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## G = C14×C3⋊S3order 252 = 22·32·7

### Direct product of C14 and C3⋊S3

Aliases: C14×C3⋊S3, C423S3, C218D6, C6⋊(S3×C7), C32(S3×C14), (C3×C42)⋊5C2, (C3×C6)⋊2C14, C323(C2×C14), (C3×C21)⋊10C22, SmallGroup(252,44)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C14×C3⋊S3
 Chief series C1 — C3 — C32 — C3×C21 — C7×C3⋊S3 — C14×C3⋊S3
 Lower central C32 — C14×C3⋊S3
 Upper central C1 — C14

Generators and relations for C14×C3⋊S3
G = < a,b,c,d | a14=b3=c3=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b-1, dcd=c-1 >

Subgroups: 156 in 60 conjugacy classes, 30 normal (10 characteristic)
C1, C2, C2, C3, C22, S3, C6, C7, C32, D6, C14, C14, C3⋊S3, C3×C6, C21, C2×C14, C2×C3⋊S3, S3×C7, C42, C3×C21, S3×C14, C7×C3⋊S3, C3×C42, C14×C3⋊S3
Quotients: C1, C2, C22, S3, C7, D6, C14, C3⋊S3, C2×C14, C2×C3⋊S3, S3×C7, S3×C14, C7×C3⋊S3, C14×C3⋊S3

Smallest permutation representation of C14×C3⋊S3
On 126 points
Generators in S126
(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)
(1 21 56)(2 22 43)(3 23 44)(4 24 45)(5 25 46)(6 26 47)(7 27 48)(8 28 49)(9 15 50)(10 16 51)(11 17 52)(12 18 53)(13 19 54)(14 20 55)(29 60 104)(30 61 105)(31 62 106)(32 63 107)(33 64 108)(34 65 109)(35 66 110)(36 67 111)(37 68 112)(38 69 99)(39 70 100)(40 57 101)(41 58 102)(42 59 103)(71 92 121)(72 93 122)(73 94 123)(74 95 124)(75 96 125)(76 97 126)(77 98 113)(78 85 114)(79 86 115)(80 87 116)(81 88 117)(82 89 118)(83 90 119)(84 91 120)
(1 116 102)(2 117 103)(3 118 104)(4 119 105)(5 120 106)(6 121 107)(7 122 108)(8 123 109)(9 124 110)(10 125 111)(11 126 112)(12 113 99)(13 114 100)(14 115 101)(15 74 35)(16 75 36)(17 76 37)(18 77 38)(19 78 39)(20 79 40)(21 80 41)(22 81 42)(23 82 29)(24 83 30)(25 84 31)(26 71 32)(27 72 33)(28 73 34)(43 88 59)(44 89 60)(45 90 61)(46 91 62)(47 92 63)(48 93 64)(49 94 65)(50 95 66)(51 96 67)(52 97 68)(53 98 69)(54 85 70)(55 86 57)(56 87 58)
(15 50)(16 51)(17 52)(18 53)(19 54)(20 55)(21 56)(22 43)(23 44)(24 45)(25 46)(26 47)(27 48)(28 49)(29 89)(30 90)(31 91)(32 92)(33 93)(34 94)(35 95)(36 96)(37 97)(38 98)(39 85)(40 86)(41 87)(42 88)(57 79)(58 80)(59 81)(60 82)(61 83)(62 84)(63 71)(64 72)(65 73)(66 74)(67 75)(68 76)(69 77)(70 78)(99 113)(100 114)(101 115)(102 116)(103 117)(104 118)(105 119)(106 120)(107 121)(108 122)(109 123)(110 124)(111 125)(112 126)

