metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.234D14, (Q8×Dic7)⋊19C2, (D4×Dic7)⋊30C2, C4.4D4⋊19D7, (D7×C42)⋊10C2, D14⋊3Q8⋊32C2, D14.9(C4○D4), (C2×D4).175D14, C28⋊2D4.12C2, (C2×Q8).138D14, C22⋊C4.74D14, C28.6Q8⋊20C2, Dic7⋊4D4⋊33C2, D14.D4⋊45C2, C28.125(C4○D4), C4.38(D4⋊2D7), (C4×C28).187C22, (C2×C14).224C24, (C2×C28).504C23, D14⋊C4.36C22, C23.46(C22×D7), Dic7.43(C4○D4), (D4×C14).157C22, C23.D14⋊41C2, Dic7⋊C4.70C22, C4⋊Dic7.234C22, (C22×C14).54C23, (Q8×C14).128C22, C22.245(C23×D7), C23.D7.57C22, C23.11D14⋊19C2, C7⋊9(C23.36C23), (C2×Dic7).310C23, (C4×Dic7).134C22, (C22×D7).218C23, (C22×Dic7).144C22, C2.80(D7×C4○D4), C14.191(C2×C4○D4), C2.56(C2×D4⋊2D7), (C7×C4.4D4)⋊16C2, (C2×C4×D7).298C22, (C2×C4).301(C22×D7), (C2×C7⋊D4).62C22, (C7×C22⋊C4).66C22, SmallGroup(448,1133)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 940 in 234 conjugacy classes, 99 normal (43 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×12], C22, C22 [×10], C7, C2×C4 [×3], C2×C4 [×2], C2×C4 [×17], D4 [×6], Q8 [×2], C23 [×2], C23, D7 [×2], C14 [×3], C14 [×2], C42, C42 [×5], C22⋊C4 [×4], C22⋊C4 [×6], C4⋊C4 [×10], C22×C4 [×5], C2×D4, C2×D4 [×2], C2×Q8, Dic7 [×2], Dic7 [×6], C28 [×2], C28 [×4], D14 [×2], D14 [×2], C2×C14, C2×C14 [×6], C2×C42, C42⋊C2 [×2], C4×D4 [×3], C4×Q8, C4⋊D4, C22⋊Q8, C22.D4 [×2], C4.4D4, C42.C2, C42⋊2C2 [×2], C4×D7 [×6], C2×Dic7 [×3], C2×Dic7 [×4], C2×Dic7 [×4], C7⋊D4 [×4], C2×C28 [×3], C2×C28 [×2], C7×D4 [×2], C7×Q8 [×2], C22×D7, C22×C14 [×2], C23.36C23, C4×Dic7 [×3], C4×Dic7 [×2], Dic7⋊C4 [×6], C4⋊Dic7 [×2], C4⋊Dic7 [×2], D14⋊C4 [×2], C23.D7 [×4], C4×C28, C7×C22⋊C4 [×4], C2×C4×D7 [×3], C22×Dic7 [×2], C2×C7⋊D4 [×2], D4×C14, Q8×C14, C28.6Q8, D7×C42, C23.11D14 [×2], C23.D14 [×2], Dic7⋊4D4 [×2], D14.D4 [×2], D4×Dic7, C28⋊2D4, Q8×Dic7, D14⋊3Q8, C7×C4.4D4, C42.234D14
Quotients:
C1, C2 [×15], C22 [×35], C23 [×15], D7, C4○D4 [×6], C24, D14 [×7], C2×C4○D4 [×3], C22×D7 [×7], C23.36C23, D4⋊2D7 [×2], C23×D7, C2×D4⋊2D7, D7×C4○D4 [×2], C42.234D14
Generators and relations
G = < a,b,c,d | a4=b4=c14=1, d2=a2, ab=ba, cac-1=dad-1=ab2, cbc-1=dbd-1=a2b, dcd-1=a2c-1 >
►On 224 points
Generators in S224
(1 176 157 79)(2 134 158 141)(3 178 159 81)(4 136 160 143)(5 180 161 83)(6 138 162 145)(7 182 163 71)(8 140 164 147)(9 170 165 73)(10 128 166 149)(11 172 167 75)(12 130 168 151)(13 174 155 77)(14 132 156 153)(15 69 108 212)(16 193 109 53)(17 57 110 214)(18 195 111 55)(19 59 112 216)(20 183 99 43)(21 61 100 218)(22 185 101 45)(23 63 102 220)(24 187 103 47)(25 65 104 222)(26 189 105 49)(27 67 106 224)(28 191 107 51)(29 131 210 152)(30 175 197 78)(31 133 198 154)(32 177 199 80)(33 135 200 142)(34 179 201 82)(35 137 202 144)(36 181 203 84)(37 139 204 146)(38 169 205 72)(39 127 206 148)(40 171 207 74)(41 129 208 150)(42 173 209 76)(44 92 184 124)(46 94 186 126)(48 96 188 114)(50 98 190 116)(52 86 192 118)(54 88 194 120)(56 90 196 122)(58 121 215 89)(60 123 217 91)(62 125 219 93)(64 113 221 95)(66 115 223 97)(68 117 211 85)(70 119 213 87)
