direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C7×C8○2M4(2), M4(2)⋊5C28, C28.31C42, (C2×C8)⋊9C28, (C2×C56)⋊19C4, (C4×C56)⋊30C2, (C4×C8)⋊14C14, C4.5(C4×C28), C4⋊C4.10C28, C8.23(C2×C28), C56.88(C2×C4), C8⋊C4⋊13C14, C22⋊C4.6C28, C22.5(C4×C28), C14.44(C8○D4), (C7×M4(2))⋊11C4, (C22×C56).33C2, (C22×C8).15C14, (C2×C14).12C42, C23.16(C2×C28), C42.59(C2×C14), C4.34(C22×C28), C14.35(C2×C42), C28.192(C22×C4), (C4×C28).300C22, (C2×C56).453C22, (C2×C28).980C23, C42⋊C2.14C14, (C14×M4(2)).36C2, (C2×M4(2)).17C14, C22.18(C22×C28), (C22×C28).579C22, C2.7(C2×C4×C28), C2.1(C7×C8○D4), (C7×C4⋊C4).22C4, (C7×C8⋊C4)⋊27C2, (C2×C4).35(C2×C28), (C2×C8).100(C2×C14), (C2×C28).332(C2×C4), (C7×C22⋊C4).13C4, (C22×C14).83(C2×C4), (C2×C14).230(C22×C4), (C22×C4).109(C2×C14), (C7×C42⋊C2).28C2, (C2×C4).148(C22×C14), SmallGroup(448,813)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7×C8○2M4(2)
G = < a,b,c,d | a7=b8=d2=1, c4=b4, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b4c >
Subgroups: 146 in 130 conjugacy classes, 114 normal (30 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C7, C8, C2×C4, C2×C4, C23, C14, C14, C14, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), C22×C4, C28, C28, C28, C2×C14, C2×C14, C2×C14, C4×C8, C8⋊C4, C42⋊C2, C22×C8, C2×M4(2), C56, C2×C28, C2×C28, C22×C14, C8○2M4(2), C4×C28, C7×C22⋊C4, C7×C4⋊C4, C2×C56, C2×C56, C7×M4(2), C22×C28, C4×C56, C7×C8⋊C4, C7×C42⋊C2, C22×C56, C14×M4(2), C7×C8○2M4(2)
Quotients: C1, C2, C4, C22, C7, C2×C4, C23, C14, C42, C22×C4, C28, C2×C14, C2×C42, C8○D4, C2×C28, C22×C14, C8○2M4(2), C4×C28, C22×C28, C2×C4×C28, C7×C8○D4, C7×C8○2M4(2)
►On 224 points
Generators in S224
(1 151 38 143 30 135 22)(2 152 39 144 31 136 23)(3 145 40 137 32 129 24)(4 146 33 138 25 130 17)(5 147 34 139 26 131 18)(6 148 35 140 27 132 19)(7 149 36 141 28 133 20)(8 150 37 142 29 134 21)(9 123 62 167 54 159 46)(10 124 63 168 55 160 47)(11 125 64 161 56 153 48)(12 126 57 162 49 154 41)(13 127 58 163 50 155 42)(14 128 59 164 51 156 43)(15 121 60 165 52 157 44)(16 122 61 166 53 158 45)(65 170 194 81 186 73 178)(66 171 195 82 187 74 179)(67 172 196 83 188 75 180)(68 173 197 84 189 76 181)(69 174 198 85 190 77 182)(70 175 199 86 191 78 183)(71 176 200 87 192 79 184)(72 169 193 88 185 80 177)(89 119 218 105 210 97 202)(90 120 219 106 211 98 203)(91 113 220 107 212 99 204)(92 114 221 108 213 100 205)(93 115 222 109 214 101 206)(94 116 223 110 215 102 207)(95 117 224 111 216 103 208)(96 118 217 112 209 104 201)
(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 89 169 43 5 93 173 47)(2 90 170 44 6 94 174 48)(3 91 171 45 7 95 175 41)(4 92 172 46 8 96 176 42)(9 150 118 200 13 146 114 196)(10 151 119 193 14 147 115 197)(11 152 120 194 15 148 116 198)(12 145 113 195 16 149 117 199)(17 205 67 159 21 201 71 155)(18 206 68 160 22 202 72 156)(19 207 69 153 23 203 65 157)(20 208 70 154 24 204 66 158)(25 213 75 167 29 209 79 163)(26 214 76 168 30 210 80 164)(27 215 77 161 31 211 73 165)(28 216 78 162 32 212 74 166)(33 221 83 123 37 217 87 127)(34 222 84 124 38 218 88 128)(35 223 85 125 39 219 81 121)(36 224 86 126 40 220 82 122)(49 129 99 179 53 133 103 183)(50 130 100 180 54 134 104 184)(51 131 101 181 55 135 97 177)(52 132 102 182 56 136 98 178)(57 137 107 187 61 141 111 191)(58 138 108 188 62 142 112 192)(59 139 109 189 63 143 105 185)(60 140 110 190 64 144 106 186)
