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## G = D28.4C8order 448 = 26·7

### The non-split extension by D28 of C8 acting through Inn(D28)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C14 — D28.4C8
 Chief series C1 — C7 — C14 — C28 — C56 — C8×D7 — D28.2C4 — D28.4C8
 Lower central C7 — C14 — D28.4C8
 Upper central C1 — C16 — C2×C16

Generators and relations for D28.4C8
G = < a,b,c | a28=b2=1, c8=a14, bab=a-1, ac=ca, bc=cb >

Subgroups: 276 in 84 conjugacy classes, 51 normal (31 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C8, C2×C4, C2×C4, D4, Q8, D7, C14, C14, C16, C16, C2×C8, C2×C8, M4(2), C4○D4, Dic7, C28, D14, C2×C14, C2×C16, C2×C16, M5(2), C8○D4, C7⋊C8, C56, Dic14, C4×D7, D28, C7⋊D4, C2×C28, D4○C16, C7⋊C16, C112, C8×D7, C8⋊D7, C4.Dic7, C2×C56, C4○D28, D7×C16, C16⋊D7, C28.C8, C2×C112, D28.2C4, D28.4C8
Quotients: C1, C2, C4, C22, C8, C2×C4, C23, D7, C2×C8, C22×C4, D14, C22×C8, C4×D7, C22×D7, D4○C16, C8×D7, C2×C4×D7, D7×C2×C8, D28.4C8

Smallest permutation representation of D28.4C8
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 28)(2 27)(3 26)(4 25)(5 24)(6 23)(7 22)(8 21)(9 20)(10 19)(11 18)(12 17)(13 16)(14 15)(29 54)(30 53)(31 52)(32 51)(33 50)(34 49)(35 48)(36 47)(37 46)(38 45)(39 44)(40 43)(41 42)(55 56)(57 72)(58 71)(59 70)(60 69)(61 68)(62 67)(63 66)(64 65)(73 84)(74 83)(75 82)(76 81)(77 80)(78 79)(85 102)(86 101)(87 100)(88 99)(89 98)(90 97)(91 96)(92 95)(93 94)(103 112)(104 111)(105 110)(106 109)(107 108)(113 114)(115 140)(116 139)(117 138)(118 137)(119 136)(120 135)(121 134)(122 133)(123 132)(124 131)(125 130)(126 129)(127 128)(141 156)(142 155)(143 154)(144 153)(145 152)(146 151)(147 150)(148 149)(157 168)(158 167)(159 166)(160 165)(161 164)(162 163)(169 186)(170 185)(171 184)(172 183)(173 182)(174 181)(175 180)(176 179)(177 178)(187 196)(188 195)(189 194)(190 193)(191 192)(197 220)(198 219)(199 218)(200 217)(201 216)(202 215)(203 214)(204 213)(205 212)(206 211)(207 210)(208 209)(221 224)(222 223)
(1 114 209 79 163 192 56 94 15 128 223 65 149 178 42 108)(2 115 210 80 164 193 29 95 16 129 224 66 150 179 43 109)(3 116 211 81 165 194 30 96 17 130 197 67 151 180 44 110)(4 117 212 82 166 195 31 97 18 131 198 68 152 181 45 111)(5 118 213 83 167 196 32 98 19 132 199 69 153 182 46 112)(6 119 214 84 168 169 33 99 20 133 200 70 154 183 47 85)(7 120 215 57 141 170 34 100 21 134 201 71 155 184 48 86)(8 121 216 58 142 171 35 101 22 135 202 72 156 185 49 87)(9 122 217 59 143 172 36 102 23 136 203 73 157 186 50 88)(10 123 218 60 144 173 37 103 24 137 204 74 158 187 51 89)(11 124 219 61 145 174 38 104 25 138 205 75 159 188 52 90)(12 125 220 62 146 175 39 105 26 139 206 76 160 189 53 91)(13 126 221 63 147 176 40 106 27 140 207 77 161 190 54 92)(14 127 222 64 148 177 41 107 28 113 208 78 162 191 55 93)```

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

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

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

136 conjugacy classes

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 4E 7A 7B 7C 8A 8B 8C 8D 8E 8F 8G 8H 8I 8J 14A ··· 14I 16A ··· 16H 16I 16J 16K 16L 16M ··· 16T 28A ··· 28L 56A ··· 56X 112A ··· 112AV order 1 2 2 2 2 4 4 4 4 4 7 7 7 8 8 8 8 8 8 8 8 8 8 14 ··· 14 16 ··· 16 16 16 16 16 16 ··· 16 28 ··· 28 56 ··· 56 112 ··· 112 size 1 1 2 14 14 1 1 2 14 14 2 2 2 1 1 1 1 2 2 14 14 14 14 2 ··· 2 1 ··· 1 2 2 2 2 14 ··· 14 2 ··· 2 2 ··· 2 2 ··· 2

136 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 type + + + + + + + + + image C1 C2 C2 C2 C2 C2 C4 C4 C4 C8 C8 C8 D7 D14 D14 C4×D7 C4×D7 D4○C16 C8×D7 C8×D7 D28.4C8 kernel D28.4C8 D7×C16 C16⋊D7 C28.C8 C2×C112 D28.2C4 C8⋊D7 C4.Dic7 C4○D28 Dic14 D28 C7⋊D4 C2×C16 C16 C2×C8 C8 C2×C4 C7 C4 C22 C1 # reps 1 2 2 1 1 1 4 2 2 4 4 8 3 6 3 6 6 8 12 12 48

Matrix representation of D28.4C8 in GL2(𝔽113) generated by

 104 13 100 94
,
 94 94 13 19
,
 40 0 0 40
`G:=sub<GL(2,GF(113))| [104,100,13,94],[94,13,94,19],[40,0,0,40] >;`

D28.4C8 in GAP, Magma, Sage, TeX

`D_{28}._4C_8`
`% in TeX`

`G:=Group("D28.4C8");`
`// GroupNames label`

`G:=SmallGroup(448,435);`
`// by ID`

`G=gap.SmallGroup(448,435);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,477,58,80,102,18822]);`
`// Polycyclic`

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

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