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## G = C14.SD32order 448 = 26·7

### 2nd non-split extension by C14 of SD32 acting via SD32/D8=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C56 — C14.SD32
 Chief series C1 — C7 — C14 — C28 — C2×C28 — C2×C56 — C56⋊1C4 — C14.SD32
 Lower central C7 — C14 — C28 — C56 — C14.SD32
 Upper central C1 — C22 — C2×C4 — C2×C8 — C2×D8

Generators and relations for C14.SD32
G = < a,b,c | a14=b16=c2=1, bab-1=a-1, ac=ca, cbc=a7b7 >

Subgroups: 292 in 66 conjugacy classes, 31 normal (27 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, C23, C14, C14, C16, C4⋊C4, C2×C8, D8, D8, C2×D4, Dic7, C28, C2×C14, C2×C14, C2.D8, C2×C16, C2×D8, C56, C2×Dic7, C2×C28, C7×D4, C22×C14, C2.D16, C7⋊C16, C4⋊Dic7, C2×C56, C7×D8, C7×D8, D4×C14, C2×C7⋊C16, C561C4, C14×D8, C14.SD32
Quotients: C1, C2, C4, C22, C2×C4, D4, D7, C22⋊C4, D8, SD16, Dic7, D14, D4⋊C4, D16, SD32, C2×Dic7, C7⋊D4, C2.D16, D4⋊D7, D4.D7, C23.D7, C7⋊D16, D8.D7, D4⋊Dic7, C14.SD32

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

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

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

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

64 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 7A 7B 7C 8A 8B 8C 8D 14A ··· 14I 14J ··· 14U 16A ··· 16H 28A ··· 28F 56A ··· 56L order 1 2 2 2 2 2 4 4 4 4 7 7 7 8 8 8 8 14 ··· 14 14 ··· 14 16 ··· 16 28 ··· 28 56 ··· 56 size 1 1 1 1 8 8 2 2 56 56 2 2 2 2 2 2 2 2 ··· 2 8 ··· 8 14 ··· 14 4 ··· 4 4 ··· 4

64 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + - + - + + - image C1 C2 C2 C2 C4 D4 D4 D7 SD16 D8 D14 Dic7 D16 SD32 C7⋊D4 C7⋊D4 D4.D7 D4⋊D7 C7⋊D16 D8.D7 kernel C14.SD32 C2×C7⋊C16 C56⋊1C4 C14×D8 C7×D8 C56 C2×C28 C2×D8 C28 C2×C14 C2×C8 D8 C14 C14 C8 C2×C4 C4 C22 C2 C2 # reps 1 1 1 1 4 1 1 3 2 2 3 6 4 4 6 6 3 3 6 6

Matrix representation of C14.SD32 in GL4(𝔽113) generated by

 1 0 0 0 0 1 0 0 0 0 85 102 0 0 0 4
,
 9 52 0 0 82 97 0 0 0 0 33 49 0 0 17 80
,
 1 0 0 0 106 112 0 0 0 0 1 36 0 0 0 112
G:=sub<GL(4,GF(113))| [1,0,0,0,0,1,0,0,0,0,85,0,0,0,102,4],[9,82,0,0,52,97,0,0,0,0,33,17,0,0,49,80],[1,106,0,0,0,112,0,0,0,0,1,0,0,0,36,112] >;

C14.SD32 in GAP, Magma, Sage, TeX

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

G:=Group("C14.SD32");
// GroupNames label

G:=SmallGroup(448,119);
// by ID

G=gap.SmallGroup(448,119);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,28,141,675,346,192,1684,851,102,18822]);
// Polycyclic

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

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