(9 13)(10 14)(11 15)(12 16)(41 45)(42 46)(43 47)(44 48)(49 53)(50 54)(51 55)(52 56)(57 61)(58 62)(59 63)(60 64)(89 93)(90 94)(91 95)(92 96)(97 101)(98 102)(99 103)(100 104)(105 109)(106 110)(107 111)(108 112)(113 117)(114 118)(115 119)(116 120)(121 125)(122 126)(123 127)(124 128)(153 157)(154 158)(155 159)(156 160)(161 165)(162 166)(163 167)(164 168)(201 205)(202 206)(203 207)(204 208)(209 213)(210 214)(211 215)(212 216)(217 221)(218 222)(219 223)(220 224)
G:=sub<Sym(224)| (1,151,38,143,30,135,22)(2,152,39,144,31,136,23)(3,145,40,137,32,129,24)(4,146,33,138,25,130,17)(5,147,34,139,26,131,18)(6,148,35,140,27,132,19)(7,149,36,141,28,133,20)(8,150,37,142,29,134,21)(9,123,62,167,54,159,46)(10,124,63,168,55,160,47)(11,125,64,161,56,153,48)(12,126,57,162,49,154,41)(13,127,58,163,50,155,42)(14,128,59,164,51,156,43)(15,121,60,165,52,157,44)(16,122,61,166,53,158,45)(65,170,194,81,186,73,178)(66,171,195,82,187,74,179)(67,172,196,83,188,75,180)(68,173,197,84,189,76,181)(69,174,198,85,190,77,182)(70,175,199,86,191,78,183)(71,176,200,87,192,79,184)(72,169,193,88,185,80,177)(89,119,218,105,210,97,202)(90,120,219,106,211,98,203)(91,113,220,107,212,99,204)(92,114,221,108,213,100,205)(93,115,222,109,214,101,206)(94,116,223,110,215,102,207)(95,117,224,111,216,103,208)(96,118,217,112,209,104,201), (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), (1,89,169,43,5,93,173,47)(2,90,170,44,6,94,174,48)(3,91,171,45,7,95,175,41)(4,92,172,46,8,96,176,42)(9,150,118,200,13,146,114,196)(10,151,119,193,14,147,115,197)(11,152,120,194,15,148,116,198)(12,145,113,195,16,149,117,199)(17,205,67,159,21,201,71,155)(18,206,68,160,22,202,72,156)(19,207,69,153,23,203,65,157)(20,208,70,154,24,204,66,158)(25,213,75,167,29,209,79,163)(26,214,76,168,30,210,80,164)(27,215,77,161,31,211,73,165)(28,216,78,162,32,212,74,166)(33,221,83,123,37,217,87,127)(34,222,84,124,38,218,88,128)(35,223,85,125,39,219,81,121)(36,224,86,126,40,220,82,122)(49,129,99,179,53,133,103,183)(50,130,100,180,54,134,104,184)(51,131,101,181,55,135,97,177)(52,132,102,182,56,136,98,178)(57,137,107,187,61,141,111,191)(58,138,108,188,62,142,112,192)(59,139,109,189,63,143,105,185)(60,140,110,190,64,144,106,186), (9,13)(10,14)(11,15)(12,16)(41,45)(42,46)(43,47)(44,48)(49,53)(50,54)(51,55)(52,56)(57,61)(58,62)(59,63)(60,64)(89,93)(90,94)(91,95)(92,96)(97,101)(98,102)(99,103)(100,104)(105,109)(106,110)(107,111)(108,112)(113,117)(114,118)(115,119)(116,120)(121,125)(122,126)(123,127)(124,128)(153,157)(154,158)(155,159)(156,160)(161,165)(162,166)(163,167)(164,168)(201,205)(202,206)(203,207)(204,208)(209,213)(210,214)(211,215)(212,216)(217,221)(218,222)(219,223)(220,224)>;
