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G = C7⋊C27order 189 = 33·7

The semidirect product of C7 and C27 acting via C27/C9=C3

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

Aliases: C7⋊C27, C63.C3, C21.C9, C3.(C7⋊C9), C9.(C7⋊C3), SmallGroup(189,1)

Series: Derived Chief Lower central Upper central

C1C7 — C7⋊C27
C1C7C21C63 — C7⋊C27
C7 — C7⋊C27
C1C9

Generators and relations for C7⋊C27
 G = < a,b | a7=b27=1, bab-1=a4 >

7C27

Smallest permutation representation of C7⋊C27
Regular action on 189 points
Generators in S189
(1 108 131 28 187 147 61)(2 188 82 148 132 62 29)(3 133 189 63 83 30 149)(4 84 134 31 163 150 64)(5 164 85 151 135 65 32)(6 109 165 66 86 33 152)(7 87 110 34 166 153 67)(8 167 88 154 111 68 35)(9 112 168 69 89 36 155)(10 90 113 37 169 156 70)(11 170 91 157 114 71 38)(12 115 171 72 92 39 158)(13 93 116 40 172 159 73)(14 173 94 160 117 74 41)(15 118 174 75 95 42 161)(16 96 119 43 175 162 76)(17 176 97 136 120 77 44)(18 121 177 78 98 45 137)(19 99 122 46 178 138 79)(20 179 100 139 123 80 47)(21 124 180 81 101 48 140)(22 102 125 49 181 141 55)(23 182 103 142 126 56 50)(24 127 183 57 104 51 143)(25 105 128 52 184 144 58)(26 185 106 145 129 59 53)(27 130 186 60 107 54 146)
(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)(136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162)(163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189)

G:=sub<Sym(189)| (1,108,131,28,187,147,61)(2,188,82,148,132,62,29)(3,133,189,63,83,30,149)(4,84,134,31,163,150,64)(5,164,85,151,135,65,32)(6,109,165,66,86,33,152)(7,87,110,34,166,153,67)(8,167,88,154,111,68,35)(9,112,168,69,89,36,155)(10,90,113,37,169,156,70)(11,170,91,157,114,71,38)(12,115,171,72,92,39,158)(13,93,116,40,172,159,73)(14,173,94,160,117,74,41)(15,118,174,75,95,42,161)(16,96,119,43,175,162,76)(17,176,97,136,120,77,44)(18,121,177,78,98,45,137)(19,99,122,46,178,138,79)(20,179,100,139,123,80,47)(21,124,180,81,101,48,140)(22,102,125,49,181,141,55)(23,182,103,142,126,56,50)(24,127,183,57,104,51,143)(25,105,128,52,184,144,58)(26,185,106,145,129,59,53)(27,130,186,60,107,54,146), (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)(136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162)(163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189)>;

G:=Group( (1,108,131,28,187,147,61)(2,188,82,148,132,62,29)(3,133,189,63,83,30,149)(4,84,134,31,163,150,64)(5,164,85,151,135,65,32)(6,109,165,66,86,33,152)(7,87,110,34,166,153,67)(8,167,88,154,111,68,35)(9,112,168,69,89,36,155)(10,90,113,37,169,156,70)(11,170,91,157,114,71,38)(12,115,171,72,92,39,158)(13,93,116,40,172,159,73)(14,173,94,160,117,74,41)(15,118,174,75,95,42,161)(16,96,119,43,175,162,76)(17,176,97,136,120,77,44)(18,121,177,78,98,45,137)(19,99,122,46,178,138,79)(20,179,100,139,123,80,47)(21,124,180,81,101,48,140)(22,102,125,49,181,141,55)(23,182,103,142,126,56,50)(24,127,183,57,104,51,143)(25,105,128,52,184,144,58)(26,185,106,145,129,59,53)(27,130,186,60,107,54,146), (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)(136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162)(163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189) );

G=PermutationGroup([(1,108,131,28,187,147,61),(2,188,82,148,132,62,29),(3,133,189,63,83,30,149),(4,84,134,31,163,150,64),(5,164,85,151,135,65,32),(6,109,165,66,86,33,152),(7,87,110,34,166,153,67),(8,167,88,154,111,68,35),(9,112,168,69,89,36,155),(10,90,113,37,169,156,70),(11,170,91,157,114,71,38),(12,115,171,72,92,39,158),(13,93,116,40,172,159,73),(14,173,94,160,117,74,41),(15,118,174,75,95,42,161),(16,96,119,43,175,162,76),(17,176,97,136,120,77,44),(18,121,177,78,98,45,137),(19,99,122,46,178,138,79),(20,179,100,139,123,80,47),(21,124,180,81,101,48,140),(22,102,125,49,181,141,55),(23,182,103,142,126,56,50),(24,127,183,57,104,51,143),(25,105,128,52,184,144,58),(26,185,106,145,129,59,53),(27,130,186,60,107,54,146)], [(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),(136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162),(163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189)])

C7⋊C27 is a maximal subgroup of   C7⋊C54

45 conjugacy classes

class 1 3A3B7A7B9A···9F21A21B21C21D27A···27R63A···63L
order133779···92121212127···2763···63
size111331···133337···73···3

45 irreducible representations

dim1111333
type+
imageC1C3C9C27C7⋊C3C7⋊C9C7⋊C27
kernelC7⋊C27C63C21C7C9C3C1
# reps126182412

Matrix representation of C7⋊C27 in GL3(𝔽379) generated by

37810
37801
351128
,
132117317
12430644
95376320
G:=sub<GL(3,GF(379))| [378,378,351,1,0,1,0,1,28],[132,124,95,117,306,376,317,44,320] >;

C7⋊C27 in GAP, Magma, Sage, TeX

C_7\rtimes C_{27}
% in TeX

G:=Group("C7:C27");
// GroupNames label

G:=SmallGroup(189,1);
// by ID

G=gap.SmallGroup(189,1);
# by ID

G:=PCGroup([4,-3,-3,-3,-7,12,29,867]);
// Polycyclic

G:=Group<a,b|a^7=b^27=1,b*a*b^-1=a^4>;
// generators/relations

Export

Subgroup lattice of C7⋊C27 in TeX

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