G:=sub<Sym(126)| (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), (1,21,56)(2,22,43)(3,23,44)(4,24,45)(5,25,46)(6,26,47)(7,27,48)(8,28,49)(9,15,50)(10,16,51)(11,17,52)(12,18,53)(13,19,54)(14,20,55)(29,60,104)(30,61,105)(31,62,106)(32,63,107)(33,64,108)(34,65,109)(35,66,110)(36,67,111)(37,68,112)(38,69,99)(39,70,100)(40,57,101)(41,58,102)(42,59,103)(71,92,121)(72,93,122)(73,94,123)(74,95,124)(75,96,125)(76,97,126)(77,98,113)(78,85,114)(79,86,115)(80,87,116)(81,88,117)(82,89,118)(83,90,119)(84,91,120), (1,116,102)(2,117,103)(3,118,104)(4,119,105)(5,120,106)(6,121,107)(7,122,108)(8,123,109)(9,124,110)(10,125,111)(11,126,112)(12,113,99)(13,114,100)(14,115,101)(15,74,35)(16,75,36)(17,76,37)(18,77,38)(19,78,39)(20,79,40)(21,80,41)(22,81,42)(23,82,29)(24,83,30)(25,84,31)(26,71,32)(27,72,33)(28,73,34)(43,88,59)(44,89,60)(45,90,61)(46,91,62)(47,92,63)(48,93,64)(49,94,65)(50,95,66)(51,96,67)(52,97,68)(53,98,69)(54,85,70)(55,86,57)(56,87,58), (15,50)(16,51)(17,52)(18,53)(19,54)(20,55)(21,56)(22,43)(23,44)(24,45)(25,46)(26,47)(27,48)(28,49)(29,89)(30,90)(31,91)(32,92)(33,93)(34,94)(35,95)(36,96)(37,97)(38,98)(39,85)(40,86)(41,87)(42,88)(57,79)(58,80)(59,81)(60,82)(61,83)(62,84)(63,71)(64,72)(65,73)(66,74)(67,75)(68,76)(69,77)(70,78)(99,113)(100,114)(101,115)(102,116)(103,117)(104,118)(105,119)(106,120)(107,121)(108,122)(109,123)(110,124)(111,125)(112,126)>;

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), (1,21,56)(2,22,43)(3,23,44)(4,24,45)(5,25,46)(6,26,47)(7,27,48)(8,28,49)(9,15,50)(10,16,51)(11,17,52)(12,18,53)(13,19,54)(14,20,55)(29,60,104)(30,61,105)(31,62,106)(32,63,107)(33,64,108)(34,65,109)(35,66,110)(36,67,111)(37,68,112)(38,69,99)(39,70,100)(40,57,101)(41,58,102)(42,59,103)(71,92,121)(72,93,122)(73,94,123)(74,95,124)(75,96,125)(76,97,126)(77,98,113)(78,85,114)(79,86,115)(80,87,116)(81,88,117)(82,89,118)(83,90,119)(84,91,120), (1,116,102)(2,117,103)(3,118,104)(4,119,105)(5,120,106)(6,121,107)(7,122,108)(8,123,109)(9,124,110)(10,125,111)(11,126,112)(12,113,99)(13,114,100)(14,115,101)(15,74,35)(16,75,36)(17,76,37)(18,77,38)(19,78,39)(20,79,40)(21,80,41)(22,81,42)(23,82,29)(24,83,30)(25,84,31)(26,71,32)(27,72,33)(28,73,34)(43,88,59)(44,89,60)(45,90,61)(46,91,62)(47,92,63)(48,93,64)(49,94,65)(50,95,66)(51,96,67)(52,97,68)(53,98,69)(54,85,70)(55,86,57)(56,87,58), (15,50)(16,51)(17,52)(18,53)(19,54)(20,55)(21,56)(22,43)(23,44)(24,45)(25,46)(26,47)(27,48)(28,49)(29,89)(30,90)(31,91)(32,92)(33,93)(34,94)(35,95)(36,96)(37,97)(38,98)(39,85)(40,86)(41,87)(42,88)(57,79)(58,80)(59,81)(60,82)(61,83)(62,84)(63,71)(64,72)(65,73)(66,74)(67,75)(68,76)(69,77)(70,78)(99,113)(100,114)(101,115)(102,116)(103,117)(104,118)(105,119)(106,120)(107,121)(108,122)(109,123)(110,124)(111,125)(112,126) );

