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## G = C28⋊Q8order 224 = 25·7

### The semidirect product of C28 and Q8 acting via Q8/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C14 — C28⋊Q8
 Chief series C1 — C7 — C14 — C2×C14 — C2×Dic7 — C4×Dic7 — C28⋊Q8
 Lower central C7 — C2×C14 — C28⋊Q8
 Upper central C1 — C22 — C4⋊C4

Generators and relations for C28⋊Q8
G = < a,b,c | a28=b4=1, c2=b2, bab-1=a15, cac-1=a13, cbc-1=b-1 >

Subgroups: 246 in 68 conjugacy classes, 37 normal (19 characteristic)
C1, C2, C4, C4, C22, C7, C2×C4, C2×C4, C2×C4, Q8, C14, C42, C4⋊C4, C4⋊C4, C2×Q8, Dic7, Dic7, C28, C28, C2×C14, C4⋊Q8, Dic14, C2×Dic7, C2×Dic7, C2×C28, C2×C28, C4×Dic7, Dic7⋊C4, C4⋊Dic7, C7×C4⋊C4, C2×Dic14, C28⋊Q8
Quotients: C1, C2, C22, D4, Q8, C23, D7, C2×D4, C2×Q8, D14, C4⋊Q8, Dic14, C22×D7, C2×Dic14, D4×D7, Q8×D7, C28⋊Q8

Smallest permutation representation of C28⋊Q8
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)
(1 118 205 59)(2 133 206 74)(3 120 207 61)(4 135 208 76)(5 122 209 63)(6 137 210 78)(7 124 211 65)(8 139 212 80)(9 126 213 67)(10 113 214 82)(11 128 215 69)(12 115 216 84)(13 130 217 71)(14 117 218 58)(15 132 219 73)(16 119 220 60)(17 134 221 75)(18 121 222 62)(19 136 223 77)(20 123 224 64)(21 138 197 79)(22 125 198 66)(23 140 199 81)(24 127 200 68)(25 114 201 83)(26 129 202 70)(27 116 203 57)(28 131 204 72)(29 191 97 159)(30 178 98 146)(31 193 99 161)(32 180 100 148)(33 195 101 163)(34 182 102 150)(35 169 103 165)(36 184 104 152)(37 171 105 167)(38 186 106 154)(39 173 107 141)(40 188 108 156)(41 175 109 143)(42 190 110 158)(43 177 111 145)(44 192 112 160)(45 179 85 147)(46 194 86 162)(47 181 87 149)(48 196 88 164)(49 183 89 151)(50 170 90 166)(51 185 91 153)(52 172 92 168)(53 187 93 155)(54 174 94 142)(55 189 95 157)(56 176 96 144)
(1 144 205 176)(2 157 206 189)(3 142 207 174)(4 155 208 187)(5 168 209 172)(6 153 210 185)(7 166 211 170)(8 151 212 183)(9 164 213 196)(10 149 214 181)(11 162 215 194)(12 147 216 179)(13 160 217 192)(14 145 218 177)(15 158 219 190)(16 143 220 175)(17 156 221 188)(18 141 222 173)(19 154 223 186)(20 167 224 171)(21 152 197 184)(22 165 198 169)(23 150 199 182)(24 163 200 195)(25 148 201 180)(26 161 202 193)(27 146 203 178)(28 159 204 191)(29 131 97 72)(30 116 98 57)(31 129 99 70)(32 114 100 83)(33 127 101 68)(34 140 102 81)(35 125 103 66)(36 138 104 79)(37 123 105 64)(38 136 106 77)(39 121 107 62)(40 134 108 75)(41 119 109 60)(42 132 110 73)(43 117 111 58)(44 130 112 71)(45 115 85 84)(46 128 86 69)(47 113 87 82)(48 126 88 67)(49 139 89 80)(50 124 90 65)(51 137 91 78)(52 122 92 63)(53 135 93 76)(54 120 94 61)(55 133 95 74)(56 118 96 59)```

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

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

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

44 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 7A 7B 7C 14A ··· 14I 28A ··· 28R order 1 2 2 2 4 4 4 4 4 4 4 4 4 4 7 7 7 14 ··· 14 28 ··· 28 size 1 1 1 1 2 2 4 4 14 14 14 14 28 28 2 2 2 2 ··· 2 4 ··· 4

44 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 4 4 type + + + + + + + - - + + - + - image C1 C2 C2 C2 C2 C2 D4 Q8 Q8 D7 D14 Dic14 D4×D7 Q8×D7 kernel C28⋊Q8 C4×Dic7 Dic7⋊C4 C4⋊Dic7 C7×C4⋊C4 C2×Dic14 Dic7 Dic7 C28 C4⋊C4 C2×C4 C4 C2 C2 # reps 1 1 2 1 1 2 2 2 2 3 9 12 3 3

Matrix representation of C28⋊Q8 in GL6(𝔽29)

 27 13 0 0 0 0 13 2 0 0 0 0 0 0 28 0 0 0 0 0 0 28 0 0 0 0 0 0 25 3 0 0 0 0 14 11
,
 0 1 0 0 0 0 28 0 0 0 0 0 0 0 0 1 0 0 0 0 28 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 2 16 0 0 0 0 16 27 0 0 0 0 0 0 27 13 0 0 0 0 13 2 0 0 0 0 0 0 28 1 0 0 0 0 0 1

`G:=sub<GL(6,GF(29))| [27,13,0,0,0,0,13,2,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,25,14,0,0,0,0,3,11],[0,28,0,0,0,0,1,0,0,0,0,0,0,0,0,28,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[2,16,0,0,0,0,16,27,0,0,0,0,0,0,27,13,0,0,0,0,13,2,0,0,0,0,0,0,28,0,0,0,0,0,1,1] >;`

C28⋊Q8 in GAP, Magma, Sage, TeX

`C_{28}\rtimes Q_8`
`% in TeX`

`G:=Group("C28:Q8");`
`// GroupNames label`

`G:=SmallGroup(224,83);`
`// by ID`

`G=gap.SmallGroup(224,83);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-7,48,103,218,188,50,6917]);`
`// Polycyclic`

`G:=Group<a,b,c|a^28=b^4=1,c^2=b^2,b*a*b^-1=a^15,c*a*c^-1=a^13,c*b*c^-1=b^-1>;`
`// generators/relations`

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