(1 108 31 118)(2 16 32 87)(3 110 33 120)(4 18 34 89)(5 112 35 122)(6 20 36 91)(7 100 37 124)(8 22 38 93)(9 102 39 126)(10 24 40 95)(11 104 41 114)(12 26 42 97)(13 106 29 116)(14 28 30 85)(15 198 86 157)(17 200 88 159)(19 202 90 161)(21 204 92 163)(23 206 94 165)(25 208 96 167)(27 210 98 155)(43 84 217 145)(44 182 218 139)(45 72 219 147)(46 170 220 127)(47 74 221 149)(48 172 222 129)(49 76 223 151)(50 174 224 131)(51 78 211 153)(52 176 212 133)(53 80 213 141)(54 178 214 135)(55 82 215 143)(56 180 216 137)(57 142 194 81)(58 136 195 179)(59 144 196 83)(60 138 183 181)(61 146 184 71)(62 140 185 169)(63 148 186 73)(64 128 187 171)(65 150 188 75)(66 130 189 173)(67 152 190 77)(68 132 191 175)(69 154 192 79)(70 134 193 177)(99 203 123 162)(101 205 125 164)(103 207 113 166)(105 209 115 168)(107 197 117 156)(109 199 119 158)(111 201 121 160)
(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 30 157 197)(2 210 158 29)(3 42 159 209)(4 208 160 41)(5 40 161 207)(6 206 162 39)(7 38 163 205)(8 204 164 37)(9 36 165 203)(10 202 166 35)(11 34 167 201)(12 200 168 33)(13 32 155 199)(14 198 156 31)(15 85 108 117)(16 116 109 98)(17 97 110 115)(18 114 111 96)(19 95 112 113)(20 126 99 94)(21 93 100 125)(22 124 101 92)(23 91 102 123)(24 122 103 90)(25 89 104 121)(26 120 105 88)(27 87 106 119)(28 118 107 86)(43 63 183 220)(44 219 184 62)(45 61 185 218)(46 217 186 60)(47 59 187 216)(48 215 188 58)(49 57 189 214)(50 213 190 70)(51 69 191 212)(52 211 192 68)(53 67 193 224)(54 223 194 66)(55 65 195 222)(56 221 196 64)(71 147 182 140)(72 139 169 146)(73 145 170 138)(74 137 171 144)(75 143 172 136)(76 135 173 142)(77 141 174 134)(78 133 175 154)(79 153 176 132)(80 131 177 152)(81 151 178 130)(82 129 179 150)(83 149 180 128)(84 127 181 148)
G:=sub<Sym(224)| (1,176,157,79)(2,134,158,141)(3,178,159,81)(4,136,160,143)(5,180,161,83)(6,138,162,145)(7,182,163,71)(8,140,164,147)(9,170,165,73)(10,128,166,149)(11,172,167,75)(12,130,168,151)(13,174,155,77)(14,132,156,153)(15,69,108,212)(16,193,109,53)(17,57,110,214)(18,195,111,55)(19,59,112,216)(20,183,99,43)(21,61,100,218)(22,185,101,45)(23,63,102,220)(24,187,103,47)(25,65,104,222)(26,189,105,49)(27,67,106,224)(28,191,107,51)(29,131,210,152)(30,175,197,78)(31,133,198,154)(32,177,199,80)(33,135,200,142)(34,179,201,82)(35,137,202,144)(36,181,203,84)(37,139,204,146)(38,169,205,72)(39,127,206,148)(40,171,207,74)(41,129,208,150)(42,173,209,76)(44,92,184,124)(46,94,186,126)(48,96,188,114)(50,98,190,116)(52,86,192,118)(54,88,194,120)(56,90,196,122)(58,121,215,89)(60,123,217,91)(62,125,219,93)(64,113,221,95)(66,115,223,97)(68,117,211,85)(70,119,213,87), (1,108,31,118)(2,16,32,87)(3,110,33,120)(4,18,34,89)(5,112,35,122)(6,20,36,91)(7,100,37,124)(8,22,38,93)(9,102,39,126)(10,24,40,95)(11,104,41,114)(12,26,42,97)(13,106,29,116)(14,28,30,85)(15,198,86,157)(17,200,88,159)(19,202,90,161)(21,204,92,163)(23,206,94,165)(25,208,96,167)(27,210,98,155)(43,84,217,145)(44,182,218,139)(45,72,219,147)(46,170,220,127)(47,74,221,149)(48,172,222,129)(49,76,223,151)(50,174,224,131)(51