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)], [(1,21,56),(2,22,43),(3,23,44),(4,24,45),(5,25,46),(6,26,47),(7,27,48),(8,28,49),(9,15,50),(10,16,51),(11,17,52),(12,18,53),(13,19,54),(14,20,55),(29,60,104),(30,61,105),(31,62,106),(32,63,107),(33,64,108),(34,65,109),(35,66,110),(36,67,111),(37,68,112),(38,69,99),(39,70,100),(40,57,101),(41,58,102),(42,59,103),(71,92,121),(72,93,122),(73,94,123),(74,95,124),(75,96,125),(76,97,126),(77,98,113),(78,85,114),(79,86,115),(80,87,116),(81,88,117),(82,89,118),(83,90,119),(84,91,120)], [(1,116,102),(2,117,103),(3,118,104),(4,119,105),(5,120,106),(6,121,107),(7,122,108),(8,123,109),(9,124,110),(10,125,111),(11,126,112),(12,113,99),(13,114,100),(14,115,101),(15,74,35),(16,75,36),(17,76,37),(18,77,38),(19,78,39),(20,79,40),(21,80,41),(22,81,42),(23,82,29),(24,83,30),(25,84,31),(26,71,32),(27,72,33),(28,73,34),(43,88,59),(44,89,60),(45,90,61),(46,91,62),(47,92,63),(48,93,64),(49,94,65),(50,95,66),(51,96,67),(52,97,68),(53,98,69),(54,85,70),(55,86,57),(56,87,58)], [(15,50),(16,51),(17,52),(18,53),(19,54),(20,55),(21,56),(22,43),(23,44),(24,45),(25,46),(26,47),(27,48),(28,49),(29,89),(30,90),(31,91),(32,92),(33,93),(34,94),(35,95),(36,96),(37,97),(38,98),(39,85),(40,86),(41,87),(42,88),(57,79),(58,80),(59,81),(60,82),(61,83),(62,84),(63,71),(64,72),(65,73),(66,74),(67,75),(68,76),(69,77),(70,78),(99,113),(100,114),(101,115),(102,116),(103,117),(104,118),(105,119),(106,120),(107,121),(108,122),(109,123),(110,124),(111,125),(112,126)]])

84 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 6A 6B 6C 6D 7A ··· 7F 14A ··· 14F 14G ··· 14R 21A ··· 21X 42A ··· 42X order 1 2 2 2 3 3 3 3 6 6 6 6 7 ··· 7 14 ··· 14 14 ··· 14 21 ··· 21 42 ··· 42 size 1 1 9 9 2 2 2 2 2 2 2 2 1 ··· 1 1 ··· 1 9 ··· 9 2 ··· 2 2 ··· 2

84 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 type + + + + + image C1 C2 C2 C7 C14 C14 S3 D6 S3×C7 S3×C14 kernel C14×C3⋊S3 C7×C3⋊S3 C3×C42 C2×C3⋊S3 C3⋊S3 C3×C6 C42 C21 C6 C3 # reps 1 2 1 6 12 6 4 4 24 24

Matrix representation of C14×C3⋊S3 in GL4(𝔽43) generated by

 22 0 0 0 0 22 0 0 0 0 11 0 0 0 0 11
,
 42 1 0 0 42 0 0 0 0 0 0 1 0 0 42 42
,
 1 0 0 0 0 1 0 0 0 0 0 1 0 0 42 42
,
 1 42 0 0 0 42 0 0 0 0 0 42 0 0 42 0
G:=sub<GL(4,GF(43))| [22,0,0,0,0,22,0,0,0,0,11,0,0,0,0,11],[42,42,0,0,1,0,0,0,0,0,0,42,0,0,1,42],[1,0,0,0,0,1,0,0,0,0,0,42,0,0,1,42],[1,0,0,0,42,42,0,0,0,0,0,42,0,0,42,0] >;

C14×C3⋊S3 in GAP, Magma, Sage, TeX

C_{14}\times C_3\rtimes S_3
% in TeX

G:=Group("C14xC3:S3");
// GroupNames label

G:=SmallGroup(252,44);
// by ID

G=gap.SmallGroup(252,44);
# by ID

G:=PCGroup([5,-2,-2,-7,-3,-3,1123,4204]);
// Polycyclic

G:=Group<a,b,c,d|a^14=b^3=c^3=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

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