direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C14×C4⋊C4, C4⋊2(C2×C28), (C2×C4)⋊3C28, C28⋊9(C2×C4), (C2×C28)⋊8C4, C2.2(D4×C14), (C2×C14).8Q8, C2.1(Q8×C14), C14.65(C2×D4), (C2×C14).51D4, C14.18(C2×Q8), C22.3(C7×Q8), C2.2(C22×C28), (C22×C28).5C2, (C22×C4).3C14, C22.13(C7×D4), C23.13(C2×C14), (C2×C14).71C23, C14.30(C22×C4), C22.11(C2×C28), (C2×C28).120C22, C22.5(C22×C14), (C22×C14).49C22, (C2×C4).13(C2×C14), (C2×C14).40(C2×C4), SmallGroup(224,151)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C14×C4⋊C4
G = < a,b,c | a14=b4=c4=1, ab=ba, ac=ca, cbc-1=b-1 >
Subgroups: 108 in 92 conjugacy classes, 76 normal (16 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C2×C4, C2×C4, C23, C14, C14, C4⋊C4, C22×C4, C22×C4, C28, C28, C2×C14, C2×C14, C2×C4⋊C4, C2×C28, C2×C28, C22×C14, C7×C4⋊C4, C22×C28, C22×C28, C14×C4⋊C4
Quotients: C1, C2, C4, C22, C7, C2×C4, D4, Q8, C23, C14, C4⋊C4, C22×C4, C2×D4, C2×Q8, C28, C2×C14, C2×C4⋊C4, C2×C28, C7×D4, C7×Q8, C22×C14, C7×C4⋊C4, C22×C28, D4×C14, Q8×C14, C14×C4⋊C4
►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 35 178 131)(2 36 179 132)(3 37 180 133)(4 38 181 134)(5 39 182 135)(6 40 169 136)(7 41 170 137)(8 42 171 138)(9 29 172 139)(10 30 173 140)(11 31 174 127)(12 32 175 128)(13 33 176 129)(14 34 177 130)(15 196 119 96)(16 183 120 97)(17 184 121 98)(18 185 122 85)(19 186 123 86)(20 187 124 87)(21 188 125 88)(22 189 126 89)(23 190 113 90)(24 191 114 91)(25 192 115 92)(26 193 116 93)(27 194 117 94)(28 195 118 95)(43 83 66 220)(44 84 67 221)(45 71 68 222)(46 72 69 223)(47 73 70 224)(48 74 57 211)(49 75 58 212)(50 76 59 213)(51 77 60 214)(52 78 61 215)(53 79 62 216)(54 80 63 217)(55 81 64 218)(56 82 65 219)(99 149 159 206)(100 150 160 207)(101 151 161 208)(102 152 162 209)(103 153 163 210)(104 154 164 197)(105 141 165 198)(106 142 166 199)(107 143 167 200)(108 144 168 201)(109 145 155 202)(110 146 156 203)(111 147 157 204)(112 148 158 205)
(1 188 147 83)(2 189 148 84)(3 190 149 71)(4 191 150 72)(5 192 151 73)(6 193 152 74)(7 194 153 75)(8 195 154 76)(9 196 141 77)(10 183 142 78)(11 184 143 79)(12 185 144 80)(13 186 145 81)(14 187 146 82)(15 165 51 29)(16 166 52 30)(17 167 53 31)(18 168 54 32)(19 155 55 33)(20 156 56 34)(21 157 43 35)(22 158 44 36)(23 159 45 37)(24 160 46 38)(25 161 47 39)(26 162 48 40)(27 163 49 41)(28 164 50 42)(57 136 116 102)(58 137 117 103)(59 138 118 104)(60 139 119 105)(61 140 120 106)(62 127 121 107)(63 128 122 108)(64 129 123 109)(65 130 124 110)(66 131 125 111)(67 132 126 112)(68 133 113 99)(69 134 114 100)(70 135 115 101)(85 201 217 175)(86 202 218 176)(87 203 219 177)(88 204 220 178)(89 205 221 179)(90 206 222 180)(91 207 223 181)(92 208 224 182)(93 209 211 169)(94 210 212 170)(95 197 213 171)(96 198 214 172)(97 199 215 173)(98 200 216 174)
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), (1,35,178,131)(2,36,179,132)(3,37,180,133)(4,38,181,134)(5,39,182,135)(6,40,169,136)(7,41,170,137)(8,42,171,138)(9,29,172,139)(10,30,173,140)(11,31,174,127)(12,32,175,128)(13,33,176,129)(14,34,177,130)(15,196,119,96)(16,183,120,97)(17,184,121,98)(18,185,122,85)(19,186,123,86)(20,187,124,87)(21,188,125,88)(22,189,126,89)(23,190,113,90)(24,191,114,91)(25,192,115,92)(26,193,116,93)(27,194,117,94)(28,195,118,95)(43,83,66,220)(44,84,67,221)(45,71,68,222)(46,72,69,223)(47,73,70,224)(48,74,57,211)(49,75,58,212)(50,76,59,213)(51,77,60,214)(52,