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G = C2×D60order 240 = 24·3·5

Direct product of C2 and D60

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C2×D60, C207D6, C42D30, C304D4, C61D20, C101D12, C127D10, C608C22, D305C22, C30.30C23, C22.10D30, C32(C2×D20), C52(C2×D12), (C2×C60)⋊5C2, (C2×C20)⋊3S3, (C2×C12)⋊3D5, (C2×C4)⋊2D15, C1510(C2×D4), (C2×C6).28D10, (C2×C10).28D6, (C22×D15)⋊1C2, C6.30(C22×D5), C2.4(C22×D15), (C2×C30).29C22, C10.30(C22×S3), SmallGroup(240,177)

Series: Derived Chief Lower central Upper central

C1C30 — C2×D60
C1C5C15C30D30C22×D15 — C2×D60
C15C30 — C2×D60
C1C22C2×C4

Generators and relations for C2×D60
 G = < a,b,c | a2=b60=c2=1, ab=ba, ac=ca, cbc=b-1 >

Subgroups: 680 in 108 conjugacy classes, 43 normal (17 characteristic)
C1, C2, C2 [×2], C2 [×4], C3, C4 [×2], C22, C22 [×8], C5, S3 [×4], C6, C6 [×2], C2×C4, D4 [×4], C23 [×2], D5 [×4], C10, C10 [×2], C12 [×2], D6 [×8], C2×C6, C15, C2×D4, C20 [×2], D10 [×8], C2×C10, D12 [×4], C2×C12, C22×S3 [×2], D15 [×4], C30, C30 [×2], D20 [×4], C2×C20, C22×D5 [×2], C2×D12, C60 [×2], D30 [×4], D30 [×4], C2×C30, C2×D20, D60 [×4], C2×C60, C22×D15 [×2], C2×D60
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], C2×D4, D10 [×3], D12 [×2], C22×S3, D15, D20 [×2], C22×D5, C2×D12, D30 [×3], C2×D20, D60 [×2], C22×D15, C2×D60

Smallest permutation representation of C2×D60
On 120 points
Generators in S120
(1 94)(2 95)(3 96)(4 97)(5 98)(6 99)(7 100)(8 101)(9 102)(10 103)(11 104)(12 105)(13 106)(14 107)(15 108)(16 109)(17 110)(18 111)(19 112)(20 113)(21 114)(22 115)(23 116)(24 117)(25 118)(26 119)(27 120)(28 61)(29 62)(30 63)(31 64)(32 65)(33 66)(34 67)(35 68)(36 69)(37 70)(38 71)(39 72)(40 73)(41 74)(42 75)(43 76)(44 77)(45 78)(46 79)(47 80)(48 81)(49 82)(50 83)(51 84)(52 85)(53 86)(54 87)(55 88)(56 89)(57 90)(58 91)(59 92)(60 93)
(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 66)(62 65)(63 64)(67 120)(68 119)(69 118)(70 117)(71 116)(72 115)(73 114)(74 113)(75 112)(76 111)(77 110)(78 109)(79 108)(80 107)(81 106)(82 105)(83 104)(84 103)(85 102)(86 101)(87 100)(88 99)(89 98)(90 97)(91 96)(92 95)(93 94)

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

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

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

C2×D60 is a maximal subgroup of
C60.29D4  D6012C4  D6015C4  D609C4  D608C4  M4(2)⋊D15  D2019D6  C60.38D4  C60.47D4  C60.70D4  D6017C4  D30⋊D4  D6014C4  D30.6D4  C127D20  C20⋊D12  C12⋊D20  D302D4  C606D4  D305D4  C426D15  C427D15  D3016D4  D309D4  D6011C4  D30.29D4  C4⋊D60  C8⋊D30  C6029D4  C603D4  C60.23D4  D4⋊D30  C2×D5×D12  C2×S3×D20  D2029D6  C2×D4×D15  D48D30
C2×D60 is a maximal quotient of
C608Q8  C426D15  C427D15  D3016D4  C22.D60  C4⋊D60  D306Q8  C40.69D6  C8⋊D30  C8.D30  C6029D4

66 conjugacy classes

class 1 2A2B2C2D2E2F2G 3 4A4B5A5B6A6B6C10A···10F12A12B12C12D15A15B15C15D20A···20H30A···30L60A···60P
order122222223445566610···10121212121515151520···2030···3060···60
size111130303030222222222···2222222222···22···22···2

66 irreducible representations

dim11112222222222222
type+++++++++++++++++
imageC1C2C2C2S3D4D5D6D6D10D10D12D15D20D30D30D60
kernelC2×D60D60C2×C60C22×D15C2×C20C30C2×C12C20C2×C10C12C2×C6C10C2×C4C6C4C22C2
# reps141212221424488416

Matrix representation of C2×D60 in GL5(𝔽61)

600000
01000
00100
000600
000060
,
600000
01100
060000
000234
000567
,
10000
01100
006000
0002936
0005832

G:=sub<GL(5,GF(61))| [60,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,60,0,0,0,0,0,60],[60,0,0,0,0,0,1,60,0,0,0,1,0,0,0,0,0,0,2,56,0,0,0,34,7],[1,0,0,0,0,0,1,0,0,0,0,1,60,0,0,0,0,0,29,58,0,0,0,36,32] >;

C2×D60 in GAP, Magma, Sage, TeX

C_2\times D_{60}
% in TeX

G:=Group("C2xD60");
// GroupNames label

G:=SmallGroup(240,177);
// by ID

G=gap.SmallGroup(240,177);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-5,218,50,964,6917]);
// Polycyclic

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

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