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## G = C7⋊SD64order 448 = 26·7

### The semidirect product of C7 and SD64 acting via SD64/Q32=C2

Aliases: C73SD64, Q321D7, C28.7D8, C56.11D4, C16.6D14, D112.2C2, C14.10D16, C112.4C22, C7⋊C323C2, (C7×Q32)⋊1C2, C4.3(D4⋊D7), C8.11(C7⋊D4), C2.6(C7⋊D16), SmallGroup(448,78)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C112 — C7⋊SD64
 Chief series C1 — C7 — C14 — C28 — C56 — C112 — D112 — C7⋊SD64
 Lower central C7 — C14 — C28 — C56 — C112 — C7⋊SD64
 Upper central C1 — C2 — C4 — C8 — C16 — Q32

Generators and relations for C7⋊SD64
G = < a,b,c | a7=b32=c2=1, bab-1=cac=a-1, cbc=b15 >

Smallest permutation representation of C7⋊SD64
On 224 points
Generators in S224
(1 109 218 43 81 178 137)(2 138 179 82 44 219 110)(3 111 220 45 83 180 139)(4 140 181 84 46 221 112)(5 113 222 47 85 182 141)(6 142 183 86 48 223 114)(7 115 224 49 87 184 143)(8 144 185 88 50 193 116)(9 117 194 51 89 186 145)(10 146 187 90 52 195 118)(11 119 196 53 91 188 147)(12 148 189 92 54 197 120)(13 121 198 55 93 190 149)(14 150 191 94 56 199 122)(15 123 200 57 95 192 151)(16 152 161 96 58 201 124)(17 125 202 59 65 162 153)(18 154 163 66 60 203 126)(19 127 204 61 67 164 155)(20 156 165 68 62 205 128)(21 97 206 63 69 166 157)(22 158 167 70 64 207 98)(23 99 208 33 71 168 159)(24 160 169 72 34 209 100)(25 101 210 35 73 170 129)(26 130 171 74 36 211 102)(27 103 212 37 75 172 131)(28 132 173 76 38 213 104)(29 105 214 39 77 174 133)(30 134 175 78 40 215 106)(31 107 216 41 79 176 135)(32 136 177 80 42 217 108)
(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)(193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224)
(2 16)(3 31)(4 14)(5 29)(6 12)(7 27)(8 10)(9 25)(11 23)(13 21)(15 19)(18 32)(20 30)(22 28)(24 26)(33 91)(34 74)(35 89)(36 72)(37 87)(38 70)(39 85)(40 68)(41 83)(42 66)(43 81)(44 96)(45 79)(46 94)(47 77)(48 92)(49 75)(50 90)(51 73)(52 88)(53 71)(54 86)(55 69)(56 84)(57 67)(58 82)(59 65)(60 80)(61 95)(62 78)(63 93)(64 76)(97 149)(98 132)(99 147)(100 130)(101 145)(102 160)(103 143)(104 158)(105 141)(106 156)(107 139)(108 154)(109 137)(110 152)(111 135)(112 150)(113 133)(114 148)(115 131)(116 146)(117 129)(118 144)(119 159)(120 142)(121 157)(122 140)(123 155)(124 138)(125 153)(126 136)(127 151)(128 134)(161 219)(162 202)(163 217)(164 200)(165 215)(166 198)(167 213)(168 196)(169 211)(170 194)(171 209)(172 224)(173 207)(174 222)(175 205)(176 220)(177 203)(178 218)(179 201)(180 216)(181 199)(182 214)(183 197)(184 212)(185 195)(186 210)(187 193)(188 208)(189 223)(190 206)(191 221)(192 204)

