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G = C3×C26.C6order 468 = 22·32·13

Direct product of C3 and C26.C6

direct product, metacyclic, supersoluble, monomial, A-group

Aliases: C3×C26.C6, C394C12, C78.4C6, Dic13⋊C32, C26.(C3×C6), C13⋊C32C12, C132(C3×C12), C6.4(C13⋊C6), (C3×Dic13)⋊C3, C2.(C3×C13⋊C6), (C3×C13⋊C3)⋊4C4, (C2×C13⋊C3).C6, (C6×C13⋊C3).2C2, SmallGroup(468,19)

Series: Derived Chief Lower central Upper central

C1C13 — C3×C26.C6
C1C13C26C78C6×C13⋊C3 — C3×C26.C6
C13 — C3×C26.C6
C1C6

Generators and relations for C3×C26.C6
 G = < a,b,c | a3=b26=1, c6=b13, ab=ba, ac=ca, cbc-1=b23 >

13C3
13C3
13C3
13C4
13C6
13C6
13C6
13C32
13C12
13C12
13C12
13C12
13C3×C6
13C3×C12

Smallest permutation representation of C3×C26.C6
On 156 points
Generators in S156
(1 53 45)(2 54 46)(3 55 47)(4 56 48)(5 57 49)(6 58 50)(7 59 51)(8 60 52)(9 61 27)(10 62 28)(11 63 29)(12 64 30)(13 65 31)(14 66 32)(15 67 33)(16 68 34)(17 69 35)(18 70 36)(19 71 37)(20 72 38)(21 73 39)(22 74 40)(23 75 41)(24 76 42)(25 77 43)(26 78 44)(79 143 126)(80 144 127)(81 145 128)(82 146 129)(83 147 130)(84 148 105)(85 149 106)(86 150 107)(87 151 108)(88 152 109)(89 153 110)(90 154 111)(91 155 112)(92 156 113)(93 131 114)(94 132 115)(95 133 116)(96 134 117)(97 135 118)(98 136 119)(99 137 120)(100 138 121)(101 139 122)(102 140 123)(103 141 124)(104 142 125)
(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)(131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156)
(1 122 66 88 45 139 14 109 53 101 32 152)(2 113 69 87 28 136 15 126 56 100 41 149)(3 130 72 86 37 133 16 117 59 99 50 146)(4 121 75 85 46 156 17 108 62 98 33 143)(5 112 78 84 29 153 18 125 65 97 42 140)(6 129 55 83 38 150 19 116 68 96 51 137)(7 120 58 82 47 147 20 107 71 95 34 134)(8 111 61 81 30 144 21 124 74 94 43 131)(9 128 64 80 39 141 22 115 77 93 52 154)(10 119 67 79 48 138 23 106 54 92 35 151)(11 110 70 104 31 135 24 123 57 91 44 148)(12 127 73 103 40 132 25 114 60 90 27 145)(13 118 76 102 49 155 26 105 63 89 36 142)

G:=sub<Sym(156)| (1,53,45)(2,54,46)(3,55,47)(4,56,48)(5,57,49)(6,58,50)(7,59,51)(8,60,52)(9,61,27)(10,62,28)(11,63,29)(12,64,30)(13,65,31)(14,66,32)(15,67,33)(16,68,34)(17,69,35)(18,70,36)(19,71,37)(20,72,38)(21,73,39)(22,74,40)(23,75,41)(24,76,42)(25,77,43)(26,78,44)(79,143,126)(80,144,127)(81,145,128)(82,146,129)(83,147,130)(84,148,105)(85,149,106)(86,150,107)(87,151,108)(88,152,109)(89,153,110)(90,154,111)(91,155,112)(92,156,113)(93,131,114)(94,132,115)(95,133,116)(96,134,117)(97,135,118)(98,136,119)(99,137,120)(100,138,121)(101,139,122)(102,140,123)(103,141,124)(104,142,125), (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)(131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156), (1,122,66,88,45,139,14,109,53,101,32,152)(2,113,69,87,28,136,15,126,56,100,41,149)(3,130,72,86,37,133,16,117,59,99,50,146)(4,121,75,85,46,156,17,108,62,98,33,143)(5,112,78,84,29,153,18,125,65,97,42,140)(6,129,55,83,38,150,19,116,68,96,51,137)(7,120,58,82,47,147,20,107,71,95,34,134)(8,111,61,81,30,144,21,124,74,94,43,131)(9,128,64,80,39,141,22,115,77,93,52,154)(10,119,67,79,48,138,23,106,54,92,35,151)(11,110,70,104,31,135,24,123,57,91,44,148)(12,127,73,103,40,132,25,114,60,90,27,145)(13,118,76,102,49,155,26,105,63,89,36,142)>;

