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## G = (C2×C14)⋊8Q16order 448 = 26·7

### 2nd semidirect product of C2×C14 and Q16 acting via Q16/Q8=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C28 — (C2×C14)⋊8Q16
 Chief series C1 — C7 — C14 — C28 — C2×C28 — C2×Dic14 — C28.48D4 — (C2×C14)⋊8Q16
 Lower central C7 — C14 — C2×C28 — (C2×C14)⋊8Q16
 Upper central C1 — C22 — C22×C4 — C22×Q8

Generators and relations for (C2×C14)⋊8Q16
G = < a,b,c,d | a2=b14=c8=1, d2=c4, ab=ba, cac-1=ab7, ad=da, cbc-1=b-1, bd=db, dcd-1=c-1 >

Subgroups: 500 in 148 conjugacy classes, 51 normal (25 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C7, C8, C2×C4, C2×C4, Q8, Q8, C23, C14, C14, C22⋊C4, C4⋊C4, C2×C8, Q16, C22×C4, C22×C4, C2×Q8, C2×Q8, Dic7, C28, C28, C2×C14, C2×C14, C2×C14, C22⋊C8, Q8⋊C4, C22⋊Q8, C2×Q16, C22×Q8, C7⋊C8, Dic14, C2×Dic7, C2×C28, C2×C28, C7×Q8, C7×Q8, C22×C14, C22⋊Q16, C2×C7⋊C8, Dic7⋊C4, C4⋊Dic7, C7⋊Q16, C23.D7, C2×Dic14, C22×C28, C22×C28, Q8×C14, Q8×C14, C28.55D4, Q8⋊Dic7, C28.48D4, C2×C7⋊Q16, Q8×C2×C14, (C2×C14)⋊8Q16
Quotients: C1, C2, C22, D4, C23, D7, Q16, C2×D4, D14, C22≀C2, C2×Q16, C8.C22, C7⋊D4, C22×D7, C22⋊Q16, C7⋊Q16, C2×C7⋊D4, C28.C23, C2×C7⋊Q16, C24⋊D7, (C2×C14)⋊8Q16

Smallest permutation representation of (C2×C14)⋊8Q16
On 224 points
Generators in S224
(29 36)(30 37)(31 38)(32 39)(33 40)(34 41)(35 42)(57 64)(58 65)(59 66)(60 67)(61 68)(62 69)(63 70)(71 78)(72 79)(73 80)(74 81)(75 82)(76 83)(77 84)(99 106)(100 107)(101 108)(102 109)(103 110)(104 111)(105 112)(113 120)(114 121)(115 122)(116 123)(117 124)(118 125)(119 126)(183 190)(184 191)(185 192)(186 193)(187 194)(188 195)(189 196)(197 204)(198 205)(199 206)(200 207)(201 208)(202 209)(203 210)(211 218)(212 219)(213 220)(214 221)(215 222)(216 223)(217 224)
(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)
(1 220 22 68 48 204 91 101)(2 219 23 67 49 203 92 100)(3 218 24 66 50 202 93 99)(4 217 25 65 51 201 94 112)(5 216 26 64 52 200 95 111)(6 215 27 63 53 199 96 110)(7 214 28 62 54 198 97 109)(8 213 15 61 55 197 98 108)(9 212 16 60 56 210 85 107)(10 211 17 59 43 209 86 106)(11 224 18 58 44 208 87 105)(12 223 19 57 45 207 88 104)(13 222 20 70 46 206 89 103)(14 221 21 69 47 205 90 102)(29 180 75 159 186 136 126 146)(30 179 76 158 187 135 113 145)(31 178 77 157 188 134 114 144)(32 177 78 156 189 133 115 143)(33 176 79 155 190 132 116 142)(34 175 80 168 191 131 117 141)(35 174 81 167 192 130 118 154)(36 173 82 166 193 129 119 153)(37 172 83 165 194 128 120 152)(38 171 84 164 195 127 121 151)(39 170 71 163 196 140 122 150)(40 169 72 162 183 139 123 149)(41 182 73 161 184 138 124 148)(42 181 74 160 185 137 125 147)
(1 164 48 151)(2 165 49 152)(3 166 50 153)(4 167 51 154)(5 168 52 141)(6 155 53 142)(7 156 54 143)(8 157 55 144)(9 158 56 145)(10 159 43 146)(11 160 44 147)(12 161 45 148)(13 162 46 149)(14 163 47 150)(15 178 98 134)(16 179 85 135)(17 180 86 136)(18 181 87 137)(19 182 88 138)(20 169 89 139)(21 170 90 140)(22 171 91 127)(23 172 92 128)(24 173 93 129)(25 174 94 130)(26 175 95 131)(27 176 96 132)(28 177 97 133)(29 106 186 59)(30 107 187 60)(31 108 188 61)(32 109 189 62)(33 110 190 63)(34 111 191 64)(35 112 192 65)(36 99 193 66)(37 100 194 67)(38 101 195 68)(39 102 196 69)(40 103 183 70)(41 104 184 57)(42 105 185 58)(71 205 122 221)(72 206 123 222)(73 207 124 223)(74 208 125 224)(75 209 126 211)(76 210 113 212)(77 197 114 213)(78 198 115 214)(79 199 116 215)(80 200 117 216)(81 201 118 217)(82 202 119 218)(83 203 120 219)(84 204 121 220)

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

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

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

79 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C ··· 4G 4H 4I 7A 7B 7C 8A 8B 8C 8D 14A ··· 14U 28A ··· 28AJ order 1 2 2 2 2 2 4 4 4 ··· 4 4 4 7 7 7 8 8 8 8 14 ··· 14 28 ··· 28 size 1 1 1 1 2 2 2 2 4 ··· 4 56 56 2 2 2 28 28 28 28 2 ··· 2 4 ··· 4

79 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + - + + - - image C1 C2 C2 C2 C2 C2 D4 D4 D4 D7 Q16 D14 D14 C7⋊D4 C7⋊D4 C7⋊D4 C8.C22 C7⋊Q16 C28.C23 kernel (C2×C14)⋊8Q16 C28.55D4 Q8⋊Dic7 C28.48D4 C2×C7⋊Q16 Q8×C2×C14 C2×C28 C7×Q8 C22×C14 C22×Q8 C2×C14 C22×C4 C2×Q8 C2×C4 Q8 C23 C14 C22 C2 # reps 1 1 2 1 2 1 1 4 1 3 4 3 6 6 24 6 1 6 6

Matrix representation of (C2×C14)⋊8Q16 in GL4(𝔽113) generated by

 1 0 0 0 0 112 0 0 0 0 1 0 0 0 0 1
,
 85 0 0 0 0 4 0 0 0 0 1 0 0 0 0 1
,
 0 76 0 0 58 0 0 0 0 0 62 77 0 0 22 0
,
 112 0 0 0 0 112 0 0 0 0 66 35 0 0 66 47
G:=sub<GL(4,GF(113))| [1,0,0,0,0,112,0,0,0,0,1,0,0,0,0,1],[85,0,0,0,0,4,0,0,0,0,1,0,0,0,0,1],[0,58,0,0,76,0,0,0,0,0,62,22,0,0,77,0],[112,0,0,0,0,112,0,0,0,0,66,66,0,0,35,47] >;

(C2×C14)⋊8Q16 in GAP, Magma, Sage, TeX

(C_2\times C_{14})\rtimes_8Q_{16}
% in TeX

G:=Group("(C2xC14):8Q16");
// GroupNames label

G:=SmallGroup(448,762);
// by ID

G=gap.SmallGroup(448,762);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,224,253,254,184,1123,297,136,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^14=c^8=1,d^2=c^4,a*b=b*a,c*a*c^-1=a*b^7,a*d=d*a,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

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