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G = C2×C61⋊C3order 366 = 2·3·61

Direct product of C2 and C61⋊C3

direct product, metacyclic, supersoluble, monomial, Z-group, 3-hyperelementary

Aliases: C2×C61⋊C3, C122⋊C3, C612C6, SmallGroup(366,2)

Series: Derived Chief Lower central Upper central

C1C61 — C2×C61⋊C3
C1C61C61⋊C3 — C2×C61⋊C3
C61 — C2×C61⋊C3
C1C2

Generators and relations for C2×C61⋊C3
 G = < a,b,c | a2=b61=c3=1, ab=ba, ac=ca, cbc-1=b13 >

61C3
61C6

Smallest permutation representation of C2×C61⋊C3
On 122 points
Generators in S122
(1 62)(2 63)(3 64)(4 65)(5 66)(6 67)(7 68)(8 69)(9 70)(10 71)(11 72)(12 73)(13 74)(14 75)(15 76)(16 77)(17 78)(18 79)(19 80)(20 81)(21 82)(22 83)(23 84)(24 85)(25 86)(26 87)(27 88)(28 89)(29 90)(30 91)(31 92)(32 93)(33 94)(34 95)(35 96)(36 97)(37 98)(38 99)(39 100)(40 101)(41 102)(42 103)(43 104)(44 105)(45 106)(46 107)(47 108)(48 109)(49 110)(50 111)(51 112)(52 113)(53 114)(54 115)(55 116)(56 117)(57 118)(58 119)(59 120)(60 121)(61 122)
(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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122)
(2 48 14)(3 34 27)(4 20 40)(5 6 53)(7 39 18)(8 25 31)(9 11 44)(10 58 57)(12 30 22)(13 16 35)(15 49 61)(17 21 26)(19 54 52)(23 59 43)(24 45 56)(28 50 47)(29 36 60)(32 55 38)(33 41 51)(37 46 42)(63 109 75)(64 95 88)(65 81 101)(66 67 114)(68 100 79)(69 86 92)(70 72 105)(71 119 118)(73 91 83)(74 77 96)(76 110 122)(78 82 87)(80 115 113)(84 120 104)(85 106 117)(89 111 108)(90 97 121)(93 116 99)(94 102 112)(98 107 103)

G:=sub<Sym(122)| (1,62)(2,63)(3,64)(4,65)(5,66)(6,67)(7,68)(8,69)(9,70)(10,71)(11,72)(12,73)(13,74)(14,75)(15,76)(16,77)(17,78)(18,79)(19,80)(20,81)(21,82)(22,83)(23,84)(24,85)(25,86)(26,87)(27,88)(28,89)(29,90)(30,91)(31,92)(32,93)(33,94)(34,95)(35,96)(36,97)(37,98)(38,99)(39,100)(40,101)(41,102)(42,103)(43,104)(44,105)(45,106)(46,107)(47,108)(48,109)(49,110)(50,111)(51,112)(52,113)(53,114)(54,115)(55,116)(56,117)(57,118)(58,119)(59,120)(60,121)(61,122), (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)(62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122), (2,48,14)(3,34,27)(4,20,40)(5,6,53)(7,39,18)(8,25,31)(9,11,44)(10,58,57)(12,30,22)(13,16,35)(15,49,61)(17,21,26)(19,54,52)(23,59,43)(24,45,56)(28,50,47)(29,36,60)(32,55,38)(33,41,51)(37,46,42)(63,109,75)(64,95,88)(65,81,101)(66,67,114)(68,100,79)(69,86,92)(70,72,105)(71,119,118)(73,91,83)(74,77,96)(76,110,122)(78,82,87)(80,115,113)(84,120,104)(85,106,117)(89,111,108)(90,97,121)(93,116,99)(94,102,112)(98,107,103)>;

