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## G = C3⋊S3×C21order 378 = 2·33·7

### Direct product of C21 and C3⋊S3

Aliases: C3⋊S3×C21, C332C14, C324C42, C3⋊(S3×C21), C217(C3×S3), (C3×C21)⋊8S3, (C3×C21)⋊19C6, C323(S3×C7), (C32×C21)⋊6C2, SmallGroup(378,56)

Series: Derived Chief Lower central Upper central

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

Generators and relations for C3⋊S3×C21
G = < a,b,c,d | a21=b3=c3=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b-1, dcd=c-1 >

Subgroups: 144 in 64 conjugacy classes, 28 normal (12 characteristic)
C1, C2, C3, C3, C3, S3, C6, C7, C32, C32, C32, C14, C3×S3, C3⋊S3, C21, C21, C21, C33, S3×C7, C42, C3×C3⋊S3, C3×C21, C3×C21, C3×C21, S3×C21, C7×C3⋊S3, C32×C21, C3⋊S3×C21
Quotients: C1, C2, C3, S3, C6, C7, C14, C3×S3, C3⋊S3, C21, S3×C7, C42, C3×C3⋊S3, S3×C21, C7×C3⋊S3, C3⋊S3×C21

Smallest permutation representation of C3⋊S3×C21
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 75 34)(2 76 35)(3 77 36)(4 78 37)(5 79 38)(6 80 39)(7 81 40)(8 82 41)(9 83 42)(10 84 22)(11 64 23)(12 65 24)(13 66 25)(14 67 26)(15 68 27)(16 69 28)(17 70 29)(18 71 30)(19 72 31)(20 73 32)(21 74 33)(43 122 88)(44 123 89)(45 124 90)(46 125 91)(47 126 92)(48 106 93)(49 107 94)(50 108 95)(51 109 96)(52 110 97)(53 111 98)(54 112 99)(55 113 100)(56 114 101)(57 115 102)(58 116 103)(59 117 104)(60 118 105)(61 119 85)(62 120 86)(63 121 87)
(1 41 68)(2 42 69)(3 22 70)(4 23 71)(5 24 72)(6 25 73)(7 26 74)(8 27 75)(9 28 76)(10 29 77)(11 30 78)(12 31 79)(13 32 80)(14 33 81)(15 34 82)(16 35 83)(17 36 84)(18 37 64)(19 38 65)(20 39 66)(21 40 67)(43 102 108)(44 103 109)(45 104 110)(46 105 111)(47 85 112)(48 86 113)(49 87 114)(50 88 115)(51 89 116)(52 90 117)(53 91 118)(54 92 119)(55 93 120)(56 94 121)(57 95 122)(58 96 123)(59 97 124)(60 98 125)(61 99 126)(62 100 106)(63 101 107)
(1 60)(2 61)(3 62)(4 63)(5 43)(6 44)(7 45)(8 46)(9 47)(10 48)(11 49)(12 50)(13 51)(14 52)(15 53)(16 54)(17 55)(18 56)(19 57)(20 58)(21 59)(22 106)(23 107)(24 108)(25 109)(26 110)(27 111)(28 112)(29 113)(30 114)(31 115)(32 116)(33 117)(34 118)(35 119)(36 120)(37 121)(38 122)(39 123)(40 124)(41 125)(42 126)(64 94)(65 95)(66 96)(67 97)(68 98)(69 99)(70 100)(71 101)(72 102)(73 103)(74 104)(75 105)(76 85)(77 86)(78 87)(79 88)(80 89)(81 90)(82 91)(83 92)(84 93)

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

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

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

126 conjugacy classes

 class 1 2 3A 3B 3C ··· 3N 6A 6B 7A ··· 7F 14A ··· 14F 21A ··· 21L 21M ··· 21CF 42A ··· 42L order 1 2 3 3 3 ··· 3 6 6 7 ··· 7 14 ··· 14 21 ··· 21 21 ··· 21 42 ··· 42 size 1 9 1 1 2 ··· 2 9 9 1 ··· 1 9 ··· 9 1 ··· 1 2 ··· 2 9 ··· 9

126 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 type + + + image C1 C2 C3 C6 C7 C14 C21 C42 S3 C3×S3 S3×C7 S3×C21 kernel C3⋊S3×C21 C32×C21 C7×C3⋊S3 C3×C21 C3×C3⋊S3 C33 C3⋊S3 C32 C3×C21 C21 C32 C3 # reps 1 1 2 2 6 6 12 12 4 8 24 48

Matrix representation of C3⋊S3×C21 in GL4(𝔽43) generated by

 23 0 0 0 0 23 0 0 0 0 13 0 0 0 0 13
,
 6 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 36 0 0 0 0 6
,
 0 1 0 0 1 0 0 0 0 0 0 1 0 0 1 0
G:=sub<GL(4,GF(43))| [23,0,0,0,0,23,0,0,0,0,13,0,0,0,0,13],[6,0,0,0,0,36,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,36,0,0,0,0,6],[0,1,0,0,1,0,0,0,0,0,0,1,0,0,1,0] >;

C3⋊S3×C21 in GAP, Magma, Sage, TeX

C_3\rtimes S_3\times C_{21}
% in TeX

G:=Group("C3:S3xC21");
// GroupNames label

G:=SmallGroup(378,56);
// by ID

G=gap.SmallGroup(378,56);
# by ID

G:=PCGroup([5,-2,-3,-7,-3,-3,1683,6304]);
// Polycyclic

G:=Group<a,b,c,d|a^21=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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