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G = C133⋊C3order 399 = 3·7·19

3rd semidirect product of C133 and C3 acting faithfully

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

Aliases: C1333C3, C71(C19⋊C3), C191(C7⋊C3), SmallGroup(399,3)

Series: Derived Chief Lower central Upper central

C1C133 — C133⋊C3
C1C19C133 — C133⋊C3
C133 — C133⋊C3
C1

Generators and relations for C133⋊C3
 G = < a,b | a133=b3=1, bab-1=a11 >

133C3
19C7⋊C3
7C19⋊C3

Smallest permutation representation of C133⋊C3
On 133 points
Generators in S133
(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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133)
(2 122 12)(3 110 23)(4 98 34)(5 86 45)(6 74 56)(7 62 67)(8 50 78)(9 38 89)(10 26 100)(11 14 111)(13 123 133)(15 99 22)(16 87 33)(17 75 44)(18 63 55)(19 51 66)(20 39 77)(21 27 88)(24 124 121)(25 112 132)(28 76 32)(29 64 43)(30 52 54)(31 40 65)(35 125 109)(36 113 120)(37 101 131)(41 53 42)(46 126 97)(47 114 108)(48 102 119)(49 90 130)(57 127 85)(58 115 96)(59 103 107)(60 91 118)(61 79 129)(68 128 73)(69 116 84)(70 104 95)(71 92 106)(72 80 117)(81 105 83)(82 93 94)

G:=sub<Sym(133)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133), (2,122,12)(3,110,23)(4,98,34)(5,86,45)(6,74,56)(7,62,67)(8,50,78)(9,38,89)(10,26,100)(11,14,111)(13,123,133)(15,99,22)(16,87,33)(17,75,44)(18,63,55)(19,51,66)(20,39,77)(21,27,88)(24,124,121)(25,112,132)(28,76,32)(29,64,43)(30,52,54)(31,40,65)(35,125,109)(36,113,120)(37,101,131)(41,53,42)(46,126,97)(47,114,108)(48,102,119)(49,90,130)(57,127,85)(58,115,96)(59,103,107)(60,91,118)(61,79,129)(68,128,73)(69,116,84)(70,104,95)(71,92,106)(72,80,117)(81,105,83)(82,93,94)>;

G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133), (2,122,12)(3,110,23)(4,98,34)(5,86,45)(6,74,56)(7,62,67)(8,50,78)(9,38,89)(10,26,100)(11,14,111)(13,123,133)(15,99,22)(16,87,33)(17,75,44)(18,63,55)(19,51,66)(20,39,77)(21,27,88)(24,124,121)(25,112,132)(28,76,32)(29,64,43)(30,52,54)(31,40,65)(35,125,109)(36,113,120)(37,101,131)(41,53,42)(46,126,97)(47,114,108)(48,102,119)(49,90,130)(57,127,85)(58,115,96)(59,103,107)(60,91,118)(61,79,129)(68,128,73)(69,116,84)(70,104,95)(71,92,106)(72,80,117)(81,105,83)(82,93,94) );

G=PermutationGroup([(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133)], [(2,122,12),(3,110,23),(4,98,34),(5,86,45),(6,74,56),(7,62,67),(8,50,78),(9,38,89),(10,26,100),(11,14,111),(13,123,133),(15,99,22),(16,87,33),(17,75,44),(18,63,55),(19,51,66),(20,39,77),(21,27,88),(24,124,121),(25,112,132),(28,76,32),(29,64,43),(30,52,54),(31,40,65),(35,125,109),(36,113,120),(37,101,131),(41,53,42),(46,126,97),(47,114,108),(48,102,119),(49,90,130),(57,127,85),(58,115,96),(59,103,107),(60,91,118),(61,79,129),(68,128,73),(69,116,84),(70,104,95),(71,92,106),(72,80,117),(81,105,83),(82,93,94)])

47 conjugacy classes

class 1 3A3B7A7B19A···19F133A···133AJ
order1337719···19133···133
size1133133333···33···3

47 irreducible representations

dim11333
type+
imageC1C3C7⋊C3C19⋊C3C133⋊C3
kernelC133⋊C3C133C19C7C1
# reps122636

Matrix representation of C133⋊C3 in GL3(𝔽11) generated by

804
406
613
,
120
0101
0100
G:=sub<GL(3,GF(11))| [8,4,6,0,0,1,4,6,3],[1,0,0,2,10,10,0,1,0] >;

C133⋊C3 in GAP, Magma, Sage, TeX

C_{133}\rtimes C_3
% in TeX

G:=Group("C133:C3");
// GroupNames label

G:=SmallGroup(399,3);
// by ID

G=gap.SmallGroup(399,3);
# by ID

G:=PCGroup([3,-3,-7,-19,37,1325]);
// Polycyclic

G:=Group<a,b|a^133=b^3=1,b*a*b^-1=a^11>;
// generators/relations

Export

Subgroup lattice of C133⋊C3 in TeX

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