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G = D28.4C8order 448 = 26·7

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

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D28.4C8, C16.18D14, C56.65C23, Dic14.4C8, C112.28C22, (C2×C16)⋊9D7, (D7×C16)⋊6C2, C71(D4○C16), C8.23(C4×D7), C4.10(C8×D7), C7⋊D4.2C8, (C2×C112)⋊16C2, C16⋊D77C2, C28.20(C2×C8), C56.62(C2×C4), C4○D28.6C4, D14.1(C2×C8), C8⋊D7.3C4, C22.2(C8×D7), (C2×C8).325D14, C7⋊C16.11C22, Dic7.3(C2×C8), C28.C815C2, C8.59(C22×D7), C4.Dic7.7C4, C14.14(C22×C8), (C8×D7).17C22, C28.130(C22×C4), (C2×C56).410C22, D28.2C4.6C2, C2.15(D7×C2×C8), C7⋊C8.13(C2×C4), C4.104(C2×C4×D7), (C2×C14).16(C2×C8), (C4×D7).21(C2×C4), (C2×C4).105(C4×D7), (C2×C28).232(C2×C4), SmallGroup(448,435)

Series: Derived Chief Lower central Upper central

C1C14 — D28.4C8
C1C7C14C28C56C8×D7D28.2C4 — D28.4C8
C7C14 — D28.4C8
C1C16C2×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 [×3], C4 [×2], C4 [×2], C22, C22 [×2], C7, C8 [×2], C8 [×2], C2×C4, C2×C4 [×2], D4 [×3], Q8, D7 [×2], C14, C14, C16 [×2], C16 [×2], C2×C8, C2×C8 [×2], M4(2) [×3], C4○D4, Dic7 [×2], C28 [×2], D14 [×2], C2×C14, C2×C16, C2×C16 [×2], M5(2) [×3], C8○D4, C7⋊C8 [×2], C56 [×2], Dic14, C4×D7 [×2], D28, C7⋊D4 [×2], C2×C28, D4○C16, C7⋊C16 [×2], C112 [×2], C8×D7 [×2], C8⋊D7 [×2], C4.Dic7, C2×C56, C4○D28, D7×C16 [×2], C16⋊D7 [×2], C28.C8, C2×C112, D28.2C4, D28.4C8
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C8 [×4], C2×C4 [×6], C23, D7, C2×C8 [×6], C22×C4, D14 [×3], C22×C8, C4×D7 [×2], C22×D7, D4○C16, C8×D7 [×2], 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 56)(30 55)(31 54)(32 53)(33 52)(34 51)(35 50)(36 49)(37 48)(38 47)(39 46)(40 45)(41 44)(42 43)(57 84)(58 83)(59 82)(60 81)(61 80)(62 79)(63 78)(64 77)(65 76)(66 75)(67 74)(68 73)(69 72)(70 71)(85 92)(86 91)(87 90)(88 89)(93 112)(94 111)(95 110)(96 109)(97 108)(98 107)(99 106)(100 105)(101 104)(102 103)(113 126)(114 125)(115 124)(116 123)(117 122)(118 121)(119 120)(127 140)(128 139)(129 138)(130 137)(131 136)(132 135)(133 134)(141 142)(143 168)(144 167)(145 166)(146 165)(147 164)(148 163)(149 162)(150 161)(151 160)(152 159)(153 158)(154 157)(155 156)(169 192)(170 191)(171 190)(172 189)(173 188)(174 187)(175 186)(176 185)(177 184)(178 183)(179 182)(180 181)(193 196)(194 195)(197 200)(198 199)(201 224)(202 223)(203 222)(204 221)(205 220)(206 219)(207 218)(208 217)(209 216)(210 215)(211 214)(212 213)
(1 199 103 134 57 195 156 29 15 213 89 120 71 181 142 43)(2 200 104 135 58 196 157 30 16 214 90 121 72 182 143 44)(3 201 105 136 59 169 158 31 17 215 91 122 73 183 144 45)(4 202 106 137 60 170 159 32 18 216 92 123 74 184 145 46)(5 203 107 138 61 171 160 33 19 217 93 124 75 185 146 47)(6 204 108 139 62 172 161 34 20 218 94 125 76 186 147 48)(7 205 109 140 63 173 162 35 21 219 95 126 77 187 148 49)(8 206 110 113 64 174 163 36 22 220 96 127 78 188 149 50)(9 207 111 114 65 175 164 37 23 221 97 128 79 189 150 51)(10 208 112 115 66 176 165 38 24 222 98 129 80 190 151 52)(11 209 85 116 67 177 166 39 25 223 99 130 81 191 152 53)(12 210 86 117 68 178 167 40 26 224 100 131 82 192 153 54)(13 211 87 118 69 179 168 41 27 197 101 132 83 193 154 55)(14 212 88 119 70 180 141 42 28 198 102 133 84 194 155 56)

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

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

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

136 conjugacy classes

class 1 2A2B2C2D4A4B4C4D4E7A7B7C8A8B8C8D8E8F8G8H8I8J14A···14I16A···16H16I16J16K16L16M···16T28A···28L56A···56X112A···112AV
order1222244444777888888888814···1416···161616161616···1628···2856···56112···112
size11214141121414222111122141414142···21···1222214···142···22···22···2

136 irreducible representations

dim111111111111222222222
type+++++++++
imageC1C2C2C2C2C2C4C4C4C8C8C8D7D14D14C4×D7C4×D7D4○C16C8×D7C8×D7D28.4C8
kernelD28.4C8D7×C16C16⋊D7C28.C8C2×C112D28.2C4C8⋊D7C4.Dic7C4○D28Dic14D28C7⋊D4C2×C16C16C2×C8C8C2×C4C7C4C22C1
# reps122111422448363668121248

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

10413
10094
,
9494
1319
,
400
040
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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