Copied to
clipboard

## G = C3×D37order 222 = 2·3·37

### Direct product of C3 and D37

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C37 — C3×D37
 Chief series C1 — C37 — C111 — C3×D37
 Lower central C37 — C3×D37
 Upper central C1 — C3

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

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

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

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

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

C3×D37 is a maximal subgroup of   C37⋊Dic3

60 conjugacy classes

 class 1 2 3A 3B 6A 6B 37A ··· 37R 111A ··· 111AJ order 1 2 3 3 6 6 37 ··· 37 111 ··· 111 size 1 37 1 1 37 37 2 ··· 2 2 ··· 2

60 irreducible representations

 dim 1 1 1 1 2 2 type + + + image C1 C2 C3 C6 D37 C3×D37 kernel C3×D37 C111 D37 C37 C3 C1 # reps 1 1 2 2 18 36

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

 183 0 0 183
,
 44 1 6 137
,
 137 222 36 86
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

׿
×
𝔽