direct product, metabelian, supersoluble, monomial, A-group
Aliases: C2×C72⋊C3, C72⋊9C6, (C7×C14)⋊2C3, C14⋊1(C7⋊C3), C7⋊3(C2×C7⋊C3), SmallGroup(294,16)
Series: Derived ►Chief ►Lower central ►Upper central
| C72 — C2×C72⋊C3 |
Generators and relations for C2×C72⋊C3
G = < a,b,c,d | a2=b7=c7=d3=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b4, dcd-1=c4 >
Smallest permutation representation of C2×C72⋊C3
►On 98 points
Generators in S98
(1 34)(2 35)(3 29)(4 30)(5 31)(6 32)(7 33)(8 56)(9 50)(10 51)(11 52)(12 53)(13 54)(14 55)(15 63)(16 57)(17 58)(18 59)(19 60)(20 61)(21 62)(22 70)(23 64)(24 65)(25 66)(26 67)(27 68)(28 69)(36 84)(37 78)(38 79)(39 80)(40 81)(41 82)(42 83)(43 91)(44 85)(45 86)(46 87)(47 88)(48 89)(49 90)(71 94)(72 95)(73 96)(74 97)(75 98)(76 92)(77 93)
(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)
(1 90 61 68 82 73 54)(2 91 62 69 83 74 55)(3 85 63 70 84 75 56)(4 86 57 64 78 76 50)(5 87 58 65 79 77 51)(6 88 59 66 80 71 52)(7 89 60 67 81 72 53)(8 29 44 15 22 36 98)(9 30 45 16 23 37 92)(10 31 46 17 24 38 93)(11 32 47 18 25 39 94)(12 33 48 19 26 40 95)(13 34 49 20 27 41 96)(14 35 43 21 28 42 97)
(2 3 5)(4 7 6)(8 93 28)(9 95 25)(10 97 22)(11 92 26)(12 94 23)(13 96 27)(14 98 24)(15 38 43)(16 40 47)(17 42 44)(18 37 48)(19 39 45)(20 41 49)(21 36 46)(29 31 35)(30 33 32)(50 72 66)(51 74 70)(52 76 67)(53 71 64)(54 73 68)(55 75 65)(56 77 69)(57 81 88)(58 83 85)(59 78 89)(60 80 86)(61 82 90)(62 84 87)(63 79 91)
G:=sub<Sym(98)| (1,34)(2,35)(3,29)(4,30)(5,31)(6,32)(7,33)(8,56)(9,50)(10,51)(11,52)(12,53)(13,54)(14,55)(15,63)(16,57)(17,58)(18,59)(19,60)(20,61)(21,62)(22,70)(23,64)(24,65)(25,66)(26,67)(27,68)(28,69)(36,84)(37,78)(38,79)(39,80)(40,81)(41,82)(42,83)(43,91)(44,85)(45,86)(46,87)(47,88)(48,89)(49,90)(71,94)(72,95)(73,96)(74,97)(75,98)(76,92)(77,93), (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), (1,90,61,68,82,73,54)(2,91,62,69,83,74,55)(3,85,63,70,84,75,56)(4,86,57,64,78,76,50)(5,87,58,65,79,77,51)(6,88,59,66,80,71,52)(7,89,60,67,81,72,53)(8,29,44,15,22,36,98)(9,30,45,16,23,37,92)(10,31,46,17,24,38,93)(11,32,47,18,25,39,94)(12,33,48,19,26,40,95)(13,34,49,20,27,41,96)(14,35,43,21,28,42,97), (2,3,5)(4,7,6)(8,93,28)(9,95,25)(10,97,22)(11,92,26)(12,94,23)(13,96,27)(14,98,24)(15,38,43)(16,40,47)(17,42,44)(18,37,48)(19,39,45)(20,41,49)(21,36,46)(29,31,35)(30,33,32)(50,72,66)(51,74,70)(52,76,67)(53,71,64)(54,73,68)(55,75,65)(56,77,69)(57,81,88)(58,83,85)(59,78,89)(60,80,86)(61,82,90)(62,84,87)(63,79,91)>;
