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## G = C14×SD32order 448 = 26·7

### Direct product of C14 and SD32

direct product, metabelian, nilpotent (class 4), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C8 — C14×SD32
 Chief series C1 — C2 — C4 — C8 — C56 — C7×Q16 — C7×SD32 — C14×SD32
 Lower central C1 — C2 — C4 — C8 — C14×SD32
 Upper central C1 — C2×C14 — C2×C28 — C2×C56 — C14×SD32

Generators and relations for C14×SD32
G = < a,b,c | a14=b16=c2=1, ab=ba, ac=ca, cbc=b7 >

Subgroups: 210 in 90 conjugacy classes, 50 normal (26 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, Q8, C23, C14, C14, C14, C16, C2×C8, D8, D8, Q16, Q16, C2×D4, C2×Q8, C28, C28, C2×C14, C2×C14, C2×C16, SD32, C2×D8, C2×Q16, C56, C2×C28, C2×C28, C7×D4, C7×Q8, C22×C14, C2×SD32, C112, C2×C56, C7×D8, C7×D8, C7×Q16, C7×Q16, D4×C14, Q8×C14, C2×C112, C7×SD32, C14×D8, C14×Q16, C14×SD32
Quotients: C1, C2, C22, C7, D4, C23, C14, D8, C2×D4, C2×C14, SD32, C2×D8, C7×D4, C22×C14, C2×SD32, C7×D8, D4×C14, C7×SD32, C14×D8, C14×SD32

Smallest permutation representation of C14×SD32
On 224 points
Generators in S224
(1 182 174 79 151 94 64 33 205 107 142 212 127 23)(2 183 175 80 152 95 49 34 206 108 143 213 128 24)(3 184 176 65 153 96 50 35 207 109 144 214 113 25)(4 185 161 66 154 81 51 36 208 110 129 215 114 26)(5 186 162 67 155 82 52 37 193 111 130 216 115 27)(6 187 163 68 156 83 53 38 194 112 131 217 116 28)(7 188 164 69 157 84 54 39 195 97 132 218 117 29)(8 189 165 70 158 85 55 40 196 98 133 219 118 30)(9 190 166 71 159 86 56 41 197 99 134 220 119 31)(10 191 167 72 160 87 57 42 198 100 135 221 120 32)(11 192 168 73 145 88 58 43 199 101 136 222 121 17)(12 177 169 74 146 89 59 44 200 102 137 223 122 18)(13 178 170 75 147 90 60 45 201 103 138 224 123 19)(14 179 171 76 148 91 61 46 202 104 139 209 124 20)(15 180 172 77 149 92 62 47 203 105 140 210 125 21)(16 181 173 78 150 93 63 48 204 106 141 211 126 22)
(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 41)(2 48)(3 39)(4 46)(5 37)(6 44)(7 35)(8 42)(9 33)(10 40)(11 47)(12 38)(13 45)(14 36)(15 43)(16 34)(17 62)(18 53)(19 60)(20 51)(21 58)(22 49)(23 56)(24 63)(25 54)(26 61)(27 52)(28 59)(29 50)(30 57)(31 64)(32 55)(65 132)(66 139)(67 130)(68 137)(69 144)(70 135)(71 142)(72 133)(73 140)(74 131)(75 138)(76 129)(77 136)(78 143)(79 134)(80 141)(81 124)(82 115)(83 122)(84 113)(85 120)(86 127)(87 118)(88 125)(89 116)(90 123)(91 114)(92 121)(93 128)(94 119)(95 126)(96 117)(97 176)(98 167)(99 174)(100 165)(101 172)(102 163)(103 170)(104 161)(105 168)(106 175)(107 166)(108 173)(109 164)(110 171)(111 162)(112 169)(145 210)(146 217)(147 224)(148 215)(149 222)(150 213)(151 220)(152 211)(153 218)(154 209)(155 216)(156 223)(157 214)(158 221)(159 212)(160 219)(177 194)(178 201)(179 208)(180 199)(181 206)(182 197)(183 204)(184 195)(185 202)(186 193)(187 200)(188 207)(189 198)(190 205)(191 196)(192 203)

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

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

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

154 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 7A ··· 7F 8A 8B 8C 8D 14A ··· 14R 14S ··· 14AD 16A ··· 16H 28A ··· 28L 28M ··· 28X 56A ··· 56X 112A ··· 112AV order 1 2 2 2 2 2 4 4 4 4 7 ··· 7 8 8 8 8 14 ··· 14 14 ··· 14 16 ··· 16 28 ··· 28 28 ··· 28 56 ··· 56 112 ··· 112 size 1 1 1 1 8 8 2 2 8 8 1 ··· 1 2 2 2 2 1 ··· 1 8 ··· 8 2 ··· 2 2 ··· 2 8 ··· 8 2 ··· 2 2 ··· 2

154 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + + image C1 C2 C2 C2 C2 C7 C14 C14 C14 C14 D4 D4 D8 D8 SD32 C7×D4 C7×D4 C7×D8 C7×D8 C7×SD32 kernel C14×SD32 C2×C112 C7×SD32 C14×D8 C14×Q16 C2×SD32 C2×C16 SD32 C2×D8 C2×Q16 C56 C2×C28 C28 C2×C14 C14 C8 C2×C4 C4 C22 C2 # reps 1 1 4 1 1 6 6 24 6 6 1 1 2 2 8 6 6 12 12 48

Matrix representation of C14×SD32 in GL3(𝔽113) generated by

 7 0 0 0 7 0 0 0 7
,
 112 0 0 0 9 7 0 53 16
,
 1 0 0 0 51 62 0 82 62
G:=sub<GL(3,GF(113))| [7,0,0,0,7,0,0,0,7],[112,0,0,0,9,53,0,7,16],[1,0,0,0,51,82,0,62,62] >;

C14×SD32 in GAP, Magma, Sage, TeX

C_{14}\times {\rm SD}_{32}
% in TeX

G:=Group("C14xSD32");
// GroupNames label

G:=SmallGroup(448,914);
// by ID

G=gap.SmallGroup(448,914);
# by ID

G:=PCGroup([7,-2,-2,-2,-7,-2,-2,-2,1568,813,5884,2951,242,14117,7068,124]);
// Polycyclic

G:=Group<a,b,c|a^14=b^16=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^7>;
// generators/relations

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