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## G = D7×C3×C6order 252 = 22·32·7

### Direct product of C3×C6 and D7

Aliases: D7×C3×C6, C426C6, C73C62, C218(C2×C6), C143(C3×C6), (C3×C42)⋊3C2, (C3×C21)⋊8C22, SmallGroup(252,41)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C7 — D7×C3×C6
 Chief series C1 — C7 — C21 — C3×C21 — C32×D7 — D7×C3×C6
 Lower central C7 — D7×C3×C6
 Upper central C1 — C3×C6

Generators and relations for D7×C3×C6
G = < a,b,c,d | a3=b6=c7=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 168 in 60 conjugacy classes, 42 normal (10 characteristic)
C1, C2, C2, C3, C22, C6, C6, C7, C32, C2×C6, D7, C14, C3×C6, C3×C6, C21, D14, C62, C3×D7, C42, C3×C21, C6×D7, C32×D7, C3×C42, D7×C3×C6
Quotients: C1, C2, C3, C22, C6, C32, C2×C6, D7, C3×C6, D14, C62, C3×D7, C6×D7, C32×D7, D7×C3×C6

Smallest permutation representation of D7×C3×C6
On 126 points
Generators in S126
(1 48 27)(2 49 28)(3 43 22)(4 44 23)(5 45 24)(6 46 25)(7 47 26)(8 50 29)(9 51 30)(10 52 31)(11 53 32)(12 54 33)(13 55 34)(14 56 35)(15 57 36)(16 58 37)(17 59 38)(18 60 39)(19 61 40)(20 62 41)(21 63 42)(64 106 85)(65 107 86)(66 108 87)(67 109 88)(68 110 89)(69 111 90)(70 112 91)(71 113 92)(72 114 93)(73 115 94)(74 116 95)(75 117 96)(76 118 97)(77 119 98)(78 120 99)(79 121 100)(80 122 101)(81 123 102)(82 124 103)(83 125 104)(84 126 105)
(1 76 20 69 13 83)(2 77 21 70 14 84)(3 71 15 64 8 78)(4 72 16 65 9 79)(5 73 17 66 10 80)(6 74 18 67 11 81)(7 75 19 68 12 82)(22 92 36 85 29 99)(23 93 37 86 30 100)(24 94 38 87 31 101)(25 95 39 88 32 102)(26 96 40 89 33 103)(27 97 41 90 34 104)(28 98 42 91 35 105)(43 113 57 106 50 120)(44 114 58 107 51 121)(45 115 59 108 52 122)(46 116 60 109 53 123)(47 117 61 110 54 124)(48 118 62 111 55 125)(49 119 63 112 56 126)
(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 68)(2 67)(3 66)(4 65)(5 64)(6 70)(7 69)(8 73)(9 72)(10 71)(11 77)(12 76)(13 75)(14 74)(15 80)(16 79)(17 78)(18 84)(19 83)(20 82)(21 81)(22 87)(23 86)(24 85)(25 91)(26 90)(27 89)(28 88)(29 94)(30 93)(31 92)(32 98)(33 97)(34 96)(35 95)(36 101)(37 100)(38 99)(39 105)(40 104)(41 103)(42 102)(43 108)(44 107)(45 106)(46 112)(47 111)(48 110)(49 109)(50 115)(51 114)(52 113)(53 119)(54 118)(55 117)(56 116)(57 122)(58 121)(59 120)(60 126)(61 125)(62 124)(63 123)

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

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

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

90 conjugacy classes

 class 1 2A 2B 2C 3A ··· 3H 6A ··· 6H 6I ··· 6X 7A 7B 7C 14A 14B 14C 21A ··· 21X 42A ··· 42X order 1 2 2 2 3 ··· 3 6 ··· 6 6 ··· 6 7 7 7 14 14 14 21 ··· 21 42 ··· 42 size 1 1 7 7 1 ··· 1 1 ··· 1 7 ··· 7 2 2 2 2 2 2 2 ··· 2 2 ··· 2

90 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 type + + + + + image C1 C2 C2 C3 C6 C6 D7 D14 C3×D7 C6×D7 kernel D7×C3×C6 C32×D7 C3×C42 C6×D7 C3×D7 C42 C3×C6 C32 C6 C3 # reps 1 2 1 8 16 8 3 3 24 24

Matrix representation of D7×C3×C6 in GL4(𝔽43) generated by

 6 0 0 0 0 36 0 0 0 0 1 0 0 0 0 1
,
 7 0 0 0 0 36 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 0 1 0 0 42 19
,
 1 0 0 0 0 42 0 0 0 0 0 1 0 0 1 0
G:=sub<GL(4,GF(43))| [6,0,0,0,0,36,0,0,0,0,1,0,0,0,0,1],[7,0,0,0,0,36,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,0,42,0,0,1,19],[1,0,0,0,0,42,0,0,0,0,0,1,0,0,1,0] >;

D7×C3×C6 in GAP, Magma, Sage, TeX

D_7\times C_3\times C_6
% in TeX

G:=Group("D7xC3xC6");
// GroupNames label

G:=SmallGroup(252,41);
// by ID

G=gap.SmallGroup(252,41);
# by ID

G:=PCGroup([5,-2,-2,-3,-3,-7,5404]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^6=c^7=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

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