78,61,215)(53,79,62,216)(54,80,63,217)(55,81,64,218)(56,82,65,219)(99,149,159,206)(100,150,160,207)(101,151,161,208)(102,152,162,209)(103,153,163,210)(104,154,164,197)(105,141,165,198)(106,142,166,199)(107,143,167,200)(108,144,168,201)(109,145,155,202)(110,146,156,203)(111,147,157,204)(112,148,158,205), (1,188,147,83)(2,189,148,84)(3,190,149,71)(4,191,150,72)(5,192,151,73)(6,193,152,74)(7,194,153,75)(8,195,154,76)(9,196,141,77)(10,183,142,78)(11,184,143,79)(12,185,144,80)(13,186,145,81)(14,187,146,82)(15,165,51,29)(16,166,52,30)(17,167,53,31)(18,168,54,32)(19,155,55,33)(20,156,56,34)(21,157,43,35)(22,158,44,36)(23,159,45,37)(24,160,46,38)(25,161,47,39)(26,162,48,40)(27,163,49,41)(28,164,50,42)(57,136,116,102)(58,137,117,103)(59,138,118,104)(60,139,119,105)(61,140,120,106)(62,127,121,107)(63,128,122,108)(64,129,123,109)(65,130,124,110)(66,131,125,111)(67,132,126,112)(68,133,113,99)(69,134,114,100)(70,135,115,101)(85,201,217,175)(86,202,218,176)(87,203,219,177)(88,204,220,178)(89,205,221,179)(90,206,222,180)(91,207,223,181)(92,208,224,182)(93,209,211,169)(94,210,212,170)(95,197,213,171)(96,198,214,172)(97,199,215,173)(98,200,216,174)>;
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), (1,35,178,131)(2,36,179,132)(3,37,180,133)(4,38,181,134)(5,39,182,135)(6,40,169,136)(7,41,170,137)(8,42,171,138)(9,29,172,139)(10,30,173,140)(11,31,174,127)(12,32,175,128)(13,33,176,129)(14,34,177,130)(15,196,119,96)(16,183,120,97)(17,184,121,98)(18,185,122,85)(19,186,123,86)(20,187,124,87)(21,188,125,88)(22,189,126,89)(23,190,113,90)(24,191,114,91)(25,192,115,92)(26,193,116,93)(27,194,117,94)(28,195,118,95)(43,83,66,220)(44,84,67,221)(45,71,68,222)(46,72,69,223)(47,73,70,224)(48,74,57,211)(49,75,58,212)(50,76,59,213)(51,77,60,214)(52,78,61,215)(53,79,62,216)(54,80,63,217)(55,81,64,218)(56,82,65,219)(99,149,159,206)(100,150,160,207)(101,151,161,208)(102,152,162,209)(103,153,163,210)(104,154,164,197)(105,141,165,198)(106,142,166,199)(107,143,167,200)(108,144,168,201)(109,145,155,202)(110,146,156,203)(111,147,157,204)(112,148,158,205), (1,188,147,83)(2,189,148,84)(3,190,149,71)(4,191,150,72)(5,192,151,73)(6,193,152,74)(7,194,153,75)(8,195,154,76)(9,196,141,77)(10,183,142,78)(11,184,143,79)(12,185,144,80)(13,186,145,81)(14,187,146,82)(15,165,51,29)(16,166,52,30)(17,167,53,31)(18,168,54,32)(19,155,55,33)(20,156,56,34)(21,157,43,35)(22,158,44,36)(23,159,45,37)(24,160,46,38)(25,161,47,39)(26,162,48,40)(27,163,49,41)(28,164,50,42)(57,136,116,102)(58,137,117,103)(59,138,118,104)(60,139,119,105)(61,140,120,106)(62,127,121,107)(63,128,122,108)(64,129,123,109)(65,130,124,110)(66,131,125,111)(67,132,126,112)(68,133,113,99)(69,134,114,100)(70,135,115,101)(85,201,217,175)(86,202,218,176)(87,203,219,177)(88,204,220,178)(89,205,221,179)(90,206,222,180)(91,207,223,181)(92,208,224,182)(93,209,211,169)(94,210,212,170)(95,197,213,171)(96,198,214,172)(97,199,215,173)(98,200,216,174) );