G:=Group( (1,62)(2,63)(3,64)(4,65)(5,66)(6,67)(7,68)(8,69)(9,70)(10,71)(11,72)(12,73)(13,74)(14,75)(15,76)(16,77)(17,78)(18,79)(19,80)(20,81)(21,82)(22,83)(23,84)(24,85)(25,86)(26,87)(27,88)(28,89)(29,90)(30,91)(31,92)(32,93)(33,94)(34,95)(35,96)(36,97)(37,98)(38,99)(39,100)(40,101)(41,102)(42,103)(43,104)(44,105)(45,106)(46,107)(47,108)(48,109)(49,110)(50,111)(51,112)(52,113)(53,114)(54,115)(55,116)(56,117)(57,118)(58,119)(59,120)(60,121)(61,122), (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)(62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122), (2,48,14)(3,34,27)(4,20,40)(5,6,53)(7,39,18)(8,25,31)(9,11,44)(10,58,57)(12,30,22)(13,16,35)(15,49,61)(17,21,26)(19,54,52)(23,59,43)(24,45,56)(28,50,47)(29,36,60)(32,55,38)(33,41,51)(37,46,42)(63,109,75)(64,95,88)(65,81,101)(66,67,114)(68,100,79)(69,86,92)(70,72,105)(71,119,118)(73,91,83)(74,77,96)(76,110,122)(78,82,87)(80,115,113)(84,120,104)(85,106,117)(89,111,108)(90,97,121)(93,116,99)(94,102,112)(98,107,103) );

G=PermutationGroup([(1,62),(2,63),(3,64),(4,65),(5,66),(6,67),(7,68),(8,69),(9,70),(10,71),(11,72),(12,73),(13,74),(14,75),(15,76),(16,77),(17,78),(18,79),(19,80),(20,81),(21,82),(22,83),(23,84),(24,85),(25,86),(26,87),(27,88),(28,89),(29,90),(30,91),(31,92),(32,93),(33,94),(34,95),(35,96),(36,97),(37,98),(38,99),(39,100),(40,101),(41,102),(42,103),(43,104),(44,105),(45,106),(46,107),(47,108),(48,109),(49,110),(50,111),(51,112),(52,113),(53,114),(54,115),(55,116),(56,117),(57,118),(58,119),(59,120),(60,121),(61,122)], [(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),(62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122)], [(2,48,14),(3,34,27),(4,20,40),(5,6,53),(7,39,18),(8,25,31),(9,11,44),(10,58,57),(12,30,22),(13,16,35),(15,49,61),(17,21,26),(19,54,52),(23,59,43),(24,45,56),(28,50,47),(29,36,60),(32,55,38),(33,41,51),(37,46,42),(63,109,75),(64,95,88),(65,81,101),(66,67,114),(68,100,79),(69,86,92),(70,72,105),(71,119,118),(73,91,83),(74,77,96),(76,110,122),(78,82,87),(80,115,113),(84,120,104),(85,106,117),(89,111,108),(90,97,121),(93,116,99),(94,102,112),(98,107,103)])

46 conjugacy classes

class 1  2 3A3B6A6B61A···61T122A···122T
order12336661···61122···122
size11616161613···33···3

46 irreducible representations

dim111133
type++
imageC1C2C3C6C61⋊C3C2×C61⋊C3
kernelC2×C61⋊C3C61⋊C3C122C61C2C1
# reps11222020

Matrix representation of C2×C61⋊C3 in GL3(𝔽13) generated by

1200
0120
0012
,
340
121
040
,
109
0012
0112
G:=sub<GL(3,GF(13))| [12,0,0,0,12,0,0,0,12],[3,1,0,4,2,4,0,1,0],[1,0,0,0,0,1,9,12,12] >;

C2×C61⋊C3 in GAP, Magma, Sage, TeX

C_2\times C_{61}\rtimes C_3
% in TeX

G:=Group("C2xC61:C3");
// GroupNames label

G:=SmallGroup(366,2);
// by ID

G=gap.SmallGroup(366,2);
# by ID

G:=PCGroup([3,-2,-3,-61,1274]);
// Polycyclic

G:=Group<a,b,c|a^2=b^61=c^3=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^13>;
// generators/relations

Export

Subgroup lattice of C2×C61⋊C3 in TeX

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