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## G = Q8×C2×C14order 224 = 25·7

### Direct product of C2×C14 and Q8

direct product, metabelian, nilpotent (class 2), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — Q8×C2×C14
 Chief series C1 — C2 — C14 — C28 — C7×Q8 — Q8×C14 — Q8×C2×C14
 Lower central C1 — C2 — Q8×C2×C14
 Upper central C1 — C22×C14 — Q8×C2×C14

Generators and relations for Q8×C2×C14
G = < a,b,c,d | a2=b14=c4=1, d2=c2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 156, all normal (8 characteristic)
C1, C2, C2, C4, C22, C7, C2×C4, Q8, C23, C14, C14, C22×C4, C2×Q8, C28, C2×C14, C22×Q8, C2×C28, C7×Q8, C22×C14, C22×C28, Q8×C14, Q8×C2×C14
Quotients: C1, C2, C22, C7, Q8, C23, C14, C2×Q8, C24, C2×C14, C22×Q8, C7×Q8, C22×C14, Q8×C14, C23×C14, Q8×C2×C14

Smallest permutation representation of Q8×C2×C14
Regular action on 224 points
Generators in S224
(1 107)(2 108)(3 109)(4 110)(5 111)(6 112)(7 99)(8 100)(9 101)(10 102)(11 103)(12 104)(13 105)(14 106)(15 42)(16 29)(17 30)(18 31)(19 32)(20 33)(21 34)(22 35)(23 36)(24 37)(25 38)(26 39)(27 40)(28 41)(43 125)(44 126)(45 113)(46 114)(47 115)(48 116)(49 117)(50 118)(51 119)(52 120)(53 121)(54 122)(55 123)(56 124)(57 199)(58 200)(59 201)(60 202)(61 203)(62 204)(63 205)(64 206)(65 207)(66 208)(67 209)(68 210)(69 197)(70 198)(71 92)(72 93)(73 94)(74 95)(75 96)(76 97)(77 98)(78 85)(79 86)(80 87)(81 88)(82 89)(83 90)(84 91)(127 150)(128 151)(129 152)(130 153)(131 154)(132 141)(133 142)(134 143)(135 144)(136 145)(137 146)(138 147)(139 148)(140 149)(155 178)(156 179)(157 180)(158 181)(159 182)(160 169)(161 170)(162 171)(163 172)(164 173)(165 174)(166 175)(167 176)(168 177)(183 223)(184 224)(185 211)(186 212)(187 213)(188 214)(189 215)(190 216)(191 217)(192 218)(193 219)(194 220)(195 221)(196 222)
(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)
(1 191 33 64)(2 192 34 65)(3 193 35 66)(4 194 36 67)(5 195 37 68)(6 196 38 69)(7 183 39 70)(8 184 40 57)(9 185 41 58)(10 186 42 59)(11 187 29 60)(12 188 30 61)(13 189 31 62)(14 190 32 63)(15 201 102 212)(16 202 103 213)(17 203 104 214)(18 204 105 215)(19 205 106 216)(20 206 107 217)(21 207 108 218)(22 208 109 219)(23 209 110 220)(24 210 111 221)(25 197 112 222)(26 198 99 223)(27 199 100 224)(28 200 101 211)(43 136 84 179)(44 137 71 180)(45 138 72 181)(46 139 73 182)(47 140 74 169)(48 127 75 170)(49 128 76 171)(50 129 77 172)(51 130 78 173)(52 131 79 174)(53 132 80 175)(54 133 81 176)(55 134 82 177)(56 135 83 178)(85 164 119 153)(86 165 120 154)(87 166 121 141)(88 167 122 142)(89 168 123 143)(90 155 124 144)(91 156 125 145)(92 157 126 146)(93 158 113 147)(94 159 114 148)(95 160 115 149)(96 161 116 150)(97 162 117 151)(98 163 118 152)
(1 164 33 153)(2 165 34 154)(3 166 35 141)(4 167 36 142)(5 168 37 143)(6 155 38 144)(7 156 39 145)(8 157 40 146)(9 158 41 147)(10 159 42 148)(11 160 29 149)(12 161 30 150)(13 162 31 151)(14 163 32 152)(15 139 102 182)(16 140 103 169)(17 127 104 170)(18 128 105 171)(19 129 106 172)(20 130 107 173)(21 131 108 174)(22 132 109 175)(23 133 110 176)(24 134 111 177)(25 135 112 178)(26 136 99 179)(27 137 100 180)(28 138 101 181)(43 223 84 198)(44 224 71 199)(45 211 72 200)(46 212 73 201)(47 213 74 202)(48 214 75 203)(49 215 76 204)(50 216 77 205)(51 217 78 206)(52 218 79 207)(53 219 80 208)(54 220 81 209)(55 221 82 210)(56 222 83 197)(57 126 184 92)(58 113 185 93)(59 114 186 94)(60 115 187 95)(61 116 188 96)(62 117 189 97)(63 118 190 98)(64 119 191 85)(65 120 192 86)(66 121 193 87)(67 122 194 88)(68 123 195 89)(69 124 196 90)(70 125 183 91)

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

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

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

Q8×C2×C14 is a maximal subgroup of
(Q8×C14)⋊6C4  (C7×Q8)⋊13D4  (C2×C14)⋊8Q16  C14.C22≀C2  (Q8×C14)⋊7C4  (C22×Q8)⋊D7  C14.422- 1+4  C14.442- 1+4  C14.452- 1+4

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 2 2 type + + + - image C1 C2 C2 C7 C14 C14 Q8 C7×Q8 kernel Q8×C2×C14 C22×C28 Q8×C14 C22×Q8 C22×C4 C2×Q8 C2×C14 C22 # reps 1 3 12 6 18 72 4 24

Matrix representation of Q8×C2×C14 in GL4(𝔽29) generated by

 1 0 0 0 0 28 0 0 0 0 28 0 0 0 0 28
,
 28 0 0 0 0 1 0 0 0 0 25 0 0 0 0 25
,
 1 0 0 0 0 1 0 0 0 0 17 0 0 0 0 12
,
 1 0 0 0 0 1 0 0 0 0 0 1 0 0 28 0
G:=sub<GL(4,GF(29))| [1,0,0,0,0,28,0,0,0,0,28,0,0,0,0,28],[28,0,0,0,0,1,0,0,0,0,25,0,0,0,0,25],[1,0,0,0,0,1,0,0,0,0,17,0,0,0,0,12],[1,0,0,0,0,1,0,0,0,0,0,28,0,0,1,0] >;

Q8×C2×C14 in GAP, Magma, Sage, TeX

Q_8\times C_2\times C_{14}
% in TeX

G:=Group("Q8xC2xC14");
// GroupNames label

G:=SmallGroup(224,191);
// by ID

G=gap.SmallGroup(224,191);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-7,-2,672,1369,679]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^14=c^4=1,d^2=c^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

׿
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𝔽