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## G = C28.38SD16order 448 = 26·7

### 4th non-split extension by C28 of SD16 acting via SD16/D4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C28 — C28.38SD16
 Chief series C1 — C7 — C14 — C28 — C2×C28 — C4⋊Dic7 — C28⋊2Q8 — C28.38SD16
 Lower central C7 — C14 — C2×C28 — C28.38SD16
 Upper central C1 — C22 — C42 — C4×D4

Generators and relations for C28.38SD16
G = < a,b,c | a28=b8=c2=1, bab-1=a-1, ac=ca, cbc=a14b3 >

Subgroups: 436 in 108 conjugacy classes, 47 normal (31 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, D4, Q8, C23, C14, C14, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C2×D4, C2×Q8, Dic7, C28, C28, C28, C2×C14, C2×C14, D4⋊C4, C4⋊C8, C4.Q8, C4×D4, C4⋊Q8, C7⋊C8, Dic14, C2×Dic7, C2×C28, C2×C28, C7×D4, C7×D4, C22×C14, D42Q8, C2×C7⋊C8, C4⋊Dic7, C4⋊Dic7, C4×C28, C7×C22⋊C4, C7×C4⋊C4, C2×Dic14, C22×C28, D4×C14, C28⋊C8, C4.Dic14, D4⋊Dic7, C282Q8, D4×C28, C28.38SD16
Quotients: C1, C2, C22, D4, Q8, C23, D7, SD16, C2×D4, C2×Q8, C4○D4, D14, C22⋊Q8, C2×SD16, C8⋊C22, Dic14, C7⋊D4, C22×D7, D42Q8, D4.D7, C2×Dic14, C4○D28, C2×C7⋊D4, C28.48D4, C2×D4.D7, D4⋊D14, C28.38SD16

Smallest permutation representation of C28.38SD16
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)
(1 131 193 153 72 216 94 42)(2 130 194 152 73 215 95 41)(3 129 195 151 74 214 96 40)(4 128 196 150 75 213 97 39)(5 127 169 149 76 212 98 38)(6 126 170 148 77 211 99 37)(7 125 171 147 78 210 100 36)(8 124 172 146 79 209 101 35)(9 123 173 145 80 208 102 34)(10 122 174 144 81 207 103 33)(11 121 175 143 82 206 104 32)(12 120 176 142 83 205 105 31)(13 119 177 141 84 204 106 30)(14 118 178 168 57 203 107 29)(15 117 179 167 58 202 108 56)(16 116 180 166 59 201 109 55)(17 115 181 165 60 200 110 54)(18 114 182 164 61 199 111 53)(19 113 183 163 62 198 112 52)(20 140 184 162 63 197 85 51)(21 139 185 161 64 224 86 50)(22 138 186 160 65 223 87 49)(23 137 187 159 66 222 88 48)(24 136 188 158 67 221 89 47)(25 135 189 157 68 220 90 46)(26 134 190 156 69 219 91 45)(27 133 191 155 70 218 92 44)(28 132 192 154 71 217 93 43)
(1 15)(2 16)(3 17)(4 18)(5 19)(6 20)(7 21)(8 22)(9 23)(10 24)(11 25)(12 26)(13 27)(14 28)(29 203)(30 204)(31 205)(32 206)(33 207)(34 208)(35 209)(36 210)(37 211)(38 212)(39 213)(40 214)(41 215)(42 216)(43 217)(44 218)(45 219)(46 220)(47 221)(48 222)(49 223)(50 224)(51 197)(52 198)(53 199)(54 200)(55 201)(56 202)(57 71)(58 72)(59 73)(60 74)(61 75)(62 76)(63 77)(64 78)(65 79)(66 80)(67 81)(68 82)(69 83)(70 84)(85 170)(86 171)(87 172)(88 173)(89 174)(90 175)(91 176)(92 177)(93 178)(94 179)(95 180)(96 181)(97 182)(98 183)(99 184)(100 185)(101 186)(102 187)(103 188)(104 189)(105 190)(106 191)(107 192)(108 193)(109 194)(110 195)(111 196)(112 169)(113 163)(114 164)(115 165)(116 166)(117 167)(118 168)(119 141)(120 142)(121 143)(122 144)(123 145)(124 146)(125 147)(126 148)(127 149)(128 150)(129 151)(130 152)(131 153)(132 154)(133 155)(134 156)(135 157)(136 158)(137 159)(138 160)(139 161)(140 162)

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

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

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

79 conjugacy classes

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

79 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + - + + + + - + - + image C1 C2 C2 C2 C2 C2 D4 Q8 D7 SD16 C4○D4 D14 D14 D14 C7⋊D4 Dic14 C4○D28 C8⋊C22 D4.D7 D4⋊D14 kernel C28.38SD16 C28⋊C8 C4.Dic14 D4⋊Dic7 C28⋊2Q8 D4×C28 C2×C28 C7×D4 C4×D4 C28 C28 C42 C4⋊C4 C2×D4 C2×C4 D4 C4 C14 C4 C2 # reps 1 1 2 2 1 1 2 2 3 4 2 3 3 3 12 12 12 1 6 6

Matrix representation of C28.38SD16 in GL4(𝔽113) generated by

 49 77 0 0 46 70 0 0 0 0 1 0 0 0 0 1
,
 16 52 0 0 32 97 0 0 0 0 26 38 0 0 3 0
,
 112 0 0 0 0 112 0 0 0 0 1 0 0 0 35 112
G:=sub<GL(4,GF(113))| [49,46,0,0,77,70,0,0,0,0,1,0,0,0,0,1],[16,32,0,0,52,97,0,0,0,0,26,3,0,0,38,0],[112,0,0,0,0,112,0,0,0,0,1,35,0,0,0,112] >;

C28.38SD16 in GAP, Magma, Sage, TeX

C_{28}._{38}{\rm SD}_{16}
% in TeX

G:=Group("C28.38SD16");
// GroupNames label

G:=SmallGroup(448,542);
// by ID

G=gap.SmallGroup(448,542);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,336,253,120,254,1123,297,136,18822]);
// Polycyclic

G:=Group<a,b,c|a^28=b^8=c^2=1,b*a*b^-1=a^-1,a*c=c*a,c*b*c=a^14*b^3>;
// generators/relations

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