G:=Group( (1,53,45)(2,54,46)(3,55,47)(4,56,48)(5,57,49)(6,58,50)(7,59,51)(8,60,52)(9,61,27)(10,62,28)(11,63,29)(12,64,30)(13,65,31)(14,66,32)(15,67,33)(16,68,34)(17,69,35)(18,70,36)(19,71,37)(20,72,38)(21,73,39)(22,74,40)(23,75,41)(24,76,42)(25,77,43)(26,78,44)(79,143,126)(80,144,127)(81,145,128)(82,146,129)(83,147,130)(84,148,105)(85,149,106)(86,150,107)(87,151,108)(88,152,109)(89,153,110)(90,154,111)(91,155,112)(92,156,113)(93,131,114)(94,132,115)(95,133,116)(96,134,117)(97,135,118)(98,136,119)(99,137,120)(100,138,121)(101,139,122)(102,140,123)(103,141,124)(104,142,125), (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)(131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156), (1,122,66,88,45,139,14,109,53,101,32,152)(2,113,69,87,28,136,15,126,56,100,41,149)(3,130,72,86,37,133,16,117,59,99,50,146)(4,121,75,85,46,156,17,108,62,98,33,143)(5,112,78,84,29,153,18,125,65,97,42,140)(6,129,55,83,38,150,19,116,68,96,51,137)(7,120,58,82,47,147,20,107,71,95,34,134)(8,111,61,81,30,144,21,124,74,94,43,131)(9,128,64,80,39,141,22,115,77,93,52,154)(10,119,67,79,48,138,23,106,54,92,35,151)(11,110,70,104,31,135,24,123,57,91,44,148)(12,127,73,103,40,132,25,114,60,90,27,145)(13,118,76,102,49,155,26,105,63,89,36,142) );

G=PermutationGroup([[(1,53,45),(2,54,46),(3,55,47),(4,56,48),(5,57,49),(6,58,50),(7,59,51),(8,60,52),(9,61,27),(10,62,28),(11,63,29),(12,64,30),(13,65,31),(14,66,32),(15,67,33),(16,68,34),(17,69,35),(18,70,36),(19,71,37),(20,72,38),(21,73,39),(22,74,40),(23,75,41),(24,76,42),(25,77,43),(26,78,44),(79,143,126),(80,144,127),(81,145,128),(82,146,129),(83,147,130),(84,148,105),(85,149,106),(86,150,107),(87,151,108),(88,152,109),(89,153,110),(90,154,111),(91,155,112),(92,156,113),(93,131,114),(94,132,115),(95,133,116),(96,134,117),(97,135,118),(98,136,119),(99,137,120),(100,138,121),(101,139,122),(102,140,123),(103,141,124),(104,142,125)], [(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),(131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156)], [(1,122,66,88,45,139,14,109,53,101,32,152),(2,113,69,87,28,136,15,126,56,100,41,149),(3,130,72,86,37,133,16,117,59,99,50,146),(4,121,75,85,46,156,17,108,62,98,33,143),(5,112,78,84,29,153,18,125,65,97,42,140),(6,129,55,83,38,150,19,116,68,96,51,137),(7,120,58,82,47,147,20,107,71,95,34,134),(8,111,61,81,30,144,21,124,74,94,43,131),(9,128,64,80,39,141,22,115,77,93,52,154),(10,119,67,79,48,138,23,106,54,92,35,151),(11,110,70,104,31,135,24,123,57,91,44,148),(12,127,73,103,40,132,25,114,60,90,27,145),(13,118,76,102,49,155,26,105,63,89,36,142)]])

48 conjugacy classes

class 1  2 3A3B3C···3H4A4B6A6B6C···6H12A···12P13A13B26A26B39A39B39C39D78A78B78C78D
order12333···344666···612···12131326263939393978787878
size111113···1313131113···1313···13666666666666

48 irreducible representations

dim1111111116666
type+++-
imageC1C2C3C3C4C6C6C12C12C13⋊C6C26.C6C3×C13⋊C6C3×C26.C6
kernelC3×C26.C6C6×C13⋊C3C26.C6C3×Dic13C3×C13⋊C3C2×C13⋊C3C78C13⋊C3C39C6C3C2C1
# reps11622621242244

Matrix representation of C3×C26.C6 in GL7(𝔽157)

1000000
014400000
001440000
000144000
000014400
000001440
000000144
,
156000000
0906692659167
092642513391155
0906791155168
01155916790155
09113325649267
09165926690156
,
107000000
04113169762988
015544186811788
0150656979191
0451321091099588
034436928120139
0107135691162688

G:=sub<GL(7,GF(157))| [1,0,0,0,0,0,0,0,144,0,0,0,0,0,0,0,144,0,0,0,0,0,0,0,144,0,0,0,0,0,0,0,144,0,0,0,0,0,0,0,144,0,0,0,0,0,0,0,144],[156,0,0,0,0,0,0,0,90,92,90,1,91,91,0,66,64,67,155,133,65,0,92,25,91,91,25,92,0,65,133,155,67,64,66,0,91,91,1,90,92,90,0,67,155,68,155,67,156],[107,0,0,0,0,0,0,0,41,155,150,45,34,107,0,131,44,65,132,43,135,0,69,18,69,109,69,69,0,76,68,7,109,28,116,0,29,117,91,95,120,26,0,88,88,91,88,139,88] >;

C3×C26.C6 in GAP, Magma, Sage, TeX

C_3\times C_{26}.C_6
% in TeX

G:=Group("C3xC26.C6");
// GroupNames label

G:=SmallGroup(468,19);
// by ID

G=gap.SmallGroup(468,19);
# by ID

G:=PCGroup([5,-2,-3,-3,-2,-13,90,10804,1359]);
// Polycyclic

G:=Group<a,b,c|a^3=b^26=1,c^6=b^13,a*b=b*a,a*c=c*a,c*b*c^-1=b^23>;
// generators/relations

Export

Subgroup lattice of C3×C26.C6 in TeX

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