Copied to
clipboard

## G = C4×C7⋊D7order 392 = 23·72

### Direct product of C4 and C7⋊D7

Aliases: C4×C7⋊D7, C282D7, C14.13D14, C72(C4×D7), (C7×C28)⋊4C2, C725(C2×C4), C7⋊Dic74C2, (C7×C14).12C22, C2.1(C2×C7⋊D7), (C2×C7⋊D7).3C2, SmallGroup(392,29)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C72 — C4×C7⋊D7
 Chief series C1 — C7 — C72 — C7×C14 — C2×C7⋊D7 — C4×C7⋊D7
 Lower central C72 — C4×C7⋊D7
 Upper central C1 — C4

Generators and relations for C4×C7⋊D7
G = < a,b,c,d | a4=b7=c7=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b-1, dcd=c-1 >

Subgroups: 560 in 80 conjugacy classes, 35 normal (9 characteristic)
C1, C2, C2, C4, C4, C22, C7, C2×C4, D7, C14, Dic7, C28, D14, C72, C4×D7, C7⋊D7, C7×C14, C7⋊Dic7, C7×C28, C2×C7⋊D7, C4×C7⋊D7
Quotients: C1, C2, C4, C22, C2×C4, D7, D14, C4×D7, C7⋊D7, C2×C7⋊D7, C4×C7⋊D7

Smallest permutation representation of C4×C7⋊D7
On 196 points
Generators in S196
(1 16 106 155)(2 17 107 156)(3 18 108 157)(4 19 109 158)(5 20 110 159)(6 21 111 160)(7 15 112 161)(8 105 154 63)(9 99 148 57)(10 100 149 58)(11 101 150 59)(12 102 151 60)(13 103 152 61)(14 104 153 62)(22 119 168 70)(23 113 162 64)(24 114 163 65)(25 115 164 66)(26 116 165 67)(27 117 166 68)(28 118 167 69)(29 126 175 77)(30 120 169 71)(31 121 170 72)(32 122 171 73)(33 123 172 74)(34 124 173 75)(35 125 174 76)(36 133 182 84)(37 127 176 78)(38 128 177 79)(39 129 178 80)(40 130 179 81)(41 131 180 82)(42 132 181 83)(43 140 189 91)(44 134 183 85)(45 135 184 86)(46 136 185 87)(47 137 186 88)(48 138 187 89)(49 139 188 90)(50 193 95 144)(51 194 96 145)(52 195 97 146)(53 196 98 147)(54 190 92 141)(55 191 93 142)(56 192 94 143)
(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)
(1 53 83 68 60 87 72)(2 54 84 69 61 88 73)(3 55 78 70 62 89 74)(4 56 79 64 63 90 75)(5 50 80 65 57 91 76)(6 51 81 66 58 85 77)(7 52 82 67 59 86 71)(8 49 34 19 192 38 23)(9 43 35 20 193 39 24)(10 44 29 21 194 40 25)(11 45 30 15 195 41 26)(12 46 31 16 196 42 27)(13 47 32 17 190 36 28)(14 48 33 18 191 37 22)(92 133 118 103 137 122 107)(93 127 119 104 138 123 108)(94 128 113 105 139 124 109)(95 129 114 99 140 125 110)(96 130 115 100 134 126 111)(97 131 116 101 135 120 112)(98 132 117 102 136 121 106)(141 182 167 152 186 171 156)(142 176 168 153 187 172 157)(143 177 162 154 188 173 158)(144 178 163 148 189 174 159)(145 179 164 149 183 175 160)(146 180 165 150 184 169 161)(147 181 166 151 185 170 155)
(1 72)(2 71)(3 77)(4 76)(5 75)(6 74)(7 73)(8 39)(9 38)(10 37)(11 36)(12 42)(13 41)(14 40)(15 32)(16 31)(17 30)(18 29)(19 35)(20 34)(21 33)(22 25)(23 24)(26 28)(43 192)(44 191)(45 190)(46 196)(47 195)(48 194)(49 193)(50 90)(51 89)(52 88)(53 87)(54 86)(55 85)(56 91)(57 79)(58 78)(59 84)(60 83)(61 82)(62 81)(63 80)(64 65)(66 70)(67 69)(92 135)(93 134)(94 140)(95 139)(96 138)(97 137)(98 136)(99 128)(100 127)(101 133)(102 132)(103 131)(104 130)(105 129)(106 121)(107 120)(108 126)(109 125)(110 124)(111 123)(112 122)(113 114)(115 119)(116 118)(141 184)(142 183)(143 189)(144 188)(145 187)(146 186)(147 185)(148 177)(149 176)(150 182)(151 181)(152 180)(153 179)(154 178)(155 170)(156 169)(157 175)(158 174)(159 173)(160 172)(161 171)(162 163)(164 168)(165 167)

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

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

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

104 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 7A ··· 7X 14A ··· 14X 28A ··· 28AV order 1 2 2 2 4 4 4 4 7 ··· 7 14 ··· 14 28 ··· 28 size 1 1 49 49 1 1 49 49 2 ··· 2 2 ··· 2 2 ··· 2

104 irreducible representations

 dim 1 1 1 1 1 2 2 2 type + + + + + + image C1 C2 C2 C2 C4 D7 D14 C4×D7 kernel C4×C7⋊D7 C7⋊Dic7 C7×C28 C2×C7⋊D7 C7⋊D7 C28 C14 C7 # reps 1 1 1 1 4 24 24 48

Matrix representation of C4×C7⋊D7 in GL4(𝔽29) generated by

 1 0 0 0 0 1 0 0 0 0 17 0 0 0 0 17
,
 28 3 0 0 26 8 0 0 0 0 8 26 0 0 3 28
,
 0 1 0 0 28 3 0 0 0 0 0 1 0 0 28 3
,
 0 1 0 0 1 0 0 0 0 0 28 0 0 0 26 1
G:=sub<GL(4,GF(29))| [1,0,0,0,0,1,0,0,0,0,17,0,0,0,0,17],[28,26,0,0,3,8,0,0,0,0,8,3,0,0,26,28],[0,28,0,0,1,3,0,0,0,0,0,28,0,0,1,3],[0,1,0,0,1,0,0,0,0,0,28,26,0,0,0,1] >;

C4×C7⋊D7 in GAP, Magma, Sage, TeX

C_4\times C_7\rtimes D_7
% in TeX

G:=Group("C4xC7:D7");
// GroupNames label

G:=SmallGroup(392,29);
// by ID

G=gap.SmallGroup(392,29);
# by ID

G:=PCGroup([5,-2,-2,-2,-7,-7,26,963,8404]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^7=c^7=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations

׿
×
𝔽