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G = C1334C3order 399 = 3·7·19

4th semidirect product of C133 and C3 acting faithfully

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

Aliases: C1334C3, C72(C19⋊C3), C192(C7⋊C3), SmallGroup(399,4)

Series: Derived Chief Lower central Upper central

C1C133 — C1334C3
C1C19C133 — C1334C3
C133 — C1334C3
C1

Generators and relations for C1334C3
 G = < a,b | a133=b3=1, bab-1=a30 >

133C3
19C7⋊C3
7C19⋊C3

Smallest permutation representation of C1334C3
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)
(2 103 31)(3 72 61)(4 41 91)(5 10 121)(6 112 18)(7 81 48)(8 50 78)(9 19 108)(11 90 35)(12 59 65)(13 28 95)(14 130 125)(15 99 22)(16 68 52)(17 37 82)(20 77 39)(21 46 69)(23 117 129)(24 86 26)(25 55 56)(27 126 116)(29 64 43)(30 33 73)(32 104 133)(34 42 60)(36 113 120)(38 51 47)(40 122 107)(44 131 94)(45 100 124)(49 109 111)(53 118 98)(54 87 128)(57 127 85)(58 96 115)(62 105 102)(63 74 132)(66 114 89)(67 83 119)(70 123 76)(71 92 106)(75 101 93)(79 110 80)(84 88 97)

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

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

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

47 conjugacy classes

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

47 irreducible representations

dim11333
type+
imageC1C3C7⋊C3C19⋊C3C1334C3
kernelC1334C3C133C19C7C1
# reps122636

Matrix representation of C1334C3 in GL3(𝔽1597) generated by

153669830
8301242397
397410577
,
100
135211231414
1719473
G:=sub<GL(3,GF(1597))| [153,830,397,669,1242,410,830,397,577],[1,1352,1,0,1123,719,0,1414,473] >;

C1334C3 in GAP, Magma, Sage, TeX

C_{133}\rtimes_4C_3
% in TeX

G:=Group("C133:4C3");
// GroupNames label

G:=SmallGroup(399,4);
// by ID

G=gap.SmallGroup(399,4);
# by ID

G:=PCGroup([3,-3,-7,-19,73,1325]);
// Polycyclic

G:=Group<a,b|a^133=b^3=1,b*a*b^-1=a^30>;
// generators/relations

Export

Subgroup lattice of C1334C3 in TeX

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