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G = C2xD60  order 240 = 24·3·5

Direct product of C2 and D60

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

Aliases: C2xD60, C20:7D6, C4:2D30, C30:4D4, C6:1D20, C10:1D12, C12:7D10, C60:8C22, D30:5C22, C30.30C23, C22.10D30, C3:2(C2xD20), C5:2(C2xD12), (C2xC60):5C2, (C2xC20):3S3, (C2xC12):3D5, (C2xC4):2D15, C15:10(C2xD4), (C2xC6).28D10, (C2xC10).28D6, (C22xD15):1C2, C6.30(C22xD5), C2.4(C22xD15), (C2xC30).29C22, C10.30(C22xS3), SmallGroup(240,177)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C30 — C2xD60
Chief series
C1 — C5 — C15 — C30 — D30 — C22xD15 — C2xD60
Lower central
C15 — C30 — C2xD60
Upper central
C1 — C22 — C2xC4

Generators and relations for C2xD60
 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, C2, C3, C4, C22, C22, C5, S3, C6, C6, C2xC4, D4, C23, D5, C10, C10, C12, D6, C2xC6, C15, C2xD4, C20, D10, C2xC10, D12, C2xC12, C22xS3, D15, C30, C30, D20, C2xC20, C22xD5, C2xD12, C60, D30, D30, C2xC30, C2xD20, D60, C2xC60, C22xD15, C2xD60
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2xD4, D10, D12, C22xS3, D15, D20, C22xD5, C2xD12, D30, C2xD20, D60, C22xD15, C2xD60

Smallest permutation representation of C2xD60
►On 120 points
Generators in S120
(1 91)(2 92)(3 93)(4 94)(5 95)(6 96)(7 97)(8 98)(9 99)(10 100)(11 101)(12 102)(13 103)(14 104)(15 105)(16 106)(17 107)(18 108)(19 109)(20 110)(21 111)(22 112)(23 113)(24 114)(25 115)(26 116)(27 117)(28 118)(29 119)(30 120)(31 61)(32 62)(33 63)(34 64)(35 65)(36 66)(37 67)(38 68)(39 69)(40 70)(41 71)(42 72)(43 73)(44 74)(45 75)(46 76)(47 77)(48 78)(49 79)(50 80)(51 81)(52 82)(53 83)(54 84)(55 85)(56 86)(57 87)(58 88)(59 89)(60 90)

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

C2xD60 is a maximal subgroup of
 C60.29D4  D60:12C4  D60:15C4  D60:9C4  D60:8C4  M4(2):D15  D20:19D6  C60.38D4  C60.47D4  C60.70D4  D60:17C4  D30:D4  D60:14C4  D30.6D4  C12:7D20  C20:D12  C12:D20  D30:2D4  C60:6D4  D30:5D4  C42:6D15  C42:7D15  D30:16D4  D30:9D4  D60:11C4  D30.29D4  C4:D60  C8:D30  C60:29D4  C60:3D4  C60.23D4  D4:D30  C2xD5xD12  C2xS3xD20  D20:29D6  C2xD4xD15  D4:8D30
C2xD60 is a maximal quotient of
 C60:8Q8  C42:6D15  C42:7D15  D30:16D4  C22.D60  C4:D60  D30:6Q8  C40.69D6  C8:D30  C8.D30  C60:29D4

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
kernelC2xD60D60C2xC60C22xD15C2xC20C30C2xC12C20C2xC10C12C2xC6C10C2xC4C6C4C22C2
# reps141212221424488416

Matrix representation of C2xD60 ►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] >;
 

C2xD60 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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