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## G = C7⋊Dic7order 196 = 22·72

### The semidirect product of C7 and Dic7 acting via Dic7/C14=C2

Aliases: C7⋊Dic7, C723C4, C14.3D7, C2.(C7⋊D7), (C7×C14).2C2, SmallGroup(196,6)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C72 — C7⋊Dic7
 Chief series C1 — C7 — C72 — C7×C14 — C7⋊Dic7
 Lower central C72 — C7⋊Dic7
 Upper central C1 — C2

Generators and relations for C7⋊Dic7
G = < a,b,c | a7=b14=1, c2=b7, ab=ba, cac-1=a-1, cbc-1=b-1 >

Smallest permutation representation of C7⋊Dic7
Regular action on 196 points
Generators in S196
(1 156 131 116 143 178 52)(2 157 132 117 144 179 53)(3 158 133 118 145 180 54)(4 159 134 119 146 181 55)(5 160 135 120 147 182 56)(6 161 136 121 148 169 43)(7 162 137 122 149 170 44)(8 163 138 123 150 171 45)(9 164 139 124 151 172 46)(10 165 140 125 152 173 47)(11 166 127 126 153 174 48)(12 167 128 113 154 175 49)(13 168 129 114 141 176 50)(14 155 130 115 142 177 51)(15 61 97 33 189 77 104)(16 62 98 34 190 78 105)(17 63 85 35 191 79 106)(18 64 86 36 192 80 107)(19 65 87 37 193 81 108)(20 66 88 38 194 82 109)(21 67 89 39 195 83 110)(22 68 90 40 196 84 111)(23 69 91 41 183 71 112)(24 70 92 42 184 72 99)(25 57 93 29 185 73 100)(26 58 94 30 186 74 101)(27 59 95 31 187 75 102)(28 60 96 32 188 76 103)
(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)(183 184 185 186 187 188 189 190 191 192 193 194 195 196)
(1 15 8 22)(2 28 9 21)(3 27 10 20)(4 26 11 19)(5 25 12 18)(6 24 13 17)(7 23 14 16)(29 154 36 147)(30 153 37 146)(31 152 38 145)(32 151 39 144)(33 150 40 143)(34 149 41 142)(35 148 42 141)(43 70 50 63)(44 69 51 62)(45 68 52 61)(46 67 53 60)(47 66 54 59)(48 65 55 58)(49 64 56 57)(71 130 78 137)(72 129 79 136)(73 128 80 135)(74 127 81 134)(75 140 82 133)(76 139 83 132)(77 138 84 131)(85 169 92 176)(86 182 93 175)(87 181 94 174)(88 180 95 173)(89 179 96 172)(90 178 97 171)(91 177 98 170)(99 168 106 161)(100 167 107 160)(101 166 108 159)(102 165 109 158)(103 164 110 157)(104 163 111 156)(105 162 112 155)(113 192 120 185)(114 191 121 184)(115 190 122 183)(116 189 123 196)(117 188 124 195)(118 187 125 194)(119 186 126 193)

