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G = D60⋊C22order 480 = 25·3·5

6th semidirect product of D60 and C22 acting faithfully

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D30.12D4, D20.12D6, D606C22, D12.12D10, C60.30C23, Dic15.44D4, C3⋊C812D10, Q8⋊D53S3, (C5×Q8)⋊7D6, Q87(S3×D5), C5⋊D246C2, C52C812D6, C3⋊D406C2, C54(Q83D6), (C3×Q8)⋊4D10, C6.76(D4×D5), C20⋊D63C2, C34(D40⋊C2), Q82S33D5, C10.77(S3×D4), Q83D151C2, C1524(C8⋊C22), C30.192(C2×D4), (Q8×C15)⋊6C22, D30.5C45C2, C20.30(C22×S3), C12.30(C22×D5), (C3×D20).11C22, (C4×D15).10C22, (C5×D12).11C22, C2.29(D10⋊D6), C4.30(C2×S3×D5), (C3×Q8⋊D5)⋊4C2, (C5×C3⋊C8)⋊14C22, (C5×Q82S3)⋊4C2, (C3×C52C8)⋊14C22, SmallGroup(480,582)

Series: Derived Chief Lower central Upper central

C1C60 — D60⋊C22
C1C5C15C30C60C3×D20C20⋊D6 — D60⋊C22
C15C30C60 — D60⋊C22
C1C2C4Q8

Generators and relations for D60⋊C22
 G = < a,b,c,d | a60=b2=c2=d2=1, bab=a-1, cac=a11, dad=a29, cbc=a55b, dbd=a58b, cd=dc >

Subgroups: 1020 in 136 conjugacy classes, 38 normal (all characteristic)
C1, C2, C2 [×4], C3, C4, C4 [×2], C22 [×6], C5, S3 [×3], C6, C6, C8 [×2], C2×C4 [×2], D4 [×5], Q8, C23, D5 [×3], C10, C10, Dic3, C12, C12, D6 [×5], C2×C6, C15, M4(2), D8 [×2], SD16 [×2], C2×D4, C4○D4, Dic5, C20, C20, D10 [×5], C2×C10, C3⋊C8, C24, C4×S3 [×2], D12, D12 [×2], C3⋊D4, C3×D4, C3×Q8, C22×S3, C5×S3, C3×D5, D15 [×2], C30, C8⋊C22, C52C8, C40, C4×D5 [×2], D20, D20 [×2], C5⋊D4, C5×D4, C5×Q8, C22×D5, C8⋊S3, D24, D4⋊S3, Q82S3, C3×SD16, S3×D4, Q83S3, Dic15, C60, C60, S3×D5 [×2], C6×D5, S3×C10, D30, D30, C8⋊D5, D40, D4⋊D5, Q8⋊D5, C5×SD16, D4×D5, Q82D5, Q83D6, C5×C3⋊C8, C3×C52C8, C15⋊D4, C3×D20, C5×D12, C4×D15, C4×D15, D60, D60, Q8×C15, C2×S3×D5, D40⋊C2, D30.5C4, C3⋊D40, C5⋊D24, C3×Q8⋊D5, C5×Q82S3, C20⋊D6, Q83D15, D60⋊C22
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], C2×D4, D10 [×3], C22×S3, C8⋊C22, C22×D5, S3×D4, S3×D5, D4×D5, Q83D6, C2×S3×D5, D40⋊C2, D10⋊D6, 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)
(1 112)(2 111)(3 110)(4 109)(5 108)(6 107)(7 106)(8 105)(9 104)(10 103)(11 102)(12 101)(13 100)(14 99)(15 98)(16 97)(17 96)(18 95)(19 94)(20 93)(21 92)(22 91)(23 90)(24 89)(25 88)(26 87)(27 86)(28 85)(29 84)(30 83)(31 82)(32 81)(33 80)(34 79)(35 78)(36 77)(37 76)(38 75)(39 74)(40 73)(41 72)(42 71)(43 70)(44 69)(45 68)(46 67)(47 66)(48 65)(49 64)(50 63)(51 62)(52 61)(53 120)(54 119)(55 118)(56 117)(57 116)(58 115)(59 114)(60 113)
(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)
(2 30)(3 59)(4 28)(5 57)(6 26)(7 55)(8 24)(9 53)(10 22)(11 51)(12 20)(13 49)(14 18)(15 47)(17 45)(19 43)(21 41)(23 39)(25 37)(27 35)(29 33)(32 60)(34 58)(36 56)(38 54)(40 52)(42 50)(44 48)(61 75)(62 104)(63 73)(64 102)(65 71)(66 100)(67 69)(68 98)(70 96)(72 94)(74 92)(76 90)(77 119)(78 88)(79 117)(80 86)(81 115)(82 84)(83 113)(85 111)(87 109)(89 107)(91 105)(93 103)(95 101)(97 99)(106 120)(108 118)(110 116)(112 114)