,78,211,153)(52,176,212,133)(53,80,213,141)(54,178,214,135)(55,82,215,143)(56,180,216,137)(57,142,194,81)(58,136,195,179)(59,144,196,83)(60,138,183,181)(61,146,184,71)(62,140,185,169)(63,148,186,73)(64,128,187,171)(65,150,188,75)(66,130,189,173)(67,152,190,77)(68,132,191,175)(69,154,192,79)(70,134,193,177)(99,203,123,162)(101,205,125,164)(103,207,113,166)(105,209,115,168)(107,197,117,156)(109,199,119,158)(111,201,121,160), (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), (1,30,157,197)(2,210,158,29)(3,42,159,209)(4,208,160,41)(5,40,161,207)(6,206,162,39)(7,38,163,205)(8,204,164,37)(9,36,165,203)(10,202,166,35)(11,34,167,201)(12,200,168,33)(13,32,155,199)(14,198,156,31)(15,85,108,117)(16,116,109,98)(17,97,110,115)(18,114,111,96)(19,95,112,113)(20,126,99,94)(21,93,100,125)(22,124,101,92)(23,91,102,123)(24,122,103,90)(25,89,104,121)(26,120,105,88)(27,87,106,119)(28,118,107,86)(43,63,183,220)(44,219,184,62)(45,61,185,218)(46,217,186,60)(47,59,187,216)(48,215,188,58)(49,57,189,214)(50,213,190,70)(51,69,191,212)(52,211,192,68)(53,67,193,224)(54,223,194,66)(55,65,195,222)(56,221,196,64)(71,147,182,140)(72,139,169,146)(73,145,170,138)(74,137,171,144)(75,143,172,136)(76,135,173,142)(77,141,174,134)(78,133,175,154)(79,153,176,132)(80,131,177,152)(81,151,178,130)(82,129,179,150)(83,149,180,128)(84,127,181,148)>;
G:=Group( (1,176,157,79)(2,134,158,141)(3,178,159,81)(4,136,160,143)(5,180,161,83)(6,138,162,145)(7,182,163,71)(8,140,164,147)(9,170,165,73)(10,128,166,149)(11,172,167,75)(12,130,168,151)(13,174,155,77)(14,132,156,153)(15,69,108,212)(16,193,109,53)(17,57,110,214)(18,195,111,55)(19,59,112,216)(20,183,99,43)(21,61,100,218)(22,185,101,45)(23,63,102,220)(24,187,103,47)(25,65,104,222)(26,189,105,49)(27,67,106,224)(28,191,107,51)(29,131,210,152)(30,175,197,78)(31,133,198,154)(32,177,199,80)(33,135,200,142)(34,179,201,82)(35,137,202,144)(36,181,203,84)(37,139,204,146)(38,169,205,72)(39,127,206,148)(40,171,207,74)(41,129,208,150)(42,173,209,76)(44,92,184,124)(46,94,186,126)(48,96,188,114)(50,98,190,116)(52,86,192,118)(54,88,194,120)(56,90,196,122)(58,121,215,89)(60,123,217,91)(62,125,219,93)(64,113,221,95)(66,115,223,97)(68,117,211,85)(70,119,213,87), (1,108,31,118)(2,16,32,87)(3,110,33,120)(4,18,34,89)(5,112,35,122)(6,20,36,91)(7,100,37,124)(8,22,38,93)(9,102,39,126)(10,24,40,95)(11,104,41,114)(12,26,42,97)(13,106,29,116)(14,28,30,85)(15,198,86,157)(17,200,88,159)(19,202,90,161)(21,204,92,163)(23,206,94,165)(25,208,96,167)(27,210,98,155)(43,84,217,145)(44,182,218,139)(45,72,219,147)(46,170,220,127)(47,74,221,149)(48,172,222,129)(49,76,223,151)(50,174,224,131)(51,78,211,153)(52,176,212,133)(53,80,213,141)(54,178,214,135)(55,82,215,143)(56,180,216,137)(57,142,194,81)(58,136,195,179)(59,144,196,83)(60,138,183,181)(61,146,184,71)(62,140,185,169)(63,148,186,73)(64,128,187,171)(65,150,188,75)(66,130,189,173)(67,152,190,77)(68,132,191,175)(69,154,192,79)(70,134,193,177)(99,203,123,162)(101,205,125,164)(103,207,113,166)(105,209,115,168)(107,197,117,156