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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 [×3], C3, C4 [×2], C4 [×2], C22, C22 [×2], C6, C6 [×3], C7, C2×C4, C2×C4 [×2], D4 [×3], Q8, C12 [×2], C12 [×2], C2×C6, C2×C6 [×2], D7 [×2], C14, C14, C4○D4, C21, C2×C12, C2×C12 [×2], C3×D4 [×3], C3×Q8, Dic7 [×2], C28 [×2], D14 [×2], C2×C14, C3×D7 [×2], C42, C42, C3×C4○D4, Dic14, C4×D7 [×2], D28, C7⋊D4 [×2], C2×C28, C3×Dic7 [×2], C84 [×2], C6×D7 [×2], C2×C42, C4○D28, C3×Dic14, C12×D7 [×2], C3×D28, C3×C7⋊D4 [×2], C2×C84, C3×C4○D28
Quotients: C1, C2 [×7], C3, C22 [×7], C6 [×7], C23, C2×C6 [×7], D7, C4○D4, C22×C6, D14 [×3], C3×D7, C3×C4○D4, C22×D7, C6×D7 [×3], C4○D28, C2×C6×D7, C3×C4○D28

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

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

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

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

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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