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## G = C28.6Q8order 224 = 25·7

### 3rd non-split extension by C28 of Q8 acting via Q8/C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C14 — C28.6Q8
 Chief series C1 — C7 — C14 — C2×C14 — C2×Dic7 — Dic7⋊C4 — C28.6Q8
 Lower central C7 — C2×C14 — C28.6Q8
 Upper central C1 — C22 — C42

Generators and relations for C28.6Q8
G = < a,b,c | a28=b4=1, c2=a14b2, ab=ba, cac-1=a-1, cbc-1=a14b-1 >

Subgroups: 182 in 56 conjugacy classes, 33 normal (11 characteristic)
C1, C2, C2, C4, C4, C22, C7, C2×C4, C2×C4, C2×C4, C14, C14, C42, C4⋊C4, Dic7, C28, C28, C2×C14, C42.C2, C2×Dic7, C2×C28, C2×C28, Dic7⋊C4, C4⋊Dic7, C4×C28, C28.6Q8
Quotients: C1, C2, C22, Q8, C23, D7, C2×Q8, C4○D4, D14, C42.C2, Dic14, C22×D7, C2×Dic14, C4○D28, C28.6Q8

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

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

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

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

62 conjugacy classes

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

62 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 type + + + + - + + - image C1 C2 C2 C2 Q8 D7 C4○D4 D14 Dic14 C4○D28 kernel C28.6Q8 Dic7⋊C4 C4⋊Dic7 C4×C28 C28 C42 C14 C2×C4 C4 C2 # reps 1 4 2 1 2 3 4 9 12 24

Matrix representation of C28.6Q8 in GL4(𝔽29) generated by

 0 17 0 0 17 0 0 0 0 0 4 0 0 0 11 22
,
 0 28 0 0 28 0 0 0 0 0 12 0 0 0 24 17
,
 14 20 0 0 9 15 0 0 0 0 6 23 0 0 11 23
`G:=sub<GL(4,GF(29))| [0,17,0,0,17,0,0,0,0,0,4,11,0,0,0,22],[0,28,0,0,28,0,0,0,0,0,12,24,0,0,0,17],[14,9,0,0,20,15,0,0,0,0,6,11,0,0,23,23] >;`

C28.6Q8 in GAP, Magma, Sage, TeX

`C_{28}._6Q_8`
`% in TeX`

`G:=Group("C28.6Q8");`
`// GroupNames label`

`G:=SmallGroup(224,65);`
`// by ID`

`G=gap.SmallGroup(224,65);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-7,96,217,55,218,86,6917]);`
`// Polycyclic`

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

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