)(109,199,119,158)(111,201,121,160), (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), (1,30,157,197)(2,210,158,29)(3,42,159,209)(4,208,160,41)(5,40,161,207)(6,206,162,39)(7,38,163,205)(8,204,164,37)(9,36,165,203)(10,202,166,35)(11,34,167,201)(12,200,168,33)(13,32,155,199)(14,198,156,31)(15,85,108,117)(16,116,109,98)(17,97,110,115)(18,114,111,96)(19,95,112,113)(20,126,99,94)(21,93,100,125)(22,124,101,92)(23,91,102,123)(24,122,103,90)(25,89,104,121)(26,120,105,88)(27,87,106,119)(28,118,107,86)(43,63,183,220)(44,219,184,62)(45,61,185,218)(46,217,186,60)(47,59,187,216)(48,215,188,58)(49,57,189,214)(50,213,190,70)(51,69,191,212)(52,211,192,68)(53,67,193,224)(54,223,194,66)(55,65,195,222)(56,221,196,64)(71,147,182,140)(72,139,169,146)(73,145,170,138)(74,137,171,144)(75,143,172,136)(76,135,173,142)(77,141,174,134)(78,133,175,154)(79,153,176,132)(80,131,177,152)(81,151,178,130)(82,129,179,150)(83,149,180,128)(84,127,181,148) );
G=PermutationGroup([(1,176,157,79),(2,134,158,141),(3,178,159,81),(4,136,160,143),(5,180,161,83),(6,138,162,145),(7,182,163,71),(8,140,164,147),(9,170,165,73),(10,128,166,149),(11,172,167,75),(12,130,168,151),(13,174,155,77),(14,132,156,153),(15,69,108,212),(16,193,109,53),(17,57,110,214),(18,195,111,55),(19,59,112,216),(20,183,99,43),(21,61,100,218),(22,185,101,45),(23,63,102,220),(24,187,103,47),(25,65,104,222),(26,189,105,49),(27,67,106,224),(28,191,107,51),(29,131,210,152),(30,175,197,78),(31,133,198,154),(32,177,199,80),(33,135,200,142),(34,179,201,82),(35,137,202,144),(36,181,203,84),(37,139,204,146),(38,169,205,72),(39,127,206,148),(40,171,207,74),(41,129,208,150),(42,173,209,76),(44,92,184,124),(46,94,186,126),(48,96,188,114),(50,98,190,116),(52,86,192,118),(54,88,194,120),(56,90,196,122),(58,121,215,89),(60,123,217,91),(62,125,219,93),(64,113,221,95),(66,115,223,97),(68,117,211,85),(70,119,213,87)], [(1,108,31,118),(2,16,32,87),(3,110,33,120),(4,18,34,89),(5,112,35,122),(6,20,36,91),(7,100,37,124),(8,22,38,93),(9,102,39,126),(10,24,40,95),(11,104,41,114),(12,26,42,97),(13,106,29,116),(14,28,30,85),(15,198,86,157),(17,200,88,159),(19,202,90,161),(21,204,92,163),(23,206,94,165),(25,208,96,167),(27,210,98,155),(43,84,217,145),(44,182,218,139),(45,72,219,147),(46,170,220,127),(47,74,221,149),(48,172,222,129),(49,76,223,151),(50,174,224,131),(51,78,211,153),(52,176,212,133),(53,80,213,141),(54,178,214,135),(55,82,215,143),(56,180,216,137),(57,142,194,81),(58,136,195,179),(59,144,196,83),(60,138,183,181),(61,146,184,71),(62,140,185,169),(63,148,186,73),(64,128,187,171),(65,150,188,75),(66,130,189,173),(67,152,190,77),(68,132,191,175),(69,154,192,79),(70,134,193,177),(99,203,123,162),(101,205,125,164),(103,207,113,166),(105,209,115,168),(107,197,117,156),(109,199,119,158),(111,201,121,160)], [(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)], [(1,30,157,197),(2,210,158,29),(3,42,159,209),(4,208,160,41),(5,40,161,207),(6,206,162,39),(7,38,163,205),(8,204,164,37),(9,36,165,203),(10,202,166,35),(11,34,167,201),