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## G = C28.3Q8order 224 = 25·7

### 3rd non-split extension by C28 of Q8 acting via Q8/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C14 — C28.3Q8
 Chief series C1 — C7 — C14 — C2×C14 — C2×Dic7 — C4×Dic7 — C28.3Q8
 Lower central C7 — C2×C14 — C28.3Q8
 Upper central C1 — C22 — C4⋊C4

Generators and relations for C28.3Q8
G = < a,b,c | a4=b28=1, c2=b14, bab-1=a-1, ac=ca, cbc-1=a2b-1 >

Subgroups: 182 in 56 conjugacy classes, 33 normal (19 characteristic)
C1, C2, C4, C4, C22, C7, C2×C4, C2×C4, C2×C4, C14, C42, C4⋊C4, C4⋊C4, Dic7, C28, C28, C2×C14, C42.C2, C2×Dic7, C2×Dic7, C2×C28, C2×C28, C4×Dic7, Dic7⋊C4, C4⋊Dic7, C4⋊Dic7, C7×C4⋊C4, C28.3Q8
Quotients: C1, C2, C22, Q8, C23, D7, C2×Q8, C4○D4, D14, C42.C2, Dic14, C22×D7, C2×Dic14, D42D7, Q82D7, C28.3Q8

Smallest permutation representation of C28.3Q8
Regular action on 224 points
Generators in S224
```(1 107 46 188)(2 189 47 108)(3 109 48 190)(4 191 49 110)(5 111 50 192)(6 193 51 112)(7 85 52 194)(8 195 53 86)(9 87 54 196)(10 169 55 88)(11 89 56 170)(12 171 29 90)(13 91 30 172)(14 173 31 92)(15 93 32 174)(16 175 33 94)(17 95 34 176)(18 177 35 96)(19 97 36 178)(20 179 37 98)(21 99 38 180)(22 181 39 100)(23 101 40 182)(24 183 41 102)(25 103 42 184)(26 185 43 104)(27 105 44 186)(28 187 45 106)(57 114 164 207)(58 208 165 115)(59 116 166 209)(60 210 167 117)(61 118 168 211)(62 212 141 119)(63 120 142 213)(64 214 143 121)(65 122 144 215)(66 216 145 123)(67 124 146 217)(68 218 147 125)(69 126 148 219)(70 220 149 127)(71 128 150 221)(72 222 151 129)(73 130 152 223)(74 224 153 131)(75 132 154 197)(76 198 155 133)(77 134 156 199)(78 200 157 135)(79 136 158 201)(80 202 159 137)(81 138 160 203)(82 204 161 139)(83 140 162 205)(84 206 163 113)
(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 76 15 62)(2 154 16 168)(3 74 17 60)(4 152 18 166)(5 72 19 58)(6 150 20 164)(7 70 21 84)(8 148 22 162)(9 68 23 82)(10 146 24 160)(11 66 25 80)(12 144 26 158)(13 64 27 78)(14 142 28 156)(29 65 43 79)(30 143 44 157)(31 63 45 77)(32 141 46 155)(33 61 47 75)(34 167 48 153)(35 59 49 73)(36 165 50 151)(37 57 51 71)(38 163 52 149)(39 83 53 69)(40 161 54 147)(41 81 55 67)(42 159 56 145)(85 220 99 206)(86 126 100 140)(87 218 101 204)(88 124 102 138)(89 216 103 202)(90 122 104 136)(91 214 105 200)(92 120 106 134)(93 212 107 198)(94 118 108 132)(95 210 109 224)(96 116 110 130)(97 208 111 222)(98 114 112 128)(113 194 127 180)(115 192 129 178)(117 190 131 176)(119 188 133 174)(121 186 135 172)(123 184 137 170)(125 182 139 196)(169 217 183 203)(171 215 185 201)(173 213 187 199)(175 211 189 197)(177 209 191 223)(179 207 193 221)(181 205 195 219)```

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

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

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

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 2 2 2 2 2 4 4 type + + + + + - + + - - + image C1 C2 C2 C2 C2 Q8 D7 C4○D4 D14 Dic14 D4⋊2D7 Q8⋊2D7 kernel C28.3Q8 C4×Dic7 Dic7⋊C4 C4⋊Dic7 C7×C4⋊C4 C28 C4⋊C4 C14 C2×C4 C4 C2 C2 # reps 1 1 2 3 1 2 3 4 9 12 3 3

Matrix representation of C28.3Q8 in GL6(𝔽29)

 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 1 13 0 0 0 0 11 28
,
 25 1 0 0 0 0 27 22 0 0 0 0 0 0 18 13 0 0 0 0 4 11 0 0 0 0 0 0 1 13 0 0 0 0 0 28
,
 18 11 0 0 0 0 26 11 0 0 0 0 0 0 15 12 0 0 0 0 15 14 0 0 0 0 0 0 12 11 0 0 0 0 16 17

`G:=sub<GL(6,GF(29))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,11,0,0,0,0,13,28],[25,27,0,0,0,0,1,22,0,0,0,0,0,0,18,4,0,0,0,0,13,11,0,0,0,0,0,0,1,0,0,0,0,0,13,28],[18,26,0,0,0,0,11,11,0,0,0,0,0,0,15,15,0,0,0,0,12,14,0,0,0,0,0,0,12,16,0,0,0,0,11,17] >;`

C28.3Q8 in GAP, Magma, Sage, TeX

`C_{28}._3Q_8`
`% in TeX`

`G:=Group("C28.3Q8");`
`// GroupNames label`

`G:=SmallGroup(224,85);`
`// by ID`

`G=gap.SmallGroup(224,85);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-7,48,103,506,188,50,6917]);`
`// Polycyclic`

`G:=Group<a,b,c|a^4=b^28=1,c^2=b^14,b*a*b^-1=a^-1,a*c=c*a,c*b*c^-1=a^2*b^-1>;`
`// generators/relations`

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