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G = C282Q8order 224 = 25·7

1st semidirect product of C28 and Q8 acting via Q8/C4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C282Q8, C4.4D28, C42Dic14, C28.27D4, C42.4D7, C71(C4⋊Q8), (C4×C28).2C2, C14.1(C2×D4), C2.4(C2×D28), C14.2(C2×Q8), (C2×C4).73D14, C4⋊Dic7.4C2, C2.4(C2×Dic14), (C2×C14).10C23, (C2×C28).85C22, (C2×Dic14).2C2, (C2×Dic7).1C22, C22.34(C22×D7), SmallGroup(224,64)

Series: Derived Chief Lower central Upper central

C1C2×C14 — C282Q8
C1C7C14C2×C14C2×Dic7C2×Dic14 — C282Q8
C7C2×C14 — C282Q8
C1C22C42

Generators and relations for C282Q8
 G = < a,b,c | a28=b4=1, c2=b2, ab=ba, cac-1=a-1, cbc-1=b-1 >

Subgroups: 246 in 68 conjugacy classes, 41 normal (11 characteristic)
C1, C2, C2, C4, C4, C22, C7, C2×C4, C2×C4, C2×C4, Q8, C14, C14, C42, C4⋊C4, C2×Q8, Dic7, C28, C2×C14, C4⋊Q8, Dic14, C2×Dic7, C2×C28, C2×C28, C4⋊Dic7, C4×C28, C2×Dic14, C282Q8
Quotients: C1, C2, C22, D4, Q8, C23, D7, C2×D4, C2×Q8, D14, C4⋊Q8, Dic14, D28, C22×D7, C2×Dic14, C2×D28, C282Q8

Smallest permutation representation of C282Q8
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)
(1 155 98 52)(2 156 99 53)(3 157 100 54)(4 158 101 55)(5 159 102 56)(6 160 103 29)(7 161 104 30)(8 162 105 31)(9 163 106 32)(10 164 107 33)(11 165 108 34)(12 166 109 35)(13 167 110 36)(14 168 111 37)(15 141 112 38)(16 142 85 39)(17 143 86 40)(18 144 87 41)(19 145 88 42)(20 146 89 43)(21 147 90 44)(22 148 91 45)(23 149 92 46)(24 150 93 47)(25 151 94 48)(26 152 95 49)(27 153 96 50)(28 154 97 51)(57 173 140 214)(58 174 113 215)(59 175 114 216)(60 176 115 217)(61 177 116 218)(62 178 117 219)(63 179 118 220)(64 180 119 221)(65 181 120 222)(66 182 121 223)(67 183 122 224)(68 184 123 197)(69 185 124 198)(70 186 125 199)(71 187 126 200)(72 188 127 201)(73 189 128 202)(74 190 129 203)(75 191 130 204)(76 192 131 205)(77 193 132 206)(78 194 133 207)(79 195 134 208)(80 196 135 209)(81 169 136 210)(82 170 137 211)(83 171 138 212)(84 172 139 213)
(1 180 98 221)(2 179 99 220)(3 178 100 219)(4 177 101 218)(5 176 102 217)(6 175 103 216)(7 174 104 215)(8 173 105 214)(9 172 106 213)(10 171 107 212)(11 170 108 211)(12 169 109 210)(13 196 110 209)(14 195 111 208)(15 194 112 207)(16 193 85 206)(17 192 86 205)(18 191 87 204)(19 190 88 203)(20 189 89 202)(21 188 90 201)(22 187 91 200)(23 186 92 199)(24 185 93 198)(25 184 94 197)(26 183 95 224)(27 182 96 223)(28 181 97 222)(29 114 160 59)(30 113 161 58)(31 140 162 57)(32 139 163 84)(33 138 164 83)(34 137 165 82)(35 136 166 81)(36 135 167 80)(37 134 168 79)(38 133 141 78)(39 132 142 77)(40 131 143 76)(41 130 144 75)(42 129 145 74)(43 128 146 73)(44 127 147 72)(45 126 148 71)(46 125 149 70)(47 124 150 69)(48 123 151 68)(49 122 152 67)(50 121 153 66)(51 120 154 65)(52 119 155 64)(53 118 156 63)(54 117 157 62)(55 116 158 61)(56 115 159 60)

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

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

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

C282Q8 is a maximal subgroup of
C4.Dic28  C28.47D8  C28.2D8  C569Q8  C28.14Q16  C568Q8  C85D28  C4.5D56  C284Q16  C8⋊Dic14  C42.14D14  C42.20D14  C8.D28  D4.9D28  C28⋊SD16  D283Q8  D284Q8  C4⋊Dic28  C28.7Q16  Dic144Q8  C28.50D8  C28.38SD16  D4.2D28  C28.48SD16  C28.23Q16  C287Q16  C42.62D14  C42.65D14  C42.68D14  C42.71D14  C28.16D8  C284SD16  C28.17D8  C28.SD16  C28.Q16  C283Q16  C28.11Q16  C42.274D14  C42.276D14  C42.89D14  C42.90D14  C42.92D14  C42.99D14  D4×Dic14  C42.106D14  D46Dic14  D2824D4  D46D28  C42.117D14  Q8×Dic14  Dic1410Q8  Q86Dic14  Q8×D28  D2810Q8  C42.135D14  C42.141D14  C42.144D14  C42.148D14  C42.155D14  C42.165D14  C42.238D14  D7×C4⋊Q8  C42.241D14
C282Q8 is a maximal quotient of
(C2×Dic7)⋊Q8  C14.(C4⋊Q8)  C569Q8  C568Q8  C56.13Q8  C8⋊Dic14  C284(C4⋊C4)  (C2×C28)⋊10Q8  C428Dic7

62 conjugacy classes

class 1 2A2B2C4A···4F4G4H4I4J7A7B7C14A···14I28A···28AJ
order12224···4444477714···1428···28
size11112···2282828282222···22···2

62 irreducible representations

dim1111222222
type+++++-++-+
imageC1C2C2C2D4Q8D7D14Dic14D28
kernelC282Q8C4⋊Dic7C4×C28C2×Dic14C28C28C42C2×C4C4C4
# reps141224392412

Matrix representation of C282Q8 in GL4(𝔽29) generated by

28300
26800
002724
00512
,
21800
112700
0010
0001
,
22900
17700
00182
002711
G:=sub<GL(4,GF(29))| [28,26,0,0,3,8,0,0,0,0,27,5,0,0,24,12],[2,11,0,0,18,27,0,0,0,0,1,0,0,0,0,1],[22,17,0,0,9,7,0,0,0,0,18,27,0,0,2,11] >;

C282Q8 in GAP, Magma, Sage, TeX

C_{28}\rtimes_2Q_8
% in TeX

G:=Group("C28:2Q8");
// GroupNames label

G:=SmallGroup(224,64);
// by ID

G=gap.SmallGroup(224,64);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-7,96,217,103,218,50,6917]);
// Polycyclic

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

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