G:=sub<Sym(196)| (1,156,131,116,143,178,52)(2,157,132,117,144,179,53)(3,158,133,118,145,180,54)(4,159,134,119,146,181,55)(5,160,135,120,147,182,56)(6,161,136,121,148,169,43)(7,162,137,122,149,170,44)(8,163,138,123,150,171,45)(9,164,139,124,151,172,46)(10,165,140,125,152,173,47)(11,166,127,126,153,174,48)(12,167,128,113,154,175,49)(13,168,129,114,141,176,50)(14,155,130,115,142,177,51)(15,61,97,33,189,77,104)(16,62,98,34,190,78,105)(17,63,85,35,191,79,106)(18,64,86,36,192,80,107)(19,65,87,37,193,81,108)(20,66,88,38,194,82,109)(21,67,89,39,195,83,110)(22,68,90,40,196,84,111)(23,69,91,41,183,71,112)(24,70,92,42,184,72,99)(25,57,93,29,185,73,100)(26,58,94,30,186,74,101)(27,59,95,31,187,75,102)(28,60,96,32,188,76,103), (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)(183,184,185,186,187,188,189,190,191,192,193,194,195,196), (1,15,8,22)(2,28,9,21)(3,27,10,20)(4,26,11,19)(5,25,12,18)(6,24,13,17)(7,23,14,16)(29,154,36,147)(30,153,37,146)(31,152,38,145)(32,151,39,144)(33,150,40,143)(34,149,41,142)(35,148,42,141)(43,70,50,63)(44,69,51,62)(45,68,52,61)(46,67,53,60)(47,66,54,59)(48,65,55,58)(49,64,56,57)(71,130,78,137)(72,129,79,136)(73,128,80,135)(74,127,81,134)(75,140,82,133)(76,139,83,132)(77,138,84,131)(85,169,92,176)(86,182,93,175)(87,181,94,174)(88,180,95,173)(89,179,96,172)(90,178,97,171)(91,177,98,170)(99,168,106,161)(100,167,107,160)(101,166,108,159)(102,165,109,158)(103,164,110,157)(104,163,111,156)(105,162,112,155)(113,192,120,185)(114,191,121,184)(115,190,122,183)(116,189,123,196)(117,188,124,195)(118,187,125,194)(119,186,126,193)>;

G:=Group( (1,156,131,116,143,178,52)(2,157,132,117,144,179,53)(3,158,133,118,145,180,54)(4,159,134,119,146,181,55)(5,160,135,120,147,182,56)(6,161,136,121,148,169,43)(7,162,137,122,149,170,44)(8,163,138,123,150,171,45)(9,164,139,124,151,172,46)(10,165,140,125,152,173,47)(11,166,127,126,153,174,48)(12,167,128,113,154,175,49)(13,168,129,114,141,176,50)(14,155,130,115,142,177,51)(15,61,97,33,189,77,104)(16,62,98,34,190,78,105)(17,63,85,35,191,79,106)(18,64,86,36,192,80,107)(19,65,87,37,193,81,108)(20,66,88,38,194,82,109)(21,67,89,39,195,83,110)(22,68,90,40,196,84,111)(23,69,91,41,183,71,112)(24,70,92,42,184,72,99)(25,57,93,29,185,73,100)(26,58,94,30,186,74,101)(27,59,95,31,187,75,102)(28,60,96,32,188,76,103), (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)(183,184,185,186,187,188,189,190,191,192,193,194,195,196), (1,15,8,22)(2,28,9,21)(3,27,10,20)(4,26,11,19)(5,25,12,18)(6,24,13,17)(7,23,14,16)(29,154,36,147)(30,153,37,146)(31,152,38,145)(32,151,39,144)(33,150,40,143)(34,149,41,142)(35,148,42,141)(43,70,50,63)(44,69,51,62)(45,68,52,61)(46,67,53,60)(47,66,54,59)(48,65,55,58)(49,64,56,57)(71,130,78,137)(72,129,79,136)(73,128,80,135)(74,127,81,134)(75,140,82,133)(76,139,83,132)(77,138,84,131)(85,169,92,176)(86,182,93,175)(87,181,94,174)(88,180,95,173)(89,179,96,172)(90,178,97,171)(91,177,98,170)(99,168,106,161)(100,167,107,160)(101,166,108,159)(102,165,109,158)(103,164,110,157)(104,163,111,156)(105,162,112,155)(113,192,120,185)(114,191,121,184)(115,190,122,183)(116,189,123,196)(117,188,124,195)(118,187,125,194)(119,186,126,193) );