(12,200,168,33),(13,32,155,199),(14,198,156,31),(15,85,108,117),(16,116,109,98),(17,97,110,115),(18,114,111,96),(19,95,112,113),(20,126,99,94),(21,93,100,125),(22,124,101,92),(23,91,102,123),(24,122,103,90),(25,89,104,121),(26,120,105,88),(27,87,106,119),(28,118,107,86),(43,63,183,220),(44,219,184,62),(45,61,185,218),(46,217,186,60),(47,59,187,216),(48,215,188,58),(49,57,189,214),(50,213,190,70),(51,69,191,212),(52,211,192,68),(53,67,193,224),(54,223,194,66),(55,65,195,222),(56,221,196,64),(71,147,182,140),(72,139,169,146),(73,145,170,138),(74,137,171,144),(75,143,172,136),(76,135,173,142),(77,141,174,134),(78,133,175,154),(79,153,176,132),(80,131,177,152),(81,151,178,130),(82,129,179,150),(83,149,180,128),(84,127,181,148)])
Matrix representation ►G ⊆ GL6(𝔽29)
| 12 | 27 | 0 | 0 | 0 | 0 |
| 28 | 17 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 17 | 0 |
| 0 | 0 | 0 | 0 | 0 | 17 |
| 17 | 0 | 0 | 0 | 0 | 0 |
| 0 | 17 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
| 1 | 24 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 10 | 0 | 0 |
| 0 | 0 | 9 | 23 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 28 | 5 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 7 | 0 | 0 |
| 0 | 0 | 18 | 22 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
| 0 | 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(6,GF(29))| [12,28,0,0,0,0,27,17,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,17,0,0,0,0,0,0,17],[17,0,0,0,0,0,0,17,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,28],[1,0,0,0,0,0,24,28,0,0,0,0,0,0,9,9,0,0,0,0,10,23,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[28,0,0,0,0,0,5,1,0,0,0,0,0,0,7,18,0,0,0,0,7,22,0,0,0,0,0,0,0,1,0,0,0,0,28,0] >;
70 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | 4Q | 4R | 4S | 4T | 7A | 7B | 7C | 14A | ··· | 14I | 14J | ··· | 14O | 28A | ··· | 28R | 28S | ··· | 28X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 14 | 14 | 2 | ··· | 2 | 4 | 4 | 7 | 7 | 7 | 7 | 14 | 14 | 14 | 14 | 28 | 28 | 28 | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 8 | ··· | 8 | 4 | ··· | 4 | 8 | ··· | 8 |
70 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D7 | C4○D4 | C4○D4 | C4○D4 | D14 | D14 | D14 | D14 | D4⋊2D7 | D7×C4○D4 |
| kernel | C42.234D14 | C28.6Q8 | D7×C42 | C23.11D14 | C23.D14 | Dic7⋊4D4 | D14.D4 | D4×Dic7 | C28⋊2D4 | Q8×Dic7 | D14⋊3Q8 | C7×C4.4D4 | C4.4D4 | Dic7 | C28 | D14 | C42 | C22⋊C4 | C2×D4 | C2×Q8 | C4 | C2 |
| # reps | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 3 | 4 | 4 | 4 | 3 | 12 | 3 | 3 | 6 | 12 |
In GAP, Magma, Sage, TeX
C_4^2._{234}D_{14} % in TeX
G:=Group("C4^2.234D14"); // GroupNames label
G:=SmallGroup(448,1133);
// by ID
G=gap.SmallGroup(448,1133);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,232,100,1123,346,297,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^14=1,d^2=a^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a*b^2,c*b*c^-1=d*b*d^-1=a^2*b,d*c*d^-1=a^2*c^-1>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