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G = D56.1C4order 448 = 26·7

1st non-split extension by D56 of C4 acting via C4/C2=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D56.1C4, C4.19D56, C28.37D8, C56.85D4, Dic28.1C4, (C2×C16)⋊4D7, (C2×C112)⋊4C2, C8.20(C4×D7), C56.50(C2×C4), (C2×C4).75D28, C56.C41C2, (C2×C28).394D4, (C2×C8).312D14, C72(D8.C4), C8.42(C7⋊D4), C4.17(D14⋊C4), D567C2.1C2, (C2×C14).18SD16, C2.8(C2.D56), C28.41(C22⋊C4), (C2×C56).384C22, C22.1(C56⋊C2), C14.16(D4⋊C4), SmallGroup(448,67)

Series: Derived Chief Lower central Upper central

C1C56 — D56.1C4
C1C7C14C28C56C2×C56D567C2 — D56.1C4
C7C14C28C56 — D56.1C4
C1C4C2×C4C2×C8C2×C16

Generators and relations for D56.1C4
 G = < a,b,c | a56=b2=1, c4=a42, bab=a-1, ac=ca, cbc-1=a7b >

2C2
56C2
28C4
28C22
2C14
8D7
14Q8
14D4
28D4
28C2×C4
28C8
4D14
4Dic7
2C16
7D8
7Q16
14M4(2)
14SD16
14C4○D4
2Dic14
2D28
4C7⋊C8
4C7⋊D4
4C4×D7
7C8.C4
7C4○D8
2C112
2C4○D28
2C56⋊C2
2C4.Dic7
7D8.C4

Smallest permutation representation of D56.1C4
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)
(1 157)(2 156)(3 155)(4 154)(5 153)(6 152)(7 151)(8 150)(9 149)(10 148)(11 147)(12 146)(13 145)(14 144)(15 143)(16 142)(17 141)(18 140)(19 139)(20 138)(21 137)(22 136)(23 135)(24 134)(25 133)(26 132)(27 131)(28 130)(29 129)(30 128)(31 127)(32 126)(33 125)(34 124)(35 123)(36 122)(37 121)(38 120)(39 119)(40 118)(41 117)(42 116)(43 115)(44 114)(45 113)(46 168)(47 167)(48 166)(49 165)(50 164)(51 163)(52 162)(53 161)(54 160)(55 159)(56 158)(57 190)(58 189)(59 188)(60 187)(61 186)(62 185)(63 184)(64 183)(65 182)(66 181)(67 180)(68 179)(69 178)(70 177)(71 176)(72 175)(73 174)(74 173)(75 172)(76 171)(77 170)(78 169)(79 224)(80 223)(81 222)(82 221)(83 220)(84 219)(85 218)(86 217)(87 216)(88 215)(89 214)(90 213)(91 212)(92 211)(93 210)(94 209)(95 208)(96 207)(97 206)(98 205)(99 204)(100 203)(101 202)(102 201)(103 200)(104 199)(105 198)(106 197)(107 196)(108 195)(109 194)(110 193)(111 192)(112 191)
(1 57 50 106 43 99 36 92 29 85 22 78 15 71 8 64)(2 58 51 107 44 100 37 93 30 86 23 79 16 72 9 65)(3 59 52 108 45 101 38 94 31 87 24 80 17 73 10 66)(4 60 53 109 46 102 39 95 32 88 25 81 18 74 11 67)(5 61 54 110 47 103 40 96 33 89 26 82 19 75 12 68)(6 62 55 111 48 104 41 97 34 90 27 83 20 76 13 69)(7 63 56 112 49 105 42 98 35 91 28 84 21 77 14 70)(113 209 134 174 155 195 120 216 141 181 162 202 127 223 148 188)(114 210 135 175 156 196 121 217 142 182 163 203 128 224 149 189)(115 211 136 176 157 197 122 218 143 183 164 204 129 169 150 190)(116 212 137 177 158 198 123 219 144 184 165 205 130 170 151 191)(117 213 138 178 159 199 124 220 145 185 166 206 131 171 152 192)(118 214 139 179 160 200 125 221 146 186 167 207 132 172 153 193)(119 215 140 180 161 201 126 222 147 187 168 208 133 173 154 194)

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

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

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

118 conjugacy classes

class 1 2A2B2C4A4B4C4D7A7B7C8A8B8C8D8E8F14A···14I16A···16H28A···28L56A···56X112A···112AV
order1222444477788888814···1416···1628···2856···56112···112
size1125611256222222256562···22···22···22···22···2

118 irreducible representations

dim1111112222222222222
type+++++++++++
imageC1C2C2C2C4C4D4D4D7D8SD16D14C4×D7C7⋊D4D28D8.C4D56C56⋊C2D56.1C4
kernelD56.1C4C56.C4C2×C112D567C2D56Dic28C56C2×C28C2×C16C28C2×C14C2×C8C8C8C2×C4C7C4C22C1
# reps1111221132236668121248

Matrix representation of D56.1C4 in GL2(𝔽113) generated by

220
036
,
036
220
,
780
065
G:=sub<GL(2,GF(113))| [22,0,0,36],[0,22,36,0],[78,0,0,65] >;

D56.1C4 in GAP, Magma, Sage, TeX

D_{56}._1C_4
% in TeX

G:=Group("D56.1C4");
// GroupNames label

G:=SmallGroup(448,67);
// by ID

G=gap.SmallGroup(448,67);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,85,92,422,268,1123,1684,102,18822]);
// Polycyclic

G:=Group<a,b,c|a^56=b^2=1,c^4=a^42,b*a*b=a^-1,a*c=c*a,c*b*c^-1=a^7*b>;
// generators/relations

Export

Subgroup lattice of D56.1C4 in TeX

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