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## G = Dic56order 224 = 25·7

### Dicyclic group

Aliases: Dic56, C16.D7, C71Q32, C4.3D28, C14.3D8, C2.5D56, C112.1C2, C28.26D4, C8.15D14, C56.16C22, Dic28.1C2, SmallGroup(224,7)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C56 — Dic56
 Chief series C1 — C7 — C14 — C28 — C56 — Dic28 — Dic56
 Lower central C7 — C14 — C28 — C56 — Dic56
 Upper central C1 — C2 — C4 — C8 — C16

Generators and relations for Dic56
G = < a,b | a112=1, b2=a56, bab-1=a-1 >

Smallest permutation representation of Dic56
Regular action 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 57 156)(2 211 58 155)(3 210 59 154)(4 209 60 153)(5 208 61 152)(6 207 62 151)(7 206 63 150)(8 205 64 149)(9 204 65 148)(10 203 66 147)(11 202 67 146)(12 201 68 145)(13 200 69 144)(14 199 70 143)(15 198 71 142)(16 197 72 141)(17 196 73 140)(18 195 74 139)(19 194 75 138)(20 193 76 137)(21 192 77 136)(22 191 78 135)(23 190 79 134)(24 189 80 133)(25 188 81 132)(26 187 82 131)(27 186 83 130)(28 185 84 129)(29 184 85 128)(30 183 86 127)(31 182 87 126)(32 181 88 125)(33 180 89 124)(34 179 90 123)(35 178 91 122)(36 177 92 121)(37 176 93 120)(38 175 94 119)(39 174 95 118)(40 173 96 117)(41 172 97 116)(42 171 98 115)(43 170 99 114)(44 169 100 113)(45 168 101 224)(46 167 102 223)(47 166 103 222)(48 165 104 221)(49 164 105 220)(50 163 106 219)(51 162 107 218)(52 161 108 217)(53 160 109 216)(54 159 110 215)(55 158 111 214)(56 157 112 213)

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

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

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

Dic56 is a maximal subgroup of
C224⋊C2  Dic112  D16.D7  C7⋊Q64  D1127C2  C16.D14  D163D7  SD32⋊D7  D7×Q32
Dic56 is a maximal quotient of
C56.78D4  C1125C4

59 conjugacy classes

 class 1 2 4A 4B 4C 7A 7B 7C 8A 8B 14A 14B 14C 16A 16B 16C 16D 28A ··· 28F 56A ··· 56L 112A ··· 112X order 1 2 4 4 4 7 7 7 8 8 14 14 14 16 16 16 16 28 ··· 28 56 ··· 56 112 ··· 112 size 1 1 2 56 56 2 2 2 2 2 2 2 2 2 2 2 2 2 ··· 2 2 ··· 2 2 ··· 2

59 irreducible representations

 dim 1 1 1 2 2 2 2 2 2 2 2 type + + + + + + + - + + - image C1 C2 C2 D4 D7 D8 D14 Q32 D28 D56 Dic56 kernel Dic56 C112 Dic28 C28 C16 C14 C8 C7 C4 C2 C1 # reps 1 1 2 1 3 2 3 4 6 12 24

Matrix representation of Dic56 in GL2(𝔽113) generated by

 78 68 93 13
,
 95 78 90 18
G:=sub<GL(2,GF(113))| [78,93,68,13],[95,90,78,18] >;

Dic56 in GAP, Magma, Sage, TeX

{\rm Dic}_{56}
% in TeX

G:=Group("Dic56");
// GroupNames label

G:=SmallGroup(224,7);
// by ID

G=gap.SmallGroup(224,7);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-7,96,73,79,218,122,579,69,6917]);
// Polycyclic

G:=Group<a,b|a^112=1,b^2=a^56,b*a*b^-1=a^-1>;
// generators/relations

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