G:=sub<Sym(224)| (1,109,218,43,81,178,137)(2,138,179,82,44,219,110)(3,111,220,45,83,180,139)(4,140,181,84,46,221,112)(5,113,222,47,85,182,141)(6,142,183,86,48,223,114)(7,115,224,49,87,184,143)(8,144,185,88,50,193,116)(9,117,194,51,89,186,145)(10,146,187,90,52,195,118)(11,119,196,53,91,188,147)(12,148,189,92,54,197,120)(13,121,198,55,93,190,149)(14,150,191,94,56,199,122)(15,123,200,57,95,192,151)(16,152,161,96,58,201,124)(17,125,202,59,65,162,153)(18,154,163,66,60,203,126)(19,127,204,61,67,164,155)(20,156,165,68,62,205,128)(21,97,206,63,69,166,157)(22,158,167,70,64,207,98)(23,99,208,33,71,168,159)(24,160,169,72,34,209,100)(25,101,210,35,73,170,129)(26,130,171,74,36,211,102)(27,103,212,37,75,172,131)(28,132,173,76,38,213,104)(29,105,214,39,77,174,133)(30,134,175,78,40,215,106)(31,107,216,41,79,176,135)(32,136,177,80,42,217,108), (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)(193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224), (2,16)(3,31)(4,14)(5,29)(6,12)(7,27)(8,10)(9,25)(11,23)(13,21)(15,19)(18,32)(20,30)(22,28)(24,26)(33,91)(34,74)(35,89)(36,72)(37,87)(38,70)(39,85)(40,68)(41,83)(42,66)(43,81)(44,96)(45,79)(46,94)(47,77)(48,92)(49,75)(50,90)(51,73)(52,88)(53,71)(54,86)(55,69)(56,84)(57,67)(58,82)(59,65)(60,80)(61,95)(62,78)(63,93)(64,76)(97,149)(98,132)(99,147)(100,130)(101,145)(102,160)(103,143)(104,158)(105,141)(106,156)(107,139)(108,154)(109,137)(110,152)(111,135)(112,150)(113,133)(114,148)(115,131)(116,146)(117,129)(118,144)(119,159)(120,142)(121,157)(122,140)(123,155)(124,138)(125,153)(126,136)(127,151)(128,134)(161,219)(162,202)(163,217)(164,200)(165,215)(166,198)(167,213)(168,196)(169,211)(170,194)(171,209)(172,224)(173,207)(174,222)(175,205)(176,220)(177,203)(178,218)(179,201)(180,216)(181,199)(182,214)(183,197)(184,212)(185,195)(186,210)(187,193)(188,208)(189,223)(190,206)(191,221)(192,204)>;

G:=Group( (1,109,218,43,81,178,137)(2,138,179,82,44,219,110)(3,111,220,45,83,180,139)(4,140,181,84,46,221,112)(5,113,222,47,85,182,141)(6,142,183,86,48,223,114)(7,115,224,49,87,184,143)(8,144,185,88,50,193,116)(9,117,194,51,89,186,145)(10,146,187,90,52,195,118)(11,119,196,53,91,188,147)(12,148,189,92,54,197,120)(13,121,198,55,93,190,149)(14,150,191,94,56,199,122)(15,123,200,57,95,192,151)(16,152,161,96,58,201,124)(17,125,202,59,65,162,153)(18,154,163,66,60,203,126)(19,127,204,61,67,164,155)(20,156,165,68,62,205,128)(21,97,206,63,69,166,157)(22,158,167,70,64,207,98)(23,99,208,33,71,168,159)(24,160,169,72,34,209,100)(25,101,210,35,73,170,129)(26,130,171,74,36,211,102)(27,103,212,37,75,172,131)(28,132,173,76,38,213,104)(29,105,214,39,77,174,133)(30,134,175,78,40,215,106)(31,107,216,41,79,176,135)(32,136,177,80,42,217,108), (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)(193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224), (2,16)(3,31)(4,14)(5,29)(6,12)(7,27)(8,10)(9,25)(11,23)(13,21)(15,19)(18,32)(20,30)(22,28)(24,26)(33,91)(34,74)(35,89)(36,72)(37,87)(38,70)(39,85)(40,68)(41,83)(42,66)(43,81)(44,96)(45,79)(46,94)(47,77)(48,92)(49,75)(50,90)(51,73)(52,88)(53,71)(54,86)(55,69)(56,84)(57,67)(58,82)(59,65)(60,80)(61,95)(62,78)(63,93)(64,76)(97,149)(98,132)(99,147)(100,130)(101,145)(102,160)(103,143)(104,158)(105,141)(106,156)(107,139)(108,154)(109,137)(110,152)(111,135)(112,150)(113,133)(114,148)(115,131)(116,146)(117,129)(118,144)(119,159)(120,142)(121,157)(122,140)(123,155)(124,138)(125,153)(126,136)(127,151)(128,134)(161,219)(162,202)(163,217)(164,200)(165,215)(166,198)(167,213)(168,196)(169,211)(170,194)(171,209)(172,224)(173,207)(174,222)(175,205)(176,220)(177,203)(178,218)(179,201)(180,216)(181,199)(182,214)(183,197)(184,212)(185,195)(186,210)(187,193)(188,208)(189,223)(190,206)(191,221)(192,204) );

