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## G = C3×C4○D28order 336 = 24·3·7

### Direct product of C3 and C4○D28

Series: Derived Chief Lower central Upper central

 Derived series C1 — C14 — C3×C4○D28
 Chief series C1 — C7 — C14 — C42 — C6×D7 — C12×D7 — C3×C4○D28
 Lower central C7 — C14 — C3×C4○D28
 Upper central C1 — C12 — C2×C12

Generators and relations for C3×C4○D28
G = < a,b,c,d | a3=b4=d2=1, c14=b2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b2c13 >

Subgroups: 272 in 80 conjugacy classes, 46 normal (30 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C6, C7, C2×C4, C2×C4, D4, Q8, C12, C12, C2×C6, C2×C6, D7, C14, C14, C4○D4, C21, C2×C12, C2×C12, C3×D4, C3×Q8, Dic7, C28, D14, C2×C14, C3×D7, C42, C42, C3×C4○D4, Dic14, C4×D7, D28, C7⋊D4, C2×C28, C3×Dic7, C84, C6×D7, C2×C42, C4○D28, C3×Dic14, C12×D7, C3×D28, C3×C7⋊D4, C2×C84, C3×C4○D28
Quotients: C1, C2, C3, C22, C6, C23, C2×C6, D7, C4○D4, C22×C6, D14, C3×D7, C3×C4○D4, C22×D7, C6×D7, C4○D28, C2×C6×D7, C3×C4○D28

Smallest permutation representation of C3×C4○D28
On 168 points
Generators in S168
(1 104 49)(2 105 50)(3 106 51)(4 107 52)(5 108 53)(6 109 54)(7 110 55)(8 111 56)(9 112 29)(10 85 30)(11 86 31)(12 87 32)(13 88 33)(14 89 34)(15 90 35)(16 91 36)(17 92 37)(18 93 38)(19 94 39)(20 95 40)(21 96 41)(22 97 42)(23 98 43)(24 99 44)(25 100 45)(26 101 46)(27 102 47)(28 103 48)(57 164 115)(58 165 116)(59 166 117)(60 167 118)(61 168 119)(62 141 120)(63 142 121)(64 143 122)(65 144 123)(66 145 124)(67 146 125)(68 147 126)(69 148 127)(70 149 128)(71 150 129)(72 151 130)(73 152 131)(74 153 132)(75 154 133)(76 155 134)(77 156 135)(78 157 136)(79 158 137)(80 159 138)(81 160 139)(82 161 140)(83 162 113)(84 163 114)
(1 135 15 121)(2 136 16 122)(3 137 17 123)(4 138 18 124)(5 139 19 125)(6 140 20 126)(7 113 21 127)(8 114 22 128)(9 115 23 129)(10 116 24 130)(11 117 25 131)(12 118 26 132)(13 119 27 133)(14 120 28 134)(29 164 43 150)(30 165 44 151)(31 166 45 152)(32 167 46 153)(33 168 47 154)(34 141 48 155)(35 142 49 156)(36 143 50 157)(37 144 51 158)(38 145 52 159)(39 146 53 160)(40 147 54 161)(41 148 55 162)(42 149 56 163)(57 98 71 112)(58 99 72 85)(59 100 73 86)(60 101 74 87)(61 102 75 88)(62 103 76 89)(63 104 77 90)(64 105 78 91)(65 106 79 92)(66 107 80 93)(67 108 81 94)(68 109 82 95)(69 110 83 96)(70 111 84 97)
(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)
(1 28)(2 27)(3 26)(4 25)(5 24)(6 23)(7 22)(8 21)(9 20)(10 19)(11 18)(12 17)(13 16)(14 15)(29 40)(30 39)(31 38)(32 37)(33 36)(34 35)(41 56)(42 55)(43 54)(44 53)(45 52)(46 51)(47 50)(48 49)(57 68)(58 67)(59 66)(60 65)(61 64)(62 63)(69 84)(70 83)(71 82)(72 81)(73 80)(74 79)(75 78)(76 77)(85 94)(86 93)(87 92)(88 91)(89 90)(95 112)(96 111)(97 110)(98 109)(99 108)(100 107)(101 106)(102 105)(103 104)(113 128)(114 127)(115 126)(116 125)(117 124)(118 123)(119 122)(120 121)(129 140)(130 139)(131 138)(132 137)(133 136)(134 135)(141 142)(143 168)(144 167)(145 166)(146 165)(147 164)(148 163)(149 162)(150 161)(151 160)(152 159)(153 158)(154 157)(155 156)

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

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

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

102 conjugacy classes

 class 1 2A 2B 2C 2D 3A 3B 4A 4B 4C 4D 4E 6A 6B 6C 6D 6E 6F 6G 6H 7A 7B 7C 12A 12B 12C 12D 12E 12F 12G 12H 12I 12J 14A ··· 14I 21A ··· 21F 28A ··· 28L 42A ··· 42R 84A ··· 84X order 1 2 2 2 2 3 3 4 4 4 4 4 6 6 6 6 6 6 6 6 7 7 7 12 12 12 12 12 12 12 12 12 12 14 ··· 14 21 ··· 21 28 ··· 28 42 ··· 42 84 ··· 84 size 1 1 2 14 14 1 1 1 1 2 14 14 1 1 2 2 14 14 14 14 2 2 2 1 1 1 1 2 2 14 14 14 14 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2

102 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + + image C1 C2 C2 C2 C2 C2 C3 C6 C6 C6 C6 C6 D7 C4○D4 D14 D14 C3×D7 C3×C4○D4 C6×D7 C6×D7 C4○D28 C3×C4○D28 kernel C3×C4○D28 C3×Dic14 C12×D7 C3×D28 C3×C7⋊D4 C2×C84 C4○D28 Dic14 C4×D7 D28 C7⋊D4 C2×C28 C2×C12 C21 C12 C2×C6 C2×C4 C7 C4 C22 C3 C1 # reps 1 1 2 1 2 1 2 2 4 2 4 2 3 2 6 3 6 4 12 6 12 24

Matrix representation of C3×C4○D28 in GL3(𝔽337) generated by

 128 0 0 0 1 0 0 0 1
,
 336 0 0 0 148 0 0 0 148
,
 1 0 0 0 313 24 0 313 38
,
 336 0 0 0 313 24 0 299 24
G:=sub<GL(3,GF(337))| [128,0,0,0,1,0,0,0,1],[336,0,0,0,148,0,0,0,148],[1,0,0,0,313,313,0,24,38],[336,0,0,0,313,299,0,24,24] >;

C3×C4○D28 in GAP, Magma, Sage, TeX

C_3\times C_4\circ D_{28}
% in TeX

G:=Group("C3xC4oD28");
// GroupNames label

G:=SmallGroup(336,177);
// by ID

G=gap.SmallGroup(336,177);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-2,-7,151,506,10373]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^4=d^2=1,c^14=b^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=b^2*c^13>;
// generators/relations

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