G:=Group( (1,151,38,143,30,135,22)(2,152,39,144,31,136,23)(3,145,40,137,32,129,24)(4,146,33,138,25,130,17)(5,147,34,139,26,131,18)(6,148,35,140,27,132,19)(7,149,36,141,28,133,20)(8,150,37,142,29,134,21)(9,123,62,167,54,159,46)(10,124,63,168,55,160,47)(11,125,64,161,56,153,48)(12,126,57,162,49,154,41)(13,127,58,163,50,155,42)(14,128,59,164,51,156,43)(15,121,60,165,52,157,44)(16,122,61,166,53,158,45)(65,170,194,81,186,73,178)(66,171,195,82,187,74,179)(67,172,196,83,188,75,180)(68,173,197,84,189,76,181)(69,174,198,85,190,77,182)(70,175,199,86,191,78,183)(71,176,200,87,192,79,184)(72,169,193,88,185,80,177)(89,119,218,105,210,97,202)(90,120,219,106,211,98,203)(91,113,220,107,212,99,204)(92,114,221,108,213,100,205)(93,115,222,109,214,101,206)(94,116,223,110,215,102,207)(95,117,224,111,216,103,208)(96,118,217,112,209,104,201), (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), (1,89,169,43,5,93,173,47)(2,90,170,44,6,94,174,48)(3,91,171,45,7,95,175,41)(4,92,172,46,8,96,176,42)(9,150,118,200,13,146,114,196)(10,151,119,193,14,147,115,197)(11,152,120,194,15,148,116,198)(12,145,113,195,16,149,117,199)(17,205,67,159,21,201,71,155)(18,206,68,160,22,202,72,156)(19,207,69,153,23,203,65,157)(20,208,70,154,24,204,66,158)(25,213,75,167,29,209,79,163)(26,214,76,168,30,210,80,164)(27,215,77,161,31,211,73,165)(28,216,78,162,32,212,74,166)(33,221,83,123,37,217,87,127)(34,222,84,124,38,218,88,128)(35,223,85,125,39,219,81,121)(36,224,86,126,40,220,82,122)(49,129,99,179,53,133,103,183)(50,130,100,180,54,134,104,184)(51,131,101,181,55,135,97,177)(52,132,102,182,56,136,98,178)(57,137,107,187,61,141,111,191)(58,138,108,188,62,142,112,192)(59,139,109,189,63,143,105,185)(60,140,110,190,64,144,106,186), (9,13)(10,14)(11,15)(12,16)(41,45)(42,46)(43,47)(44,48)(49,53)(50,54)(51,55)(52,56)(57,61)(58,62)(59,63)(60,64)(89,93)(90,94)(91,95)(92,96)(97,101)(98,102)(99,103)(100,104)(105,109)(106,110)(107,111)(108,112)(113,117)(114,118)(115,119)(116,120)(121,125)(122,126)(123,127)(124,128)(153,157)(154,158)(155,159)(156,160)(161,165)(162,166)(163,167)(164,168)(201,205)(202,206)(203,207)(204,208)(209,213)(210,214)(211,215)(212,216)(217,221)(218,222)(219,223)(220,224) );
G=PermutationGroup([[(1,151,38,143,30,135,22),(2,152,39,144,31,136,23),(3,145,40,137,32,129,24),(4,146,33,138,25,130,17),(5,147,34,139,26,131,18),(6,148,35,140,27,132,19),(7,149,36,141,28,133,20),(8,150,37,142,29,134,21),(9,123,62,167,54,159,46),(10,124,63,168,55,160,47),(11,125,64,161,56,153,48),(12,126,57,162,49,154,41),(13,127,58,163,50,155,42),(14,128,59,164,51,156,43),(15,121,60,165,52,157,44),(16,122,61,166,53,158,45),(65,170,194,81,186,73,178),(66,171,195,82,187,74,179),(67,172,196,83,188,75,180),(68,173,197,84,189,76,181),(69,174,198,85,190,77,182),(70,175,199,86,191,78,183),(71,176,200,87,192,79,184),(72,169,193,88,185,80,177),(89,119,218,105,210,97,202),(90,120,219,106,211,98,203),(91,113,220,107,212,99,204),(92,114,221,108,213,100,205),(93,115,222,109,214,101,206),(94,116,223,110,215,102,207),(95,117,224,111,216,103,208),(96,118,217,112,209,104,201)], [(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)], [(1,89,169,43,5,93,173,47),(2,90,170