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G = C7⋊C54order 378 = 2·33·7

The semidirect product of C7 and C54 acting via C54/C9=C6

metacyclic, supersoluble, monomial, Z-group

Aliases: C7⋊C54, D7⋊C27, C9.F7, C63.C6, C21.C18, C7⋊C27⋊C2, C3.(C7⋊C18), (C9×D7).C3, (C3×D7).C9, SmallGroup(378,1)

Series: Derived Chief Lower central Upper central

C1C7 — C7⋊C54
C1C7C21C63C7⋊C27 — C7⋊C54
C7 — C7⋊C54
C1C9

Generators and relations for C7⋊C54
 G = < a,b | a7=b54=1, bab-1=a3 >

7C2
7C6
7C18
7C27
7C54

Smallest permutation representation of C7⋊C54
On 189 points
Generators in S189
(1 106 39 182 155 66 133)(2 183 134 40 67 107 156)(3 41 157 135 108 184 68)(4 82 69 158 185 42 109)(5 159 110 70 43 83 186)(6 71 187 111 84 160 44)(7 112 45 188 161 72 85)(8 189 86 46 73 113 162)(9 47 163 87 114 136 74)(10 88 75 164 137 48 115)(11 165 116 76 49 89 138)(12 77 139 117 90 166 50)(13 118 51 140 167 78 91)(14 141 92 52 79 119 168)(15 53 169 93 120 142 80)(16 94 81 170 143 54 121)(17 171 122 28 55 95 144)(18 29 145 123 96 172 56)(19 124 57 146 173 30 97)(20 147 98 58 31 125 174)(21 59 175 99 126 148 32)(22 100 33 176 149 60 127)(23 177 128 34 61 101 150)(24 35 151 129 102 178 62)(25 130 63 152 179 36 103)(26 153 104 64 37 131 180)(27 65 181 105 132 154 38)
(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)| (1,106,39,182,155,66,133)(2,183,134,40,67,107,156)(3,41,157,135,108,184,68)(4,82,69,158,185,42,109)(5,159,110,70,43,83,186)(6,71,187,111,84,160,44)(7,112,45,188,161,72,85)(8,189,86,46,73,113,162)(9,47,163,87,114,136,74)(10,88,75,164,137,48,115)(11,165,116,76,49,89,138)(12,77,139,117,90,166,50)(13,118,51,140,167,78,91)(14,141,92,52,79,119,168)(15,53,169,93,120,142,80)(16,94,81,170,143,54,121)(17,171,122,28,55,95,144)(18,29,145,123,96,172,56)(19,124,57,146,173,30,97)(20,147,98,58,31,125,174)(21,59,175,99,126,148,32)(22,100,33,176,149,60,127)(23,177,128,34,61,101,150)(24,35,151,129,102,178,62)(25,130,63,152,179,36,103)(26,153,104,64,37,131,180)(27,65,181,105,132,154,38), (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( (1,106,39,182,155,66,133)(2,183,134,40,67,107,156)(3,41,157,135,108,184,68)(4,82,69,158,185,42,109)(5,159,110,70,43,83,186)(6,71,187,111,84,160,44)(7,112,45,188,161,72,85)(8,189,86,46,73,113,162)(9,47,163,87,114,136,74)(10,88,75,164,137,48,115)(11,165,116,76,49,89,138)(12,77,139,117,90,166,50)(13,118,51,140,167,78,91)(14,141,92,52,79,119,168)(15,53,169,93,120,142,80)(16,94,81,170,143,54,121)(17,171,122,28,55,95,144)(18,29,145,123,96,172,56)(19,124,57,146,173,30,97)(20,147,98,58,31,125,174)(21,59,175,99,126,148,32)(22,100,33,176,149,60,127)(23,177,128,34,61,101,150)(24,35,151,129,102,178,62)(25,130,63,152,179,36,103)(26,153,104,64,37,131,180)(27,65,181,105,132,154,38), (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([(1,106,39,182,155,66,133),(2,183,134,40,67,107,156),(3,41,157,135,108,184,68),(4,82,69,158,185,42,109),(5,159,110,70,43,83,186),(6,71,187,111,84,160,44),(7,112,45,188,161,72,85),(8,189,86,46,73,113,162),(9,47,163,87,114,136,74),(10,88,75,164,137,48,115),(11,165,116,76,49,89,138),(12,77,139,117,90,166,50),(13,118,51,140,167,78,91),(14,141,92,52,79,119,168),(15,53,169,93,120,142,80),(16,94,81,170,143,54,121),(17,171,122,28,55,95,144),(18,29,145,123,96,172,56),(19,124,57,146,173,30,97),(20,147,98,58,31,125,174),(21,59,175,99,126,148,32),(22,100,33,176,149,60,127),(23,177,128,34,61,101,150),(24,35,151,129,102,178,62),(25,130,63,152,179,36,103),(26,153,104,64,37,131,180),(27,65,181,105,132,154,38)], [(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)])

63 conjugacy classes

class 1  2 3A3B6A6B 7 9A···9F18A···18F21A21B27A···27R54A···54R63A···63F
order12336679···918···18212127···2754···5463···63
size17117761···17···7667···77···76···6

63 irreducible representations

dim11111111666
type+++
imageC1C2C3C6C9C18C27C54F7C7⋊C18C7⋊C54
kernelC7⋊C54C7⋊C27C9×D7C63C3×D7C21D7C7C9C3C1
# reps1122661818126

Matrix representation of C7⋊C54 in GL7(𝔽379)

1000000
037810000
037801000
037800100
037800010
037800001
037800000
,
228000000
01032761832080171
0103680311276354
0025120831168171
0171683112082510
0354276311068103
01710208183276103

G:=sub<GL(7,GF(379))| [1,0,0,0,0,0,0,0,378,378,378,378,378,378,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0],[228,0,0,0,0,0,0,0,103,103,0,171,354,171,0,276,68,251,68,276,0,0,183,0,208,311,311,208,0,208,311,311,208,0,183,0,0,276,68,251,68,276,0,171,354,171,0,103,103] >;

C7⋊C54 in GAP, Magma, Sage, TeX

C_7\rtimes C_{54}
% in TeX

G:=Group("C7:C54");
// GroupNames label

G:=SmallGroup(378,1);
// by ID

G=gap.SmallGroup(378,1);
# by ID

G:=PCGroup([5,-2,-3,-3,-3,-7,36,57,8104,2709]);
// Polycyclic

G:=Group<a,b|a^7=b^54=1,b*a*b^-1=a^3>;
// generators/relations

Export

Subgroup lattice of C7⋊C54 in TeX

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