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G = C37⋊Dic3order 444 = 22·3·37

The semidirect product of C37 and Dic3 acting via Dic3/C3=C4

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

Aliases: C37⋊Dic3, C1111C4, D37.S3, C3⋊(C37⋊C4), (C3×D37).1C2, SmallGroup(444,10)

Series: Derived Chief Lower central Upper central

C1C111 — C37⋊Dic3
C1C37C111C3×D37 — C37⋊Dic3
C111 — C37⋊Dic3
C1

Generators and relations for C37⋊Dic3
 G = < a,b,c | a37=b6=1, c2=b3, bab-1=a-1, cac-1=a31, cbc-1=b-1 >

37C2
111C4
37C6
37Dic3
3C37⋊C4

Smallest permutation representation of C37⋊Dic3
On 111 points
Generators in S111
(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)
(1 101 69)(2 100 70 37 102 68)(3 99 71 36 103 67)(4 98 72 35 104 66)(5 97 73 34 105 65)(6 96 74 33 106 64)(7 95 38 32 107 63)(8 94 39 31 108 62)(9 93 40 30 109 61)(10 92 41 29 110 60)(11 91 42 28 111 59)(12 90 43 27 75 58)(13 89 44 26 76 57)(14 88 45 25 77 56)(15 87 46 24 78 55)(16 86 47 23 79 54)(17 85 48 22 80 53)(18 84 49 21 81 52)(19 83 50 20 82 51)
(2 7 37 32)(3 13 36 26)(4 19 35 20)(5 25 34 14)(6 31 33 8)(9 12 30 27)(10 18 29 21)(11 24 28 15)(16 17 23 22)(38 100 63 102)(39 106 62 96)(40 75 61 90)(41 81 60 84)(42 87 59 78)(43 93 58 109)(44 99 57 103)(45 105 56 97)(46 111 55 91)(47 80 54 85)(48 86 53 79)(49 92 52 110)(50 98 51 104)(64 108 74 94)(65 77 73 88)(66 83 72 82)(67 89 71 76)(68 95 70 107)(69 101)

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

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

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

33 conjugacy classes

class 1  2  3 4A4B 6 37A···37I111A···111R
order12344637···37111···111
size1372111111744···44···4

33 irreducible representations

dim1112244
type+++-+
imageC1C2C4S3Dic3C37⋊C4C37⋊Dic3
kernelC37⋊Dic3C3×D37C111D37C37C3C1
# reps11211918

Matrix representation of C37⋊Dic3 in GL4(𝔽1777) generated by

0100
0010
0001
1776102010141020
,
5276231055269
150955611551250
85412841548268
1634328618923
,
1000
923467961033
686118015381073
1578930941669
G:=sub<GL(4,GF(1777))| [0,0,0,1776,1,0,0,1020,0,1,0,1014,0,0,1,1020],[527,1509,854,1634,623,556,1284,328,1055,1155,1548,618,269,1250,268,923],[1,92,686,1578,0,346,1180,930,0,796,1538,94,0,1033,1073,1669] >;

C37⋊Dic3 in GAP, Magma, Sage, TeX

C_{37}\rtimes {\rm Dic}_3
% in TeX

G:=Group("C37:Dic3");
// GroupNames label

G:=SmallGroup(444,10);
// by ID

G=gap.SmallGroup(444,10);
# by ID

G:=PCGroup([4,-2,-2,-3,-37,8,98,1155,3463]);
// Polycyclic

G:=Group<a,b,c|a^37=b^6=1,c^2=b^3,b*a*b^-1=a^-1,c*a*c^-1=a^31,c*b*c^-1=b^-1>;
// generators/relations

Export

Subgroup lattice of C37⋊Dic3 in TeX

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