G=PermutationGroup([[(1,109,218,43,81,178,137),(2,138,179,82,44,219,110),(3,111,220,45,83,180,139),(4,140,181,84,46,221,112),(5,113,222,47,85,182,141),(6,142,183,86,48,223,114),(7,115,224,49,87,184,143),(8,144,185,88,50,193,116),(9,117,194,51,89,186,145),(10,146,187,90,52,195,118),(11,119,196,53,91,188,147),(12,148,189,92,54,197,120),(13,121,198,55,93,190,149),(14,150,191,94,56,199,122),(15,123,200,57,95,192,151),(16,152,161,96,58,201,124),(17,125,202,59,65,162,153),(18,154,163,66,60,203,126),(19,127,204,61,67,164,155),(20,156,165,68,62,205,128),(21,97,206,63,69,166,157),(22,158,167,70,64,207,98),(23,99,208,33,71,168,159),(24,160,169,72,34,209,100),(25,101,210,35,73,170,129),(26,130,171,74,36,211,102),(27,103,212,37,75,172,131),(28,132,173,76,38,213,104),(29,105,214,39,77,174,133),(30,134,175,78,40,215,106),(31,107,216,41,79,176,135),(32,136,177,80,42,217,108)], [(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),(193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224)], [(2,16),(3,31),(4,14),(5,29),(6,12),(7,27),(8,10),(9,25),(11,23),(13,21),(15,19),(18,32),(20,30),(22,28),(24,26),(33,91),(34,74),(35,89),(36,72),(37,87),(38,70),(39,85),(40,68),(41,83),(42,66),(43,81),(44,96),(45,79),(46,94),(47,77),(48,92),(49,75),(50,90),(51,73),(52,88),(53,71),(54,86),(55,69),(56,84),(57,67),(58,82),(59,65),(60,80),(61,95),(62,78),(63,93),(64,76),(97,149),(98,132),(99,147),(100,130),(101,145),(102,160),(103,143),(104,158),(105,141),(106,156),(107,139),(108,154),(109,137),(110,152),(111,135),(112,150),(113,133),(114,148),(115,131),(116,146),(117,129),(118,144),(119,159),(120,142),(121,157),(122,140),(123,155),(124,138),(125,153),(126,136),(127,151),(128,134),(161,219),(162,202),(163,217),(164,200),(165,215),(166,198),(167,213),(168,196),(169,211),(170,194),(171,209),(172,224),(173,207),(174,222),(175,205),(176,220),(177,203),(178,218),(179,201),(180,216),(181,199),(182,214),(183,197),(184,212),(185,195),(186,210),(187,193),(188,208),(189,223),(190,206),(191,221),(192,204)]])

52 conjugacy classes

 class 1 2A 2B 4A 4B 7A 7B 7C 8A 8B 14A 14B 14C 16A 16B 16C 16D 28A 28B 28C 28D ··· 28I 32A ··· 32H 56A ··· 56F 112A ··· 112L order 1 2 2 4 4 7 7 7 8 8 14 14 14 16 16 16 16 28 28 28 28 ··· 28 32 ··· 32 56 ··· 56 112 ··· 112 size 1 1 112 2 16 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 16 ··· 16 14 ··· 14 4 ··· 4 4 ··· 4

52 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + image C1 C2 C2 C2 D4 D7 D8 D14 D16 C7⋊D4 SD64 D4⋊D7 C7⋊D16 C7⋊SD64 kernel C7⋊SD64 C7⋊C32 D112 C7×Q32 C56 Q32 C28 C16 C14 C8 C7 C4 C2 C1 # reps 1 1 1 1 1 3 2 3 4 6 8 3 6 12

Matrix representation of C7⋊SD64 in GL4(𝔽449) generated by

 448 1 0 0 396 52 0 0 0 0 1 0 0 0 0 1
,
 52 448 0 0 9 397 0 0 0 0 29 184 0 0 320 64
,
 397 1 0 0 440 52 0 0 0 0 1 0 0 0 110 448
G:=sub<GL(4,GF(449))| [448,396,0,0,1,52,0,0,0,0,1,0,0,0,0,1],[52,9,0,0,448,397,0,0,0,0,29,320,0,0,184,64],[397,440,0,0,1,52,0,0,0,0,1,110,0,0,0,448] >;

C7⋊SD64 in GAP, Magma, Sage, TeX

C_7\rtimes {\rm SD}_{64}
% in TeX

G:=Group("C7:SD64");
// GroupNames label

G:=SmallGroup(448,78);
// by ID

G=gap.SmallGroup(448,78);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,85,232,254,135,142,675,346,192,1684,851,102,18822]);
// Polycyclic

G:=Group<a,b,c|a^7=b^32=c^2=1,b*a*b^-1=c*a*c=a^-1,c*b*c=b^15>;
// generators/relations

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