Copied to
clipboard

## G = C2×D60order 240 = 24·3·5

### Direct product of C2 and D60

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — C2×D60
 Chief series C1 — C5 — C15 — C30 — D30 — C22×D15 — C2×D60
 Lower central C15 — C30 — C2×D60
 Upper central C1 — C22 — C2×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)])

66 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3 4A 4B 5A 5B 6A 6B 6C 10A ··· 10F 12A 12B 12C 12D 15A 15B 15C 15D 20A ··· 20H 30A ··· 30L 60A ··· 60P order 1 2 2 2 2 2 2 2 3 4 4 5 5 6 6 6 10 ··· 10 12 12 12 12 15 15 15 15 20 ··· 20 30 ··· 30 60 ··· 60 size 1 1 1 1 30 30 30 30 2 2 2 2 2 2 2 2 2 ··· 2 2 2 2 2 2 2 2 2 2 ··· 2 2 ··· 2 2 ··· 2

66 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 S3 D4 D5 D6 D6 D10 D10 D12 D15 D20 D30 D30 D60 kernel C2×D60 D60 C2×C60 C22×D15 C2×C20 C30 C2×C12 C20 C2×C10 C12 C2×C6 C10 C2×C4 C6 C4 C22 C2 # reps 1 4 1 2 1 2 2 2 1 4 2 4 4 8 8 4 16

Matrix representation of C2×D60 in GL5(𝔽61)

 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 1 0 0 0 60 0 0 0 0 0 0 2 34 0 0 0 56 7
,
 1 0 0 0 0 0 1 1 0 0 0 0 60 0 0 0 0 0 29 36 0 0 0 58 32

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

׿
×
𝔽