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G = C7×D4○C16order 448 = 26·7

Direct product of C7 and D4○C16

direct product, metabelian, nilpotent (class 2), monomial, 2-elementary

Aliases: C7×D4○C16, D4.2C56, Q8.2C56, M5(2)⋊7C14, C56.83C23, C112.29C22, M4(2).4C28, (C2×C16)⋊9C14, C4.5(C2×C56), (C7×D4).4C8, (C2×C112)⋊19C2, (C7×Q8).4C8, C56.68(C2×C4), C28.34(C2×C8), C8.12(C2×C28), C16.8(C2×C14), C8○D4.3C14, C4○D4.3C28, C22.1(C2×C56), C2.7(C22×C56), (C7×M5(2))⋊15C2, C8.16(C22×C14), C4.36(C22×C28), C14.36(C22×C8), (C7×M4(2)).8C4, C28.194(C22×C4), (C2×C56).449C22, (C2×C14).8(C2×C8), (C7×C4○D4).7C4, (C7×C8○D4).6C2, (C2×C4).51(C2×C28), (C2×C28).272(C2×C4), (C2×C8).103(C2×C14), SmallGroup(448,912)

Series: Derived Chief Lower central Upper central

C1C2 — C7×D4○C16
C1C2C4C8C56C112C2×C112 — C7×D4○C16
C1C2 — C7×D4○C16
C1C112 — C7×D4○C16

Generators and relations for C7×D4○C16
 G = < a,b,c,d | a7=b4=c2=1, d8=b2, ab=ba, ac=ca, ad=da, cbc=b-1, bd=db, cd=dc >

Subgroups: 90 in 84 conjugacy classes, 78 normal (18 characteristic)
C1, C2, C2 [×3], C4, C4 [×3], C22 [×3], C7, C8, C8 [×3], C2×C4 [×3], D4 [×3], Q8, C14, C14 [×3], C16, C16 [×3], C2×C8 [×3], M4(2) [×3], C4○D4, C28, C28 [×3], C2×C14 [×3], C2×C16 [×3], M5(2) [×3], C8○D4, C56, C56 [×3], C2×C28 [×3], C7×D4 [×3], C7×Q8, D4○C16, C112, C112 [×3], C2×C56 [×3], C7×M4(2) [×3], C7×C4○D4, C2×C112 [×3], C7×M5(2) [×3], C7×C8○D4, C7×D4○C16
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C7, C8 [×4], C2×C4 [×6], C23, C14 [×7], C2×C8 [×6], C22×C4, C28 [×4], C2×C14 [×7], C22×C8, C56 [×4], C2×C28 [×6], C22×C14, D4○C16, C2×C56 [×6], C22×C28, C22×C56, C7×D4○C16

Smallest permutation representation of C7×D4○C16
On 224 points
Generators in S224
(1 101 143 203 18 217 149)(2 102 144 204 19 218 150)(3 103 129 205 20 219 151)(4 104 130 206 21 220 152)(5 105 131 207 22 221 153)(6 106 132 208 23 222 154)(7 107 133 193 24 223 155)(8 108 134 194 25 224 156)(9 109 135 195 26 209 157)(10 110 136 196 27 210 158)(11 111 137 197 28 211 159)(12 112 138 198 29 212 160)(13 97 139 199 30 213 145)(14 98 140 200 31 214 146)(15 99 141 201 32 215 147)(16 100 142 202 17 216 148)(33 78 183 113 174 60 94)(34 79 184 114 175 61 95)(35 80 185 115 176 62 96)(36 65 186 116 161 63 81)(37 66 187 117 162 64 82)(38 67 188 118 163 49 83)(39 68 189 119 164 50 84)(40 69 190 120 165 51 85)(41 70 191 121 166 52 86)(42 71 192 122 167 53 87)(43 72 177 123 168 54 88)(44 73 178 124 169 55 89)(45 74 179 125 170 56 90)(46 75 180 126 171 57 91)(47 76 181 127 172 58 92)(48 77 182 128 173 59 93)
(1 80 9 72)(2 65 10 73)(3 66 11 74)(4 67 12 75)(5 68 13 76)(6 69 14 77)(7 70 15 78)(8 71 16 79)(17 61 25 53)(18 62 26 54)(19 63 27 55)(20 64 28 56)(21 49 29 57)(22 50 30 58)(23 51 31 59)(24 52 32 60)(33 155 41 147)(34 156 42 148)(35 157 43 149)(36 158 44 150)(37 159 45 151)(38 160 46 152)(39 145 47 153)(40 146 48 154)(81 210 89 218)(82 211 90 219)(83 212 91 220)(84 213 92 221)(85 214 93 222)(86 215 94 223)(87 216 95 224)(88 217 96 209)(97 181 105 189)(98 182 106 190)(99 183 107 191)(100 184 108 192)(101 185 109 177)(102 186 110 178)(103 187 111 179)(104 188 112 180)(113 133 121 141)(114 134 122 142)(115 135 123 143)(116 136 124 144)(117 137 125 129)(118 138 126 130)(119 139 127 131)(120 140 128 132)(161 196 169 204)(162 197 170 205)(163 198 171 206)(164 199 172 207)(165 200 173 208)(166 201 174 193)(167 202 175 194)(168 203 176 195)
(33 41)(34 42)(35 43)(36 44)(37 45)(38 46)(39 47)(40 48)(49 57)(50 58)(51 59)(52 60)(53 61)(54 62)(55 63)(56 64)(65 73)(66 74)(67 75)(68 76)(69 77)(70 78)(71 79)(72 80)(81 89)(82 90)(83 91)(84 92)(85 93)(86 94)(87 95)(88 96)(113 121)(114 122)(115 123)(116 124)(117 125)(118 126)(119 127)(120 128)(161 169)(162 170)(163 171)(164 172)(165 173)(166 174)(167 175)(168 176)(177 185)(178 186)(179 187)(180 188)(181 189)(182 190)(183 191)(184 192)
(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)

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

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

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

280 conjugacy classes

class 1 2A2B2C2D4A4B4C4D4E7A···7F8A8B8C8D8E···8J14A···14F14G···14X16A···16H16I···16T28A···28L28M···28AD56A···56X56Y···56BH112A···112AV112AW···112DP
order12222444447···788888···814···1414···1416···1616···1628···2828···2856···5656···56112···112112···112
size11222112221···111112···21···12···21···12···21···12···21···12···21···12···2

280 irreducible representations

dim111111111111111122
type++++
imageC1C2C2C2C4C4C7C8C8C14C14C14C28C28C56C56D4○C16C7×D4○C16
kernelC7×D4○C16C2×C112C7×M5(2)C7×C8○D4C7×M4(2)C7×C4○D4D4○C16C7×D4C7×Q8C2×C16M5(2)C8○D4M4(2)C4○D4D4Q8C7C1
# reps13316261241818636127224848

Matrix representation of C7×D4○C16 in GL2(𝔽113) generated by

280
028
,
1082
1005
,
10
5112
,
780
078
G:=sub<GL(2,GF(113))| [28,0,0,28],[108,100,2,5],[1,5,0,112],[78,0,0,78] >;

C7×D4○C16 in GAP, Magma, Sage, TeX

C_7\times D_4\circ C_{16}
% in TeX

G:=Group("C7xD4oC16");
// GroupNames label

G:=SmallGroup(448,912);
// by ID

G=gap.SmallGroup(448,912);
# by ID

G:=PCGroup([7,-2,-2,-2,-7,-2,-2,-2,392,2403,102,124]);
// Polycyclic

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

׿
×
𝔽