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G = D28.C8order 448 = 26·7

1st non-split extension by D28 of C8 acting via C8/C4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D28.1C8, C8.31D28, C56.66D4, Dic14.1C8, (C2×C16)⋊2D7, C4.8(C8×D7), (C2×C112)⋊2C2, C71(D4.C8), C28.18(C2×C8), C4○D28.1C4, C28.C89C2, C2.9(D14⋊C8), (C2×C8).322D14, C8.45(C7⋊D4), C4.Dic7.1C4, C4.41(D14⋊C4), C14.8(C22⋊C8), (C2×C14).9M4(2), C28.56(C22⋊C4), (C2×C56).402C22, D28.2C4.3C2, C22.2(C8⋊D7), (C2×C4).97(C4×D7), (C2×C28).231(C2×C4), SmallGroup(448,65)

Series: Derived Chief Lower central Upper central

C1C28 — D28.C8
C1C7C14C28C56C2×C56D28.2C4 — D28.C8
C7C14C28 — D28.C8
C1C8C2×C8C2×C16

Generators and relations for D28.C8
 G = < a,b,c | a28=b2=1, c8=a14, bab=a-1, ac=ca, cbc-1=a7b >

2C2
28C2
14C22
14C4
2C14
4D7
7D4
7Q8
14C8
14D4
14C2×C4
2Dic7
2D14
2C16
7M4(2)
7C4○D4
14C16
14M4(2)
14C2×C8
2C7⋊D4
2C4×D7
2C7⋊C8
7M5(2)
7C8○D4
2C112
2C8⋊D7
2C8×D7
2C7⋊C16
7D4.C8

Smallest permutation representation of D28.C8
On 224 points
Generators in S224
(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)
(1 137)(2 136)(3 135)(4 134)(5 133)(6 132)(7 131)(8 130)(9 129)(10 128)(11 127)(12 126)(13 125)(14 124)(15 123)(16 122)(17 121)(18 120)(19 119)(20 118)(21 117)(22 116)(23 115)(24 114)(25 113)(26 140)(27 139)(28 138)(29 145)(30 144)(31 143)(32 142)(33 141)(34 168)(35 167)(36 166)(37 165)(38 164)(39 163)(40 162)(41 161)(42 160)(43 159)(44 158)(45 157)(46 156)(47 155)(48 154)(49 153)(50 152)(51 151)(52 150)(53 149)(54 148)(55 147)(56 146)(57 178)(58 177)(59 176)(60 175)(61 174)(62 173)(63 172)(64 171)(65 170)(66 169)(67 196)(68 195)(69 194)(70 193)(71 192)(72 191)(73 190)(74 189)(75 188)(76 187)(77 186)(78 185)(79 184)(80 183)(81 182)(82 181)(83 180)(84 179)(85 208)(86 207)(87 206)(88 205)(89 204)(90 203)(91 202)(92 201)(93 200)(94 199)(95 198)(96 197)(97 224)(98 223)(99 222)(100 221)(101 220)(102 219)(103 218)(104 217)(105 216)(106 215)(107 214)(108 213)(109 212)(110 211)(111 210)(112 209)
(1 91 49 74 8 98 56 81 15 105 35 60 22 112 42 67)(2 92 50 75 9 99 29 82 16 106 36 61 23 85 43 68)(3 93 51 76 10 100 30 83 17 107 37 62 24 86 44 69)(4 94 52 77 11 101 31 84 18 108 38 63 25 87 45 70)(5 95 53 78 12 102 32 57 19 109 39 64 26 88 46 71)(6 96 54 79 13 103 33 58 20 110 40 65 27 89 47 72)(7 97 55 80 14 104 34 59 21 111 41 66 28 90 48 73)(113 213 143 186 134 206 164 179 127 199 157 172 120 220 150 193)(114 214 144 187 135 207 165 180 128 200 158 173 121 221 151 194)(115 215 145 188 136 208 166 181 129 201 159 174 122 222 152 195)(116 216 146 189 137 209 167 182 130 202 160 175 123 223 153 196)(117 217 147 190 138 210 168 183 131 203 161 176 124 224 154 169)(118 218 148 191 139 211 141 184 132 204 162 177 125 197 155 170)(119 219 149 192 140 212 142 185 133 205 163 178 126 198 156 171)

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

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

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

124 conjugacy classes

class 1 2A2B2C4A4B4C4D7A7B7C8A8B8C8D8E8F8G8H14A···14I16A···16H16I16J16K16L28A···28L56A···56X112A···112AV
order122244447778888888814···1416···161616161628···2856···56112···112
size112281122822211112228282···22···2282828282···22···22···2

124 irreducible representations

dim1111111122222222222
type++++++++
imageC1C2C2C2C4C4C8C8D4D7M4(2)D14D28C7⋊D4C4×D7D4.C8C8×D7C8⋊D7D28.C8
kernelD28.C8C28.C8C2×C112D28.2C4C4.Dic7C4○D28Dic14D28C56C2×C16C2×C14C2×C8C8C8C2×C4C7C4C22C1
# reps1111224423236668121248

Matrix representation of D28.C8 in GL2(𝔽113) generated by

320
053
,
053
320
,
650
042
G:=sub<GL(2,GF(113))| [32,0,0,53],[0,32,53,0],[65,0,0,42] >;

D28.C8 in GAP, Magma, Sage, TeX

D_{28}.C_8
% in TeX

G:=Group("D28.C8");
// GroupNames label

G:=SmallGroup(448,65);
// by ID

G=gap.SmallGroup(448,65);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,141,36,758,100,1123,136,102,18822]);
// Polycyclic

G:=Group<a,b,c|a^28=b^2=1,c^8=a^14,b*a*b=a^-1,a*c=c*a,c*b*c^-1=a^7*b>;
// generators/relations

Export

Subgroup lattice of D28.C8 in TeX

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