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)], [(1,35,178,131),(2,36,179,132),(3,37,180,133),(4,38,181,134),(5,39,182,135),(6,40,169,136),(7,41,170,137),(8,42,171,138),(9,29,172,139),(10,30,173,140),(11,31,174,127),(12,32,175,128),(13,33,176,129),(14,34,177,130),(15,196,119,96),(16,183,120,97),(17,184,121,98),(18,185,122,85),(19,186,123,86),(20,187,124,87),(21,188,125,88),(22,189,126,89),(23,190,113,90),(24,191,114,91),(25,192,115,92),(26,193,116,93),(27,194,117,94),(28,195,118,95),(43,83,66,220),(44,84,67,221),(45,71,68,222),(46,72,69,223),(47,73,70,224),(48,74,57,211),(49,75,58,212),(50,76,59,213),(51,77,60,214),(52,78,61,215),(53,79,62,216),(54,80,63,217),(55,81,64,218),(56,82,65,219),(99,149,159,206),(100,150,160,207),(101,151,161,208),(102,152,162,209),(103,153,163,210),(104,154,164,197),(105,141,165,198),(106,142,166,199),(107,143,167,200),(108,144,168,201),(109,145,155,202),(110,146,156,203),(111,147,157,204),(112,148,158,205)], [(1,188,147,83),(2,189,148,84),(3,190,149,71),(4,191,150,72),(5,192,151,73),(6,193,152,74),(7,194,153,75),(8,195,154,76),(9,196,141,77),(10,183,142,78),(11,184,143,79),(12,185,144,80),(13,186,145,81),(14,187,146,82),(15,165,51,29),(16,166,52,30),(17,167,53,31),(18,168,54,32),(19,155,55,33),(20,156,56,34),(21,157,43,35),(22,158,44,36),(23,159,45,37),(24,160,46,38),(25,161,47,39),(26,162,48,40),(27,163,49,41),(28,164,50,42),(57,136,116,102),(58,137,117,103),(59,138,118,104),(60,139,119,105),(61,140,120,106),(62,127,121,107),(63,128,122,108),(64,129,123,109),(65,130,124,110),(66,131,125,111),(67,132,126,112),(68,133,113,99),(69,134,114,100),(70,135,115,101),(85,201,217,175),(86,202,218,176),(87,203,219,177),(88,204,220,178),(89,205,221,179),(90,206,222,180),(91,207,223,181),(92,208,224,182),(93,209,211,169),(94,210,212,170),(95,197,213,171),(96,198,214,172),(97,199,215,173),(98,200,216,174)]])
C14×C4⋊C4 is a maximal subgroup of
(C2×C28)⋊C8 C28.C42 C28.(C4⋊C4) C4.Dic7⋊C4 C4○D28⋊C4 (C2×C14).40D8 C4⋊C4.228D14 C4⋊C4.230D14 C4⋊C4.231D14 Dic7⋊(C4⋊C4) C28⋊(C4⋊C4) (C2×Dic7)⋊6Q8 (C4×Dic7)⋊8C4 (C4×Dic7)⋊9C4 C22.23(Q8×D7) (C2×C4)⋊Dic14 (C2×C28).287D4 C4⋊C4⋊5Dic7 (C2×C28).288D4 (C2×C4).44D28 (C2×C28).54D4 C4⋊(C4⋊Dic7) (C2×C28).55D4 C4⋊(D14⋊C4) (C2×D28)⋊10C4 D14⋊C4⋊6C4 D14⋊C4⋊7C4 (C2×C4)⋊3D28 (C2×C28).289D4 (C2×C28).290D4 (C2×C4).45D28 C14.72+ 1+4 C14.82+ 1+4 C14.2- 1+4 C14.2+ 1+4 C14.102+ 1+4 C14.52- 1+4 C14.112+ 1+4 C14.62- 1+4 D4×C2×C28 Q8×C2×C28
140 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | ··· | 4L | 7A | ··· | 7F | 14A | ··· | 14AP | 28A | ··· | 28BT |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 7 | ··· | 7 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 |
140 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | |||||||
| image | C1 | C2 | C2 | C4 | C7 | C14 | C14 | C28 | D4 | Q8 | C7×D4 | C7×Q8 |
| kernel | C14×C4⋊C4 | C7×C4⋊C4 | C22×C28 | C2×C28 | C2×C4⋊C4 | C4⋊C4 | C22×C4 | C2×C4 | C2×C14 | C2×C14 | C22 | C22 |
| # reps | 1 | 4 | 3 | 8 | 6 | 24 | 18 | 48 | 2 | 2 | 12 | 12 |
Matrix representation of C14×C4⋊C4 ►in GL4(𝔽29) generated by
| 6 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 |
| 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 28 |
| 28 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 28 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 17 | 0 | 0 |
| 0 | 0 | 8 | 15 |
| 0 | 0 | 15 | 21 |
G:=sub<GL(4,GF(29))| [6,0,0,0,0,28,0,0,0,0,28,0,0,0,0,28],[28,0,0,0,0,28,0,0,0,0,0,28,0,0,1,0],[1,0,0,0,0,17,0,0,0,0,8,15,0,0,15,21] >;
C14×C4⋊C4 in GAP, Magma, Sage, TeX
C_{14}\times C_4\rtimes C_4 % in TeX
G:=Group("C14xC4:C4"); // GroupNames label
G:=SmallGroup(224,151);
// by ID
G=gap.SmallGroup(224,151);
# by ID
G:=PCGroup([6,-2,-2,-2,-7,-2,-2,672,697,343]);
// Polycyclic
G:=Group<a,b,c|a^14=b^4=c^4=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations