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G = C8⋊Dic7order 224 = 25·7

2nd semidirect product of C8 and Dic7 acting via Dic7/C14=C2

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: C562C4, C82Dic7, C28.4Q8, C14.2SD16, C4.4Dic14, C22.8D28, (C2×C8).6D7, C72(C4.Q8), (C2×C56).8C2, C14.5(C4⋊C4), C28.33(C2×C4), (C2×C14).13D4, (C2×C4).68D14, C4⋊Dic7.2C2, C4.6(C2×Dic7), C2.2(C56⋊C2), C2.3(C4⋊Dic7), (C2×C28).81C22, SmallGroup(224,23)

Series: Derived Chief Lower central Upper central

C1C28 — C8⋊Dic7
C1C7C14C2×C14C2×C28C4⋊Dic7 — C8⋊Dic7
C7C14C28 — C8⋊Dic7
C1C22C2×C4C2×C8

Generators and relations for C8⋊Dic7
 G = < a,b,c | a8=b14=1, c2=b7, ab=ba, cac-1=a3, cbc-1=b-1 >

28C4
28C4
14C2×C4
14C2×C4
4Dic7
4Dic7
7C4⋊C4
7C4⋊C4
2C2×Dic7
2C2×Dic7
7C4.Q8

Smallest permutation representation of C8⋊Dic7
Regular action on 224 points
Generators in S224
(1 181 34 27 74 150 221 189)(2 182 35 28 75 151 222 190)(3 169 36 15 76 152 223 191)(4 170 37 16 77 153 224 192)(5 171 38 17 78 154 211 193)(6 172 39 18 79 141 212 194)(7 173 40 19 80 142 213 195)(8 174 41 20 81 143 214 196)(9 175 42 21 82 144 215 183)(10 176 29 22 83 145 216 184)(11 177 30 23 84 146 217 185)(12 178 31 24 71 147 218 186)(13 179 32 25 72 148 219 187)(14 180 33 26 73 149 220 188)(43 208 58 136 85 160 112 126)(44 209 59 137 86 161 99 113)(45 210 60 138 87 162 100 114)(46 197 61 139 88 163 101 115)(47 198 62 140 89 164 102 116)(48 199 63 127 90 165 103 117)(49 200 64 128 91 166 104 118)(50 201 65 129 92 167 105 119)(51 202 66 130 93 168 106 120)(52 203 67 131 94 155 107 121)(53 204 68 132 95 156 108 122)(54 205 69 133 96 157 109 123)(55 206 70 134 97 158 110 124)(56 207 57 135 98 159 111 125)
(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)
(1 121 8 114)(2 120 9 113)(3 119 10 126)(4 118 11 125)(5 117 12 124)(6 116 13 123)(7 115 14 122)(15 50 22 43)(16 49 23 56)(17 48 24 55)(18 47 25 54)(19 46 26 53)(20 45 27 52)(21 44 28 51)(29 160 36 167)(30 159 37 166)(31 158 38 165)(32 157 39 164)(33 156 40 163)(34 155 41 162)(35 168 42 161)(57 170 64 177)(58 169 65 176)(59 182 66 175)(60 181 67 174)(61 180 68 173)(62 179 69 172)(63 178 70 171)(71 134 78 127)(72 133 79 140)(73 132 80 139)(74 131 81 138)(75 130 82 137)(76 129 83 136)(77 128 84 135)(85 191 92 184)(86 190 93 183)(87 189 94 196)(88 188 95 195)(89 187 96 194)(90 186 97 193)(91 185 98 192)(99 151 106 144)(100 150 107 143)(101 149 108 142)(102 148 109 141)(103 147 110 154)(104 146 111 153)(105 145 112 152)(197 220 204 213)(198 219 205 212)(199 218 206 211)(200 217 207 224)(201 216 208 223)(202 215 209 222)(203 214 210 221)

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

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

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

C8⋊Dic7 is a maximal subgroup of
C8.Dic14  C16⋊Dic7  D562C4  D82Dic7  C569Q8  C56.13Q8  C4×C56⋊C2  C8⋊Dic14  D56⋊C4  Dic28⋊C4  C23.34D28  C23.10D28  C23.38D28  C23.13D28  D4⋊Dic14  D4.Dic14  D14.SD16  C8⋊Dic7⋊C2  Q8⋊Dic14  Q8.2Dic14  D14.1SD16  (C2×C8).D14  Dic14.3Q8  D283Q8  D28.3Q8  Dic144Q8  C565Q8  C56.8Q8  D7×C4.Q8  (C8×D7)⋊C4  C564Q8  C56⋊(C2×C4)  C23.22D28  C5630D4  C23.47D28  C563D4  C56.4D4  D8⋊Dic7  C5612D4  SD16×Dic7  C5614D4  Q16⋊Dic7  C56.36D4
C8⋊Dic7 is a maximal quotient of
C562C8  C16⋊Dic7  C28.9C42

62 conjugacy classes

class 1 2A2B2C4A4B4C4D4E4F7A7B7C8A8B8C8D14A···14I28A···28L56A···56X
order1222444444777888814···1428···2856···56
size1111222828282822222222···22···22···2

62 irreducible representations

dim1111222222222
type+++-++-+-+
imageC1C2C2C4Q8D4D7SD16Dic7D14Dic14D28C56⋊C2
kernelC8⋊Dic7C4⋊Dic7C2×C56C56C28C2×C14C2×C8C14C8C2×C4C4C22C2
# reps12141134636624

Matrix representation of C8⋊Dic7 in GL5(𝔽113)

1120000
018000
006900
00098
00018104
,
1120000
01000
00100
0001121
0008725
,
150000
00100
01000
0001249
0004101

G:=sub<GL(5,GF(113))| [112,0,0,0,0,0,18,0,0,0,0,0,69,0,0,0,0,0,9,18,0,0,0,8,104],[112,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,112,87,0,0,0,1,25],[15,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,12,4,0,0,0,49,101] >;

C8⋊Dic7 in GAP, Magma, Sage, TeX

C_8\rtimes {\rm Dic}_7
% in TeX

G:=Group("C8:Dic7");
// GroupNames label

G:=SmallGroup(224,23);
// by ID

G=gap.SmallGroup(224,23);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-7,24,121,55,579,69,6917]);
// Polycyclic

G:=Group<a,b,c|a^8=b^14=1,c^2=b^7,a*b=b*a,c*a*c^-1=a^3,c*b*c^-1=b^-1>;
// generators/relations

Export

Subgroup lattice of C8⋊Dic7 in TeX

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