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G = C3×D37order 222 = 2·3·37

Direct product of C3 and D37

direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary

Aliases: C3×D37, C373C6, C1112C2, SmallGroup(222,4)

Series: Derived Chief Lower central Upper central

C1C37 — C3×D37
C1C37C111 — C3×D37
C37 — C3×D37
C1C3

Generators and relations for C3×D37
 G = < a,b,c | a3=b37=c2=1, ab=ba, ac=ca, cbc=b-1 >

37C2
37C6

Smallest permutation representation of C3×D37
On 111 points
Generators in S111
(1 110 55)(2 111 56)(3 75 57)(4 76 58)(5 77 59)(6 78 60)(7 79 61)(8 80 62)(9 81 63)(10 82 64)(11 83 65)(12 84 66)(13 85 67)(14 86 68)(15 87 69)(16 88 70)(17 89 71)(18 90 72)(19 91 73)(20 92 74)(21 93 38)(22 94 39)(23 95 40)(24 96 41)(25 97 42)(26 98 43)(27 99 44)(28 100 45)(29 101 46)(30 102 47)(31 103 48)(32 104 49)(33 105 50)(34 106 51)(35 107 52)(36 108 53)(37 109 54)
(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)
(1 37)(2 36)(3 35)(4 34)(5 33)(6 32)(7 31)(8 30)(9 29)(10 28)(11 27)(12 26)(13 25)(14 24)(15 23)(16 22)(17 21)(18 20)(38 71)(39 70)(40 69)(41 68)(42 67)(43 66)(44 65)(45 64)(46 63)(47 62)(48 61)(49 60)(50 59)(51 58)(52 57)(53 56)(54 55)(72 74)(75 107)(76 106)(77 105)(78 104)(79 103)(80 102)(81 101)(82 100)(83 99)(84 98)(85 97)(86 96)(87 95)(88 94)(89 93)(90 92)(108 111)(109 110)

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

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

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

C3×D37 is a maximal subgroup of   C37⋊Dic3

60 conjugacy classes

class 1  2 3A3B6A6B37A···37R111A···111AJ
order12336637···37111···111
size1371137372···22···2

60 irreducible representations

dim111122
type+++
imageC1C2C3C6D37C3×D37
kernelC3×D37C111D37C37C3C1
# reps11221836

Matrix representation of C3×D37 in GL2(𝔽223) generated by

1830
0183
,
441
6137
,
137222
3686
G:=sub<GL(2,GF(223))| [183,0,0,183],[44,6,1,137],[137,36,222,86] >;

C3×D37 in GAP, Magma, Sage, TeX

C_3\times D_{37}
% in TeX

G:=Group("C3xD37");
// GroupNames label

G:=SmallGroup(222,4);
// by ID

G=gap.SmallGroup(222,4);
# by ID

G:=PCGroup([3,-2,-3,-37,1946]);
// Polycyclic

G:=Group<a,b,c|a^3=b^37=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C3×D37 in TeX

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