direct product, metabelian, supersoluble, monomial, A-group, 3-hyperelementary
Aliases: C3×C6×C7⋊C3, C14⋊C33, C42⋊C32, (C3×C42)⋊3C3, C21⋊4(C3×C6), C7⋊2(C32×C6), (C3×C21)⋊14C6, SmallGroup(378,52)
Series: Derived ►Chief ►Lower central ►Upper central
| C7 — C3×C6×C7⋊C3 |
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
►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