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

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

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

42 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C5A5B6A6B8A8B10A10B10C10D12A12B15A15B20A20B20C20D24A24B30A30B40A40B40C40D60A···60F
order1222223444556688101010101212151520202020242430304040404060···60
size11122030602243022240122022242448444488202044121212128···8

42 irreducible representations

dim111111112222222222444444448
type+++++++++++++++++++++++++++
imageC1C2C2C2C2C2C2C2S3D4D4D5D6D6D6D10D10D10C8⋊C22S3×D4S3×D5D4×D5Q83D6C2×S3×D5D40⋊C2D10⋊D6D60⋊C22
kernelD60⋊C22D30.5C4C3⋊D40C5⋊D24C3×Q8⋊D5C5×Q82S3C20⋊D6Q83D15Q8⋊D5Dic15D30Q82S3C52C8D20C5×Q8C3⋊C8D12C3×Q8C15C10Q8C6C5C4C3C2C1
# reps111111111112111222112222442

Matrix representation of D60⋊C22 in GL8(𝔽241)

2401892401890000
525252520000
152000000
189189000000
000011902139
00005151102102
00002405124051
0000190190190190
,
38282032130000
22120320380000
1651852032130000
407620380000
00000023729
000000664
000023913500
000033200
,
10000000
01000000
240024000000
024002400000
00001000
00000100
000024002400
000002400240
,
10000000
189240000000
240024000000
5215210000
00001000
00005124000
00000010
00000051240

G:=sub<GL(8,GF(241))| [240,52,1,189,0,0,0,0,189,52,52,189,0,0,0,0,240,52,0,0,0,0,0,0,189,52,0,0,0,0,0,0,0,0,0,0,1,51,240,190,0,0,0,0,190,51,51,190,0,0,0,0,2,102,240,190,0,0,0,0,139,102,51,190],[38,221,165,40,0,0,0,0,28,203,185,76,0,0,0,0,203,20,203,20,0,0,0,0,213,38,213,38,0,0,0,0,0,0,0,0,0,0,239,33,0,0,0,0,0,0,135,2,0,0,0,0,237,66,0,0,0,0,0,0,29,4,0,0],[1,0,240,0,0,0,0,0,0,1,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,1,0,240,0,0,0,0,0,0,1,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,240],[1,189,240,52,0,0,0,0,0,240,0,1,0,0,0,0,0,0,240,52,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,51,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,1,51,0,0,0,0,0,0,0,240] >;

D60⋊C22 in GAP, Magma, Sage, TeX

D_{60}\rtimes C_2^2
% in TeX

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

G:=SmallGroup(480,582);
// by ID

G=gap.SmallGroup(480,582);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,254,303,100,675,346,185,80,1356,18822]);
// 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^11,d*a*d=a^29,c*b*c=a^55*b,d*b*d=a^58*b,c*d=d*c>;
// generators/relations

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