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G = D60⋊C2order 240 = 24·3·5

6th semidirect product of D60 and C2 acting faithfully

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D606C2, D6.6D10, C20.28D6, Dic105S3, C12.17D10, C30.6C23, Dic5.3D6, C60.14C22, D30.1C22, Dic3.11D10, (C4×S3)⋊2D5, C4.7(S3×D5), (S3×C20)⋊2C2, C31(C4○D20), C154(C4○D4), C5⋊D122C2, D30.C21C2, C51(Q83S3), C6.6(C22×D5), (C3×Dic10)⋊3C2, C10.6(C22×S3), (S3×C10).7C22, (C5×Dic3).9C22, (C3×Dic5).3C22, C2.10(C2×S3×D5), SmallGroup(240,130)

Series: Derived Chief Lower central Upper central

C1C30 — D60⋊C2
C1C5C15C30C3×Dic5D30.C2 — D60⋊C2
C15C30 — D60⋊C2
C1C2C4

Generators and relations for D60⋊C2
 G = < a,b,c | a60=b2=c2=1, bab=a-1, cac=a41, cbc=a10b >

Subgroups: 376 in 80 conjugacy classes, 32 normal (22 characteristic)
C1, C2, C2 [×3], C3, C4, C4 [×3], C22 [×3], C5, S3 [×3], C6, C2×C4 [×3], D4 [×3], Q8, D5 [×2], C10, C10, Dic3, C12, C12 [×2], D6, D6 [×2], C15, C4○D4, Dic5 [×2], C20, C20, D10 [×2], C2×C10, C4×S3, C4×S3 [×2], D12 [×3], C3×Q8, C5×S3, D15 [×2], C30, Dic10, C4×D5 [×2], D20, C5⋊D4 [×2], C2×C20, Q83S3, C5×Dic3, C3×Dic5 [×2], C60, S3×C10, D30 [×2], C4○D20, D30.C2 [×2], C5⋊D12 [×2], C3×Dic10, S3×C20, D60, D60⋊C2
Quotients: C1, C2 [×7], C22 [×7], S3, C23, D5, D6 [×3], C4○D4, D10 [×3], C22×S3, C22×D5, Q83S3, S3×D5, C4○D20, C2×S3×D5, D60⋊C2

Smallest permutation representation of D60⋊C2
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 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(16 60)(17 59)(18 58)(19 57)(20 56)(21 55)(22 54)(23 53)(24 52)(25 51)(26 50)(27 49)(28 48)(29 47)(30 46)(31 45)(32 44)(33 43)(34 42)(35 41)(36 40)(37 39)(61 113)(62 112)(63 111)(64 110)(65 109)(66 108)(67 107)(68 106)(69 105)(70 104)(71 103)(72 102)(73 101)(74 100)(75 99)(76 98)(77 97)(78 96)(79 95)(80 94)(81 93)(82 92)(83 91)(84 90)(85 89)(86 88)(114 120)(115 119)(116 118)
(1 65)(2 106)(3 87)(4 68)(5 109)(6 90)(7 71)(8 112)(9 93)(10 74)(11 115)(12 96)(13 77)(14 118)(15 99)(16 80)(17 61)(18 102)(19 83)(20 64)(21 105)(22 86)(23 67)(24 108)(25 89)(26 70)(27 111)(28 92)(29 73)(30 114)(31 95)(32 76)(33 117)(34 98)(35 79)(36 120)(37 101)(38 82)(39 63)(40 104)(41 85)(42 66)(43 107)(44 88)(45 69)(46 110)(47 91)(48 72)(49 113)(50 94)(51 75)(52 116)(53 97)(54 78)(55 119)(56 100)(57 81)(58 62)(59 103)(60 84)

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

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

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

D60⋊C2 is a maximal subgroup of
C401D6  Dic20⋊S3  D6.1D20  D1205C2  C60.19C23  D12.9D10  Dic10.27D6  C60.44C23  C30.C24  S3×C4○D20  D2029D6  D30.C23  D1214D10  C30.33C24  D5×Q83S3
D60⋊C2 is a maximal quotient of
(C2×C20).D6  Dic3×Dic10  Dic5.1Dic6  (S3×C20)⋊5C4  C605C4⋊C2  C4⋊Dic5⋊S3  D6⋊Dic5⋊C2  C60.46D4  C60.47D4  C10.D4⋊S3  C60.6Q8  D30.23(C2×C4)  D30⋊Q8  D30⋊D4  D6014C4  D30.6D4  D30.7D4  C1522(C4×D4)  C606D4

39 conjugacy classes

class 1 2A2B2C2D 3 4A4B4C4D4E5A5B 6 10A10B10C10D10E10F12A12B12C15A15B20A20B20C20D20E20F20G20H30A30B60A60B60C60D
order1222234444455610101010101012121215152020202020202020303060606060
size116303022331010222226666420204422226666444444

39 irreducible representations

dim1111112222222224444
type+++++++++++++++++
imageC1C2C2C2C2C2S3D5D6D6C4○D4D10D10D10C4○D20Q83S3S3×D5C2×S3×D5D60⋊C2
kernelD60⋊C2D30.C2C5⋊D12C3×Dic10S3×C20D60Dic10C4×S3Dic5C20C15Dic3C12D6C3C5C4C2C1
# reps1221111221222281224

Matrix representation of D60⋊C2 in GL4(𝔽61) generated by

29200
59200
0025
003660
,
14400
06000
005956
00252
,
144500
164700
0010
003660
G:=sub<GL(4,GF(61))| [29,59,0,0,2,2,0,0,0,0,2,36,0,0,5,60],[1,0,0,0,44,60,0,0,0,0,59,25,0,0,56,2],[14,16,0,0,45,47,0,0,0,0,1,36,0,0,0,60] >;

D60⋊C2 in GAP, Magma, Sage, TeX

D_{60}\rtimes C_2
% in TeX

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

G:=SmallGroup(240,130);
// by ID

G=gap.SmallGroup(240,130);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-5,48,55,218,50,490,6917]);
// Polycyclic

G:=Group<a,b,c|a^60=b^2=c^2=1,b*a*b=a^-1,c*a*c=a^41,c*b*c=a^10*b>;
// generators/relations

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