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## G = C28.4C42order 448 = 26·7

### 4th non-split extension by C28 of C42 acting via C42/C22=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C28 — C28.4C42
 Chief series C1 — C7 — C14 — C28 — C2×C28 — C22×C28 — C22×C7⋊C8 — C28.4C42
 Lower central C7 — C14 — C28 — C28.4C42
 Upper central C1 — C2×C4 — C22×C4 — C2×M4(2)

Generators and relations for C28.4C42
G = < a,b,c | a28=1, b4=c4=a14, bab-1=a13, cac-1=a15, cbc-1=a7b >

Subgroups: 260 in 90 conjugacy classes, 51 normal (39 characteristic)
C1, C2, C2, C2, C4, C22, C22, C7, C8, C2×C4, C23, C14, C14, C14, C2×C8, M4(2), M4(2), C22×C4, C28, C2×C14, C2×C14, C22×C8, C2×M4(2), C2×M4(2), C7⋊C8, C56, C2×C28, C22×C14, C4.C42, C2×C7⋊C8, C2×C7⋊C8, C4.Dic7, C4.Dic7, C2×C56, C7×M4(2), C7×M4(2), C22×C28, C22×C7⋊C8, C2×C4.Dic7, C14×M4(2), C28.4C42
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, D7, C42, C22⋊C4, C4⋊C4, Dic7, D14, C2.C42, C8.C4, Dic14, C4×D7, D28, C2×Dic7, C7⋊D4, C4.C42, C4×Dic7, Dic7⋊C4, C4⋊Dic7, D14⋊C4, C23.D7, C28.53D4, C14.C42, C28.4C42

Smallest permutation representation of C28.4C42
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)
(1 175 164 132 15 189 150 118)(2 188 165 117 16 174 151 131)(3 173 166 130 17 187 152 116)(4 186 167 115 18 172 153 129)(5 171 168 128 19 185 154 114)(6 184 141 113 20 170 155 127)(7 169 142 126 21 183 156 140)(8 182 143 139 22 196 157 125)(9 195 144 124 23 181 158 138)(10 180 145 137 24 194 159 123)(11 193 146 122 25 179 160 136)(12 178 147 135 26 192 161 121)(13 191 148 120 27 177 162 134)(14 176 149 133 28 190 163 119)(29 105 215 68 43 91 201 82)(30 90 216 81 44 104 202 67)(31 103 217 66 45 89 203 80)(32 88 218 79 46 102 204 65)(33 101 219 64 47 87 205 78)(34 86 220 77 48 100 206 63)(35 99 221 62 49 85 207 76)(36 112 222 75 50 98 208 61)(37 97 223 60 51 111 209 74)(38 110 224 73 52 96 210 59)(39 95 197 58 53 109 211 72)(40 108 198 71 54 94 212 57)(41 93 199 84 55 107 213 70)(42 106 200 69 56 92 214 83)
(1 64 150 87 15 78 164 101)(2 79 151 102 16 65 165 88)(3 66 152 89 17 80 166 103)(4 81 153 104 18 67 167 90)(5 68 154 91 19 82 168 105)(6 83 155 106 20 69 141 92)(7 70 156 93 21 84 142 107)(8 57 157 108 22 71 143 94)(9 72 158 95 23 58 144 109)(10 59 159 110 24 73 145 96)(11 74 160 97 25 60 146 111)(12 61 161 112 26 75 147 98)(13 76 162 99 27 62 148 85)(14 63 163 86 28 77 149 100)(29 135 215 178 43 121 201 192)(30 122 216 193 44 136 202 179)(31 137 217 180 45 123 203 194)(32 124 218 195 46 138 204 181)(33 139 219 182 47 125 205 196)(34 126 220 169 48 140 206 183)(35 113 221 184 49 127 207 170)(36 128 222 171 50 114 208 185)(37 115 223 186 51 129 209 172)(38 130 224 173 52 116 210 187)(39 117 197 188 53 131 211 174)(40 132 198 175 54 118 212 189)(41 119 199 190 55 133 213 176)(42 134 200 177 56 120 214 191)```

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

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

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

88 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 7A 7B 7C 8A 8B 8C 8D 8E ··· 8L 8M 8N 8O 8P 14A ··· 14I 14J ··· 14O 28A ··· 28L 28M ··· 28R 56A ··· 56X order 1 2 2 2 2 2 4 4 4 4 4 4 7 7 7 8 8 8 8 8 ··· 8 8 8 8 8 14 ··· 14 14 ··· 14 28 ··· 28 28 ··· 28 56 ··· 56 size 1 1 1 1 2 2 1 1 1 1 2 2 2 2 2 4 4 4 4 14 ··· 14 28 28 28 28 2 ··· 2 4 ··· 4 2 ··· 2 4 ··· 4 4 ··· 4

88 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 type + + + + + - + - + + - image C1 C2 C2 C2 C4 C4 C4 D4 Q8 D7 Dic7 D14 C8.C4 C4×D7 D28 C7⋊D4 Dic14 C28.53D4 kernel C28.4C42 C22×C7⋊C8 C2×C4.Dic7 C14×M4(2) C2×C7⋊C8 C4.Dic7 C7×M4(2) C2×C28 C22×C14 C2×M4(2) M4(2) C22×C4 C14 C2×C4 C2×C4 C2×C4 C23 C2 # reps 1 1 1 1 4 4 4 3 1 3 6 3 8 12 6 12 6 12

Matrix representation of C28.4C42 in GL5(𝔽113)

 1 0 0 0 0 0 103 9 0 0 0 14 89 0 0 0 0 0 15 0 0 0 0 112 98
,
 15 0 0 0 0 0 89 104 0 0 0 89 24 0 0 0 0 0 18 0 0 0 0 13 69
,
 112 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 15 111 0 0 0 105 98

`G:=sub<GL(5,GF(113))| [1,0,0,0,0,0,103,14,0,0,0,9,89,0,0,0,0,0,15,112,0,0,0,0,98],[15,0,0,0,0,0,89,89,0,0,0,104,24,0,0,0,0,0,18,13,0,0,0,0,69],[112,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,15,105,0,0,0,111,98] >;`

C28.4C42 in GAP, Magma, Sage, TeX

`C_{28}._4C_4^2`
`% in TeX`

`G:=Group("C28.4C4^2");`
`// GroupNames label`

`G:=SmallGroup(448,115);`
`// by ID`

`G=gap.SmallGroup(448,115);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,56,365,36,184,570,136,18822]);`
`// Polycyclic`

`G:=Group<a,b,c|a^28=1,b^4=c^4=a^14,b*a*b^-1=a^13,c*a*c^-1=a^15,c*b*c^-1=a^7*b>;`
`// generators/relations`

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