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## G = C3×C6×C7⋊C3order 378 = 2·33·7

### Direct product of C3×C6 and C7⋊C3

Aliases: C3×C6×C7⋊C3, C14⋊C33, C42⋊C32, (C3×C42)⋊3C3, C214(C3×C6), C72(C32×C6), (C3×C21)⋊14C6, SmallGroup(378,52)

Series: Derived Chief Lower central Upper central

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

Generators and relations for C3×C6×C7⋊C3
G = < a,b,c,d | a3=b6=c7=d3=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c4 >

Subgroups: 376 in 112 conjugacy classes, 68 normal (10 characteristic)
C1, C2, C3, C3, C6, C6, C7, C32, C32, C14, C3×C6, C3×C6, C7⋊C3, C21, C33, C2×C7⋊C3, C42, C32×C6, C3×C7⋊C3, C3×C21, C6×C7⋊C3, C3×C42, C32×C7⋊C3, C3×C6×C7⋊C3
Quotients: C1, C2, C3, C6, C32, C3×C6, C7⋊C3, C33, C2×C7⋊C3, C32×C6, C3×C7⋊C3, C6×C7⋊C3, C32×C7⋊C3, C3×C6×C7⋊C3

Smallest permutation representation of C3×C6×C7⋊C3
On 126 points
Generators in S126
(1 57 29)(2 58 30)(3 59 31)(4 60 32)(5 61 33)(6 62 34)(7 63 35)(8 43 36)(9 44 37)(10 45 38)(11 46 39)(12 47 40)(13 48 41)(14 49 42)(15 50 22)(16 51 23)(17 52 24)(18 53 25)(19 54 26)(20 55 27)(21 56 28)(64 120 92)(65 121 93)(66 122 94)(67 123 95)(68 124 96)(69 125 97)(70 126 98)(71 106 99)(72 107 100)(73 108 101)(74 109 102)(75 110 103)(76 111 104)(77 112 105)(78 113 85)(79 114 86)(80 115 87)(81 116 88)(82 117 89)(83 118 90)(84 119 91)
(1 71 15 64 8 78)(2 72 16 65 9 79)(3 73 17 66 10 80)(4 74 18 67 11 81)(5 75 19 68 12 82)(6 76 20 69 13 83)(7 77 21 70 14 84)(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 43 22)(2 45 26)(3 47 23)(4 49 27)(5 44 24)(6 46 28)(7 48 25)(8 50 29)(9 52 33)(10 54 30)(11 56 34)(12 51 31)(13 53 35)(14 55 32)(15 57 36)(16 59 40)(17 61 37)(18 63 41)(19 58 38)(20 60 42)(21 62 39)(64 106 85)(65 108 89)(66 110 86)(67 112 90)(68 107 87)(69 109 91)(70 111 88)(71 113 92)(72 115 96)(73 117 93)(74 119 97)(75 114 94)(76 116 98)(77 118 95)(78 120 99)(79 122 103)(80 124 100)(81 126 104)(82 121 101)(83 123 105)(84 125 102)

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

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

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

90 conjugacy classes

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

90 irreducible representations

 dim 1 1 1 1 1 1 3 3 3 3 type + + image C1 C2 C3 C3 C6 C6 C7⋊C3 C2×C7⋊C3 C3×C7⋊C3 C6×C7⋊C3 kernel C3×C6×C7⋊C3 C32×C7⋊C3 C6×C7⋊C3 C3×C42 C3×C7⋊C3 C3×C21 C3×C6 C32 C6 C3 # reps 1 1 24 2 24 2 2 2 16 16

Matrix representation of C3×C6×C7⋊C3 in GL4(𝔽43) generated by

 6 0 0 0 0 36 0 0 0 0 36 0 0 0 0 36
,
 6 0 0 0 0 37 0 0 0 0 37 0 0 0 0 37
,
 1 0 0 0 0 0 0 1 0 1 0 19 0 0 1 18
,
 6 0 0 0 0 1 0 18 0 0 0 42 0 0 1 42
G:=sub<GL(4,GF(43))| [6,0,0,0,0,36,0,0,0,0,36,0,0,0,0,36],[6,0,0,0,0,37,0,0,0,0,37,0,0,0,0,37],[1,0,0,0,0,0,1,0,0,0,0,1,0,1,19,18],[6,0,0,0,0,1,0,0,0,0,0,1,0,18,42,42] >;

C3×C6×C7⋊C3 in GAP, Magma, Sage, TeX

C_3\times C_6\times C_7\rtimes C_3
% in TeX

G:=Group("C3xC6xC7:C3");
// GroupNames label

G:=SmallGroup(378,52);
// by ID

G=gap.SmallGroup(378,52);
# by ID

G:=PCGroup([5,-2,-3,-3,-3,-7,1359]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^6=c^7=d^3=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^4>;
// generators/relations

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