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G = C563D4order 448 = 26·7

3rd semidirect product of C56 and D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C563D4, C23.18D28, C82(C7⋊D4), C74(C82D4), C8⋊Dic74C2, (C2×D56)⋊12C2, (C2×C4).52D28, (C2×C8).78D14, C287D441C2, C28.421(C2×D4), (C2×C28).298D4, (C2×M4(2))⋊2D7, C2.D5642C2, (C14×M4(2))⋊2C2, (C2×C56).64C22, C4.115(C4○D28), C28.231(C4○D4), C2.22(C287D4), C14.74(C4⋊D4), C2.23(C8⋊D14), C14.23(C8⋊C22), (C2×C28).776C23, (C2×D28).21C22, (C22×C14).104D4, (C22×C4).143D14, C22.135(C2×D28), C4⋊Dic7.27C22, (C22×C28).305C22, C4.114(C2×C7⋊D4), (C2×C14).166(C2×D4), (C2×C4).725(C22×D7), SmallGroup(448,669)

Series: Derived Chief Lower central Upper central

C1C2×C28 — C563D4
C1C7C14C2×C14C2×C28C2×D28C2×D56 — C563D4
C7C14C2×C28 — C563D4
C1C22C22×C4C2×M4(2)

Generators and relations for C563D4
 G = < a,b,c | a56=b4=c2=1, bab-1=a27, cac=a-1, cbc=b-1 >

Subgroups: 900 in 130 conjugacy classes, 43 normal (27 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C7, C8, C8, C2×C4, C2×C4, D4, C23, C23, D7, C14, C14, C14, C22⋊C4, C4⋊C4, C2×C8, M4(2), D8, C22×C4, C2×D4, Dic7, C28, C28, D14, C2×C14, C2×C14, D4⋊C4, C4.Q8, C4⋊D4, C2×M4(2), C2×D8, C56, C56, D28, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C22×D7, C22×C14, C82D4, D56, C4⋊Dic7, D14⋊C4, C2×C56, C7×M4(2), C2×D28, C2×C7⋊D4, C22×C28, C8⋊Dic7, C2.D56, C2×D56, C287D4, C14×M4(2), C563D4
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C4○D4, D14, C4⋊D4, C8⋊C22, D28, C7⋊D4, C22×D7, C82D4, C2×D28, C4○D28, C2×C7⋊D4, C8⋊D14, C287D4, C563D4

Smallest permutation representation of C563D4
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 96 159 170)(2 67 160 197)(3 94 161 224)(4 65 162 195)(5 92 163 222)(6 63 164 193)(7 90 165 220)(8 61 166 191)(9 88 167 218)(10 59 168 189)(11 86 113 216)(12 57 114 187)(13 84 115 214)(14 111 116 185)(15 82 117 212)(16 109 118 183)(17 80 119 210)(18 107 120 181)(19 78 121 208)(20 105 122 179)(21 76 123 206)(22 103 124 177)(23 74 125 204)(24 101 126 175)(25 72 127 202)(26 99 128 173)(27 70 129 200)(28 97 130 171)(29 68 131 198)(30 95 132 169)(31 66 133 196)(32 93 134 223)(33 64 135 194)(34 91 136 221)(35 62 137 192)(36 89 138 219)(37 60 139 190)(38 87 140 217)(39 58 141 188)(40 85 142 215)(41 112 143 186)(42 83 144 213)(43 110 145 184)(44 81 146 211)(45 108 147 182)(46 79 148 209)(47 106 149 180)(48 77 150 207)(49 104 151 178)(50 75 152 205)(51 102 153 176)(52 73 154 203)(53 100 155 174)(54 71 156 201)(55 98 157 172)(56 69 158 199)
(1 170)(2 169)(3 224)(4 223)(5 222)(6 221)(7 220)(8 219)(9 218)(10 217)(11 216)(12 215)(13 214)(14 213)(15 212)(16 211)(17 210)(18 209)(19 208)(20 207)(21 206)(22 205)(23 204)(24 203)(25 202)(26 201)(27 200)(28 199)(29 198)(30 197)(31 196)(32 195)(33 194)(34 193)(35 192)(36 191)(37 190)(38 189)(39 188)(40 187)(41 186)(42 185)(43 184)(44 183)(45 182)(46 181)(47 180)(48 179)(49 178)(50 177)(51 176)(52 175)(53 174)(54 173)(55 172)(56 171)(57 142)(58 141)(59 140)(60 139)(61 138)(62 137)(63 136)(64 135)(65 134)(66 133)(67 132)(68 131)(69 130)(70 129)(71 128)(72 127)(73 126)(74 125)(75 124)(76 123)(77 122)(78 121)(79 120)(80 119)(81 118)(82 117)(83 116)(84 115)(85 114)(86 113)(87 168)(88 167)(89 166)(90 165)(91 164)(92 163)(93 162)(94 161)(95 160)(96 159)(97 158)(98 157)(99 156)(100 155)(101 154)(102 153)(103 152)(104 151)(105 150)(106 149)(107 148)(108 147)(109 146)(110 145)(111 144)(112 143)

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

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

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

76 conjugacy classes

class 1 2A2B2C2D2E2F4A4B4C4D4E7A7B7C8A8B8C8D14A···14I14J···14O28A···28L28M···28R56A···56X
order122222244444777888814···1414···1428···2828···2856···56
size111145656224565622244442···24···42···24···44···4

76 irreducible representations

dim1111112222222222244
type++++++++++++++++
imageC1C2C2C2C2C2D4D4D4D7C4○D4D14D14C7⋊D4D28D28C4○D28C8⋊C22C8⋊D14
kernelC563D4C8⋊Dic7C2.D56C2×D56C287D4C14×M4(2)C56C2×C28C22×C14C2×M4(2)C28C2×C8C22×C4C8C2×C4C23C4C14C2
# reps1121212113263126612212

Matrix representation of C563D4 in GL6(𝔽113)

9800000
44150000
0049112669
0040902
00421108861
0047234580
,
1820000
7950000
0048122100
003810210564
0061610450
0060564885
,
1820000
8950000
0048122100
003810210564
00477210450
005134885

G:=sub<GL(6,GF(113))| [98,44,0,0,0,0,0,15,0,0,0,0,0,0,49,40,42,47,0,0,11,9,110,23,0,0,26,0,88,45,0,0,69,2,61,80],[18,7,0,0,0,0,2,95,0,0,0,0,0,0,48,38,6,60,0,0,1,102,16,56,0,0,22,105,104,48,0,0,100,64,50,85],[18,8,0,0,0,0,2,95,0,0,0,0,0,0,48,38,47,51,0,0,1,102,72,3,0,0,22,105,104,48,0,0,100,64,50,85] >;

C563D4 in GAP, Magma, Sage, TeX

C_{56}\rtimes_3D_4
% in TeX

G:=Group("C56:3D4");
// GroupNames label

G:=SmallGroup(448,669);
// by ID

G=gap.SmallGroup(448,669);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,253,344,254,387,1684,102,18822]);
// Polycyclic

G:=Group<a,b,c|a^56=b^4=c^2=1,b*a*b^-1=a^27,c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations

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