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## G = (C2×C28)⋊C8order 448 = 26·7

### 1st semidirect product of C2×C28 and C8 acting via C8/C2=C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C14 — (C2×C28)⋊C8
 Chief series C1 — C7 — C14 — C2×C14 — C2×C28 — C22×C28 — C28.55D4 — (C2×C28)⋊C8
 Lower central C7 — C14 — C2×C14 — (C2×C28)⋊C8
 Upper central C1 — C22 — C22×C4 — C2×C4⋊C4

Generators and relations for (C2×C28)⋊C8
G = < a,b,c | a2=b28=c8=1, ab=ba, cac-1=ab14, cbc-1=ab-1 >

Subgroups: 228 in 78 conjugacy classes, 35 normal (25 characteristic)
C1, C2, C2, C4, C22, C22, C7, C8, C2×C4, C2×C4, C23, C14, C14, C4⋊C4, C2×C8, C22×C4, C28, C2×C14, C2×C14, C22⋊C8, C2×C4⋊C4, C7⋊C8, C2×C28, C2×C28, C22×C14, C22.M4(2), C2×C7⋊C8, C7×C4⋊C4, C22×C28, C28.55D4, C14×C4⋊C4, (C2×C28)⋊C8
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, D7, C22⋊C4, C2×C8, M4(2), Dic7, D14, C22⋊C8, C23⋊C4, C4.10D4, C7⋊C8, C2×Dic7, C7⋊D4, C22.M4(2), C2×C7⋊C8, C4.Dic7, C23.D7, C28.55D4, C23⋊Dic7, C28.10D4, (C2×C28)⋊C8

Smallest permutation representation of (C2×C28)⋊C8
On 224 points
Generators in S224
(1 135)(2 136)(3 137)(4 138)(5 139)(6 140)(7 113)(8 114)(9 115)(10 116)(11 117)(12 118)(13 119)(14 120)(15 121)(16 122)(17 123)(18 124)(19 125)(20 126)(21 127)(22 128)(23 129)(24 130)(25 131)(26 132)(27 133)(28 134)(29 81)(30 82)(31 83)(32 84)(33 57)(34 58)(35 59)(36 60)(37 61)(38 62)(39 63)(40 64)(41 65)(42 66)(43 67)(44 68)(45 69)(46 70)(47 71)(48 72)(49 73)(50 74)(51 75)(52 76)(53 77)(54 78)(55 79)(56 80)(85 219)(86 220)(87 221)(88 222)(89 223)(90 224)(91 197)(92 198)(93 199)(94 200)(95 201)(96 202)(97 203)(98 204)(99 205)(100 206)(101 207)(102 208)(103 209)(104 210)(105 211)(106 212)(107 213)(108 214)(109 215)(110 216)(111 217)(112 218)(141 191)(142 192)(143 193)(144 194)(145 195)(146 196)(147 169)(148 170)(149 171)(150 172)(151 173)(152 174)(153 175)(154 176)(155 177)(156 178)(157 179)(158 180)(159 181)(160 182)(161 183)(162 184)(163 185)(164 186)(165 187)(166 188)(167 189)(168 190)
(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 66 222 194 135 56 88 158)(2 55 209 143 136 65 103 179)(3 64 224 192 137 54 90 156)(4 53 211 141 138 63 105 177)(5 62 198 190 139 52 92 154)(6 51 213 167 140 61 107 175)(7 60 200 188 113 50 94 152)(8 49 215 165 114 59 109 173)(9 58 202 186 115 48 96 150)(10 47 217 163 116 57 111 171)(11 84 204 184 117 46 98 148)(12 45 219 161 118 83 85 169)(13 82 206 182 119 44 100 146)(14 43 221 159 120 81 87 195)(15 80 208 180 121 42 102 144)(16 41 223 157 122 79 89 193)(17 78 210 178 123 40 104 142)(18 39 197 155 124 77 91 191)(19 76 212 176 125 38 106 168)(20 37 199 153 126 75 93 189)(21 74 214 174 127 36 108 166)(22 35 201 151 128 73 95 187)(23 72 216 172 129 34 110 164)(24 33 203 149 130 71 97 185)(25 70 218 170 131 32 112 162)(26 31 205 147 132 69 99 183)(27 68 220 196 133 30 86 160)(28 29 207 145 134 67 101 181)

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

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

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

82 conjugacy classes

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

82 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + - + + - image C1 C2 C2 C4 C8 D4 D7 M4(2) Dic7 D14 C7⋊C8 C7⋊D4 C4.Dic7 C23⋊C4 C4.10D4 C23⋊Dic7 C28.10D4 kernel (C2×C28)⋊C8 C28.55D4 C14×C4⋊C4 C22×C28 C2×C28 C2×C28 C2×C4⋊C4 C2×C14 C22×C4 C22×C4 C2×C4 C2×C4 C22 C14 C14 C2 C2 # reps 1 2 1 4 8 2 3 2 6 3 12 12 12 1 1 6 6

Matrix representation of (C2×C28)⋊C8 in GL6(𝔽113)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 112 0 0 0 0 0 0 112
,
 101 16 0 0 0 0 16 101 0 0 0 0 0 0 47 17 0 0 0 0 96 66 0 0 0 0 0 0 47 17 0 0 0 0 96 66
,
 62 103 0 0 0 0 10 51 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 1 0 0 0

G:=sub<GL(6,GF(113))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,112,0,0,0,0,0,0,112],[101,16,0,0,0,0,16,101,0,0,0,0,0,0,47,96,0,0,0,0,17,66,0,0,0,0,0,0,47,96,0,0,0,0,17,66],[62,10,0,0,0,0,103,51,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0] >;

(C2×C28)⋊C8 in GAP, Magma, Sage, TeX

(C_2\times C_{28})\rtimes C_8
% in TeX

G:=Group("(C2xC28):C8");
// GroupNames label

G:=SmallGroup(448,85);
// by ID

G=gap.SmallGroup(448,85);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,28,141,232,387,100,1123,18822]);
// Polycyclic

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

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