G=PermutationGroup([[(1,156,131,116,143,178,52),(2,157,132,117,144,179,53),(3,158,133,118,145,180,54),(4,159,134,119,146,181,55),(5,160,135,120,147,182,56),(6,161,136,121,148,169,43),(7,162,137,122,149,170,44),(8,163,138,123,150,171,45),(9,164,139,124,151,172,46),(10,165,140,125,152,173,47),(11,166,127,126,153,174,48),(12,167,128,113,154,175,49),(13,168,129,114,141,176,50),(14,155,130,115,142,177,51),(15,61,97,33,189,77,104),(16,62,98,34,190,78,105),(17,63,85,35,191,79,106),(18,64,86,36,192,80,107),(19,65,87,37,193,81,108),(20,66,88,38,194,82,109),(21,67,89,39,195,83,110),(22,68,90,40,196,84,111),(23,69,91,41,183,71,112),(24,70,92,42,184,72,99),(25,57,93,29,185,73,100),(26,58,94,30,186,74,101),(27,59,95,31,187,75,102),(28,60,96,32,188,76,103)], [(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),(183,184,185,186,187,188,189,190,191,192,193,194,195,196)], [(1,15,8,22),(2,28,9,21),(3,27,10,20),(4,26,11,19),(5,25,12,18),(6,24,13,17),(7,23,14,16),(29,154,36,147),(30,153,37,146),(31,152,38,145),(32,151,39,144),(33,150,40,143),(34,149,41,142),(35,148,42,141),(43,70,50,63),(44,69,51,62),(45,68,52,61),(46,67,53,60),(47,66,54,59),(48,65,55,58),(49,64,56,57),(71,130,78,137),(72,129,79,136),(73,128,80,135),(74,127,81,134),(75,140,82,133),(76,139,83,132),(77,138,84,131),(85,169,92,176),(86,182,93,175),(87,181,94,174),(88,180,95,173),(89,179,96,172),(90,178,97,171),(91,177,98,170),(99,168,106,161),(100,167,107,160),(101,166,108,159),(102,165,109,158),(103,164,110,157),(104,163,111,156),(105,162,112,155),(113,192,120,185),(114,191,121,184),(115,190,122,183),(116,189,123,196),(117,188,124,195),(118,187,125,194),(119,186,126,193)]])

C7⋊Dic7 is a maximal subgroup of   C722C8  D7×Dic7  C722D4  C722Q8  C724Q8  C4×C7⋊D7  C727D4
C7⋊Dic7 is a maximal quotient of   C724C8

52 conjugacy classes

 class 1 2 4A 4B 7A ··· 7X 14A ··· 14X order 1 2 4 4 7 ··· 7 14 ··· 14 size 1 1 49 49 2 ··· 2 2 ··· 2

52 irreducible representations

 dim 1 1 1 2 2 type + + + - image C1 C2 C4 D7 Dic7 kernel C7⋊Dic7 C7×C14 C72 C14 C7 # reps 1 1 2 24 24

Matrix representation of C7⋊Dic7 in GL4(𝔽29) generated by

 1 0 0 0 0 1 0 0 0 0 28 1 0 0 24 4
,
 0 28 0 0 1 26 0 0 0 0 18 3 0 0 14 4
,
 12 0 0 0 7 17 0 0 0 0 7 24 0 0 10 22
G:=sub<GL(4,GF(29))| [1,0,0,0,0,1,0,0,0,0,28,24,0,0,1,4],[0,1,0,0,28,26,0,0,0,0,18,14,0,0,3,4],[12,7,0,0,0,17,0,0,0,0,7,10,0,0,24,22] >;

C7⋊Dic7 in GAP, Magma, Sage, TeX

C_7\rtimes {\rm Dic}_7
% in TeX

G:=Group("C7:Dic7");
// GroupNames label

G:=SmallGroup(196,6);
// by ID

G=gap.SmallGroup(196,6);
# by ID

G:=PCGroup([4,-2,-2,-7,-7,8,290,2691]);
// Polycyclic

G:=Group<a,b,c|a^7=b^14=1,c^2=b^7,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=b^-1>;
// generators/relations

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