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G = D112⋊7C2  order 448 = 26·7

The semidirect product of D112 and C2 acting through Inn(D112)

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D112⋊7C2, C28.38D8, C8.13D28, C4.20D56, C56.63D4, Dic56⋊7C2, C16.16D14, C22.1D56, C56.57C23, D56.7C22, C112.18C22, Dic28.7C22, (C2×C16)⋊6D7, C7⋊1(C4○D16), (C2×C112)⋊10C2, C112⋊C2⋊7C2, (C2×C4).85D28, C2.13(C2×D56), (C2×C14).20D8, C4.38(C2×D28), C14.11(C2×D8), D56⋊7C2⋊1C2, (C2×C28).395D4, C28.281(C2×D4), (C2×C8).313D14, C8.47(C22×D7), (C2×C56).385C22, SmallGroup(448,438)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C56 — D112⋊7C2
Chief series
C1 — C7 — C14 — C28 — C56 — D56 — D56⋊7C2 — D112⋊7C2
Lower central
C7 — C14 — C28 — C56 — D112⋊7C2
Upper central
C1 — C4 — C2×C4 — C2×C8 — C2×C16

Generators and relations for D112⋊7C2
 G = < a,b,c | a112=b2=c2=1, bab=a-1, ac=ca, cbc=a56b >

Subgroups: 628 in 84 conjugacy classes, 35 normal (27 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, Q8, D7, C14, C14, C16, C2×C8, D8, SD16, Q16, C4○D4, Dic7, C28, D14, C2×C14, C2×C16, D16, SD32, Q32, C4○D8, C56, Dic14, C4×D7, D28, C7⋊D4, C2×C28, C4○D16, C112, C56⋊C2, D56, Dic28, C2×C56, C4○D28, D112, C112⋊C2, Dic56, C2×C112, D56⋊7C2, D112⋊7C2
Quotients: C1, C2, C22, D4, C23, D7, D8, C2×D4, D14, C2×D8, D28, C22×D7, C4○D16, D56, C2×D28, C2×D56, D112⋊7C2

Smallest permutation representation of D112⋊7C2
►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)

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

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

118 conjugacy classes

class 1 2A2B2C2D4A4B4C4D4E7A7B7C8A8B8C8D14A···14I16A···16H28A···28L56A···56X112A···112AV
order1222244444777888814···1416···1628···2856···56112···112
size1125656112565622222222···22···22···22···22···2

118 irreducible representations

dim1111112222222222222
type+++++++++++++++++
imageC1C2C2C2C2C2D4D4D7D8D8D14D14D28D28C4○D16D56D56D112⋊7C2
kernelD112⋊7C2D112C112⋊C2Dic56C2×C112D56⋊7C2C56C2×C28C2×C16C28C2×C14C16C2×C8C8C2×C4C7C4C22C1
# reps1121121132263668121248

Matrix representation of D112⋊7C2 ►in GL2(𝔽113) generated by

860
046
,
046
860
,
10
0112
G:=sub<GL(2,GF(113))| [86,0,0,46],[0,86,46,0],[1,0,0,112] >;
 

D112⋊7C2 in GAP, Magma, Sage, TeX 

D_{112}\rtimes_7C_2
 
% in TeX
 
G:=Group("D112:7C2");
 
// GroupNames label
 
G:=SmallGroup(448,438);
 
// by ID
 
G=gap.SmallGroup(448,438);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,232,254,142,675,192,1684,102,18822]);
 
// Polycyclic
 
G:=Group<a,b,c|a^112=b^2=c^2=1,b*a*b=a^-1,a*c=c*a,c*b*c=a^56*b>;
 
// generators/relations
 

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