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G = C117:3C3order 351 = 33·13

3rd semidirect product of C117 and C3 acting faithfully

metacyclic, supersoluble, monomial, 3-hyperelementary

Aliases: C117:3C3, C39.3C32, C13:23- 1+2, C13:C9:2C3, C9:2(C13:C3), C3.4(C3xC13:C3), (C3xC13:C3).2C3, SmallGroup(351,5)

Series: Derived Chief Lower central Upper central

C1C39 — C117:3C3
C1C13C39C3xC13:C3 — C117:3C3
C13C39 — C117:3C3
C1C3C9

Generators and relations for C117:3C3
 G = < a,b | a117=b3=1, bab-1=a22 >

Subgroups: 104 in 16 conjugacy classes, 10 normal (8 characteristic)
Quotients: C1, C3, C32, 3- 1+2, C13:C3, C3xC13:C3, C117:3C3
39C3
13C32
13C9
13C9
3C13:C3
133- 1+2

Smallest permutation representation of C117:3C3
On 117 points
Generators in S117
(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)
(2 17 23)(3 33 45)(4 49 67)(5 65 89)(6 81 111)(7 97 16)(8 113 38)(9 12 60)(10 28 82)(11 44 104)(13 76 31)(14 92 53)(15 108 75)(18 39 24)(19 55 46)(20 71 68)(21 87 90)(22 103 112)(25 34 61)(26 50 83)(27 66 105)(29 98 32)(30 114 54)(35 77 47)(36 93 69)(37 109 91)(41 56 62)(42 72 84)(43 88 106)(48 51 99)(52 115 70)(57 78 63)(58 94 85)(59 110 107)(64 73 100)(74 116 86)(80 95 101)(96 117 102)

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

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

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

47 conjugacy classes

class 1 3A3B3C3D9A9B9C9D9E9F13A13B13C13D39A···39H117A···117X
order133339999991313131339···39117···117
size1113939333939393933333···33···3

47 irreducible representations

dim11113333
type+
imageC1C3C3C33- 1+2C13:C3C3xC13:C3C117:3C3
kernelC117:3C3C13:C9C117C3xC13:C3C13C9C3C1
# reps142224824

Matrix representation of C117:3C3 in GL3(F937) generated by

646549548
85678505
38852250
,
76154775
153311870
143778802
G:=sub<GL(3,GF(937))| [646,85,38,549,678,852,548,505,250],[761,153,143,547,311,778,75,870,802] >;

C117:3C3 in GAP, Magma, Sage, TeX

C_{117}\rtimes_3C_3
% in TeX

G:=Group("C117:3C3");
// GroupNames label

G:=SmallGroup(351,5);
// by ID

G=gap.SmallGroup(351,5);
# by ID

G:=PCGroup([4,-3,-3,-3,-13,97,53,1299]);
// Polycyclic

G:=Group<a,b|a^117=b^3=1,b*a*b^-1=a^22>;
// generators/relations

Export

Subgroup lattice of C117:3C3 in TeX

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