,44,6,94,174,48),(3,91,171,45,7,95,175,41),(4,92,172,46,8,96,176,42),(9,150,118,200,13,146,114,196),(10,151,119,193,14,147,115,197),(11,152,120,194,15,148,116,198),(12,145,113,195,16,149,117,199),(17,205,67,159,21,201,71,155),(18,206,68,160,22,202,72,156),(19,207,69,153,23,203,65,157),(20,208,70,154,24,204,66,158),(25,213,75,167,29,209,79,163),(26,214,76,168,30,210,80,164),(27,215,77,161,31,211,73,165),(28,216,78,162,32,212,74,166),(33,221,83,123,37,217,87,127),(34,222,84,124,38,218,88,128),(35,223,85,125,39,219,81,121),(36,224,86,126,40,220,82,122),(49,129,99,179,53,133,103,183),(50,130,100,180,54,134,104,184),(51,131,101,181,55,135,97,177),(52,132,102,182,56,136,98,178),(57,137,107,187,61,141,111,191),(58,138,108,188,62,142,112,192),(59,139,109,189,63,143,105,185),(60,140,110,190,64,144,106,186)], [(9,13),(10,14),(11,15),(12,16),(41,45),(42,46),(43,47),(44,48),(49,53),(50,54),(51,55),(52,56),(57,61),(58,62),(59,63),(60,64),(89,93),(90,94),(91,95),(92,96),(97,101),(98,102),(99,103),(100,104),(105,109),(106,110),(107,111),(108,112),(113,117),(114,118),(115,119),(116,120),(121,125),(122,126),(123,127),(124,128),(153,157),(154,158),(155,159),(156,160),(161,165),(162,166),(163,167),(164,168),(201,205),(202,206),(203,207),(204,208),(209,213),(210,214),(211,215),(212,216),(217,221),(218,222),(219,223),(220,224)]])
280 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | ··· | 4N | 7A | ··· | 7F | 8A | ··· | 8H | 8I | ··· | 8T | 14A | ··· | 14R | 14S | ··· | 14AD | 28A | ··· | 28X | 28Y | ··· | 28CF | 56A | ··· | 56AV | 56AW | ··· | 56DP |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 7 | ··· | 7 | 8 | ··· | 8 | 8 | ··· | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 |
280 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | + | + | + | ||||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | C7 | C14 | C14 | C14 | C14 | C14 | C28 | C28 | C28 | C28 | C8○D4 | C7×C8○D4 |
| kernel | C7×C8○2M4(2) | C4×C56 | C7×C8⋊C4 | C7×C42⋊C2 | C22×C56 | C14×M4(2) | C7×C22⋊C4 | C7×C4⋊C4 | C2×C56 | C7×M4(2) | C8○2M4(2) | C4×C8 | C8⋊C4 | C42⋊C2 | C22×C8 | C2×M4(2) | C22⋊C4 | C4⋊C4 | C2×C8 | M4(2) | C14 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 4 | 4 | 8 | 8 | 6 | 12 | 12 | 6 | 6 | 6 | 24 | 24 | 48 | 48 | 8 | 48 |
Matrix representation of C7×C8○2M4(2) ►in GL3(𝔽113) generated by
| 1 | 0 | 0 |
| 0 | 106 | 0 |
| 0 | 0 | 106 |
| 15 | 0 | 0 |
| 0 | 44 | 0 |
| 0 | 0 | 44 |
| 1 | 0 | 0 |
| 0 | 77 | 77 |
| 0 | 45 | 36 |
| 112 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 111 | 112 |
G:=sub<GL(3,GF(113))| [1,0,0,0,106,0,0,0,106],[15,0,0,0,44,0,0,0,44],[1,0,0,0,77,45,0,77,36],[112,0,0,0,1,111,0,0,112] >;
C7×C8○2M4(2) in GAP, Magma, Sage, TeX
C_7\times C_8\circ_2M_4(2)
% in TeX
G:=Group("C7xC8o2M4(2)"); // GroupNames label
G:=SmallGroup(448,813);
// by ID
G=gap.SmallGroup(448,813);
# by ID
G:=PCGroup([7,-2,-2,-2,-7,-2,-2,-2,392,792,2403,172]);
// Polycyclic
G:=Group<a,b,c,d|a^7=b^8=d^2=1,c^4=b^4,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=b^4*c>;
// generators/relations