3⋊D20, C3×D20, C5×Dic6, S3×C20, C10×Dic3, C5×C3⋊D4, D60, D4×C15, C2×S3×D5, D4⋊D10, D6.Dic5, C30.D4, Dic6⋊D5, C3×D4⋊D5, D4⋊D15, S3×D20, C5×D42S3, D60.C22
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], C2×D4, D10 [×3], C22×S3, C8⋊C22, C5⋊D4 [×2], C22×D5, S3×D4, S3×D5, C2×C5⋊D4, D8⋊S3, C2×S3×D5, D4⋊D10, S3×C5⋊D4, D60.C22

Smallest permutation representation of D60.C22
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 60)(2 59)(3 58)(4 57)(5 56)(6 55)(7 54)(8 53)(9 52)(10 51)(11 50)(12 49)(13 48)(14 47)(15 46)(16 45)(17 44)(18 43)(19 42)(20 41)(21 40)(22 39)(23 38)(24 37)(25 36)(26 35)(27 34)(28 33)(29 32)(30 31)(61 93)(62 92)(63 91)(64 90)(65 89)(66 88)(67 87)(68 86)(69 85)(70 84)(71 83)(72 82)(73 81)(74 80)(75 79)(76 78)(94 120)(95 119)(96 118)(97 117)(98 116)(99 115)(100 114)(101 113)(102 112)(103 111)(104 110)(105 109)(106 108)
(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 64)(62 83)(63 102)(65 80)(66 99)(67 118)(68 77)(69 96)(70 115)(71 74)(72 93)(73 112)(75 90)(76 109)(78 87)(79 106)(81 84)(82 103)(85 100)(86 119)(88 97)(89 116)(91 94)(92 113)(95 110)(98 107)(101 104)(105 120)(108 117)(111 114)
(1 70)(2 101)(3 72)(4 103)(5 74)(6 105)(7 76)(8 107)(9 78)(10 109)(11 80)(12 111)(13 82)(14 113)(15 84)(16 115)(17 86)(18 117)(19 88)(20 119)(21 90)(22 61)(23 92)(24 63)(25 94)(26 65)(27 96)(28 67)(29 98)(30 69)(31 100)(32 71)(33 102)(34 73)(35 104)(36 75)(37 106)(38 77)(39 108)(40 79)(41 110)(42 81)(43 112)(44 83)(45 114)(46 85)(47 116)(48 87)(49 118)(50 89)(51 120)(52 91)(53 62)(54 93)(55 64)(56 95)(57 66)(58 97)(59 68)(60 99)

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

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

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

48 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C5A5B6A6B6C8A8B10A10B10C10D10E10F10G10H 12 15A15B20A20B20C20D20E20F20G20H20I20J24A24B30A30B30C30D30E30F60A60B
order1222223444556668810101010101010101215152020202020202020202024243030303030306060
size11462060226122228402060224444121244444666612121212202044888888

48 irreducible representations

dim1111111122222222222244444448
type++++++++++++++++++++++++
imageC1C2C2C2C2C2C2C2S3D4D4D5D6D6D6D10D10D10C5⋊D4C5⋊D4C8⋊C22S3×D4S3×D5D8⋊S3C2×S3×D5D4⋊D10S3×C5⋊D4D60.C22
kernelD60.C22D6.Dic5C30.D4Dic6⋊D5C3×D4⋊D5D4⋊D15S3×D20C5×D42S3D4⋊D5C5×Dic3S3×C10D42S3C52C8D20C5×D4Dic6C4×S3C3×D4Dic3D6C15C10D4C5C4C3C2C1
# reps1111111111121112224411222442

Matrix representation of D60.C22 in GL6(𝔽241)

189510000
18900000
0000240240
000010
001100
00240000
,
24010000
010000
0000240240
000001
0024024000
000100
,
18910000
189520000
001000
000100
00002400
00000240
,
24000000
02400000
0047944794
00147194147194
004794194147
001471949447

G:=sub<GL(6,GF(241))| [189,189,0,0,0,0,51,0,0,0,0,0,0,0,0,0,1,240,0,0,0,0,1,0,0,0,240,1,0,0,0,0,240,0,0,0],[240,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,240,0,0,0,0,0,240,1,0,0,240,0,0,0,0,0,240,1,0,0],[189,189,0,0,0,0,1,52,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,240,0,0,0,0,0,0,240],[240,0,0,0,0,0,0,240,0,0,0,0,0,0,47,147,47,147,0,0,94,194,94,194,0,0,47,147,194,94,0,0,94,194,147,47] >;

D60.C22 in GAP, Magma, Sage, TeX

D_{60}.C_2^2
% in TeX

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

G:=SmallGroup(480,556);
// by ID

G=gap.SmallGroup(480,556);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,254,219,675,185,80,1356,18822]);
// 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,d*a*d=a^31,c*b*c=a^18*b,d*b*d=a^45*b,d*c*d=a^45*c>;
// generators/relations

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