G:=Group( (1,34)(2,35)(3,29)(4,30)(5,31)(6,32)(7,33)(8,56)(9,50)(10,51)(11,52)(12,53)(13,54)(14,55)(15,63)(16,57)(17,58)(18,59)(19,60)(20,61)(21,62)(22,70)(23,64)(24,65)(25,66)(26,67)(27,68)(28,69)(36,84)(37,78)(38,79)(39,80)(40,81)(41,82)(42,83)(43,91)(44,85)(45,86)(46,87)(47,88)(48,89)(49,90)(71,94)(72,95)(73,96)(74,97)(75,98)(76,92)(77,93), (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), (1,90,61,68,82,73,54)(2,91,62,69,83,74,55)(3,85,63,70,84,75,56)(4,86,57,64,78,76,50)(5,87,58,65,79,77,51)(6,88,59,66,80,71,52)(7,89,60,67,81,72,53)(8,29,44,15,22,36,98)(9,30,45,16,23,37,92)(10,31,46,17,24,38,93)(11,32,47,18,25,39,94)(12,33,48,19,26,40,95)(13,34,49,20,27,41,96)(14,35,43,21,28,42,97), (2,3,5)(4,7,6)(8,93,28)(9,95,25)(10,97,22)(11,92,26)(12,94,23)(13,96,27)(14,98,24)(15,38,43)(16,40,47)(17,42,44)(18,37,48)(19,39,45)(20,41,49)(21,36,46)(29,31,35)(30,33,32)(50,72,66)(51,74,70)(52,76,67)(53,71,64)(54,73,68)(55,75,65)(56,77,69)(57,81,88)(58,83,85)(59,78,89)(60,80,86)(61,82,90)(62,84,87)(63,79,91) );
G=PermutationGroup([[(1,34),(2,35),(3,29),(4,30),(5,31),(6,32),(7,33),(8,56),(9,50),(10,51),(11,52),(12,53),(13,54),(14,55),(15,63),(16,57),(17,58),(18,59),(19,60),(20,61),(21,62),(22,70),(23,64),(24,65),(25,66),(26,67),(27,68),(28,69),(36,84),(37,78),(38,79),(39,80),(40,81),(41,82),(42,83),(43,91),(44,85),(45,86),(46,87),(47,88),(48,89),(49,90),(71,94),(72,95),(73,96),(74,97),(75,98),(76,92),(77,93)], [(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)], [(1,90,61,68,82,73,54),(2,91,62,69,83,74,55),(3,85,63,70,84,75,56),(4,86,57,64,78,76,50),(5,87,58,65,79,77,51),(6,88,59,66,80,71,52),(7,89,60,67,81,72,53),(8,29,44,15,22,36,98),(9,30,45,16,23,37,92),(10,31,46,17,24,38,93),(11,32,47,18,25,39,94),(12,33,48,19,26,40,95),(13,34,49,20,27,41,96),(14,35,43,21,28,42,97)], [(2,3,5),(4,7,6),(8,93,28),(9,95,25),(10,97,22),(11,92,26),(12,94,23),(13,96,27),(14,98,24),(15,38,43),(16,40,47),(17,42,44),(18,37,48),(19,39,45),(20,41,49),(21,36,46),(29,31,35),(30,33,32),(50,72,66),(51,74,70),(52,76,67),(53,71,64),(54,73,68),(55,75,65),(56,77,69),(57,81,88),(58,83,85),(59,78,89),(60,80,86),(61,82,90),(62,84,87),(63,79,91)]])
38 conjugacy classes
| class | 1 | 2 | 3A | 3B | 6A | 6B | 7A | ··· | 7P | 14A | ··· | 14P |
| order | 1 | 2 | 3 | 3 | 6 | 6 | 7 | ··· | 7 | 14 | ··· | 14 |
| size | 1 | 1 | 49 | 49 | 49 | 49 | 3 | ··· | 3 | 3 | ··· | 3 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 3 | 3 |
| type | + | + | ||||
| image | C1 | C2 | C3 | C6 | C7⋊C3 | C2×C7⋊C3 |
| kernel | C2×C72⋊C3 | C72⋊C3 | C7×C14 | C72 | C14 | C7 |
| # reps | 1 | 1 | 2 | 2 | 16 | 16 |
Matrix representation of C2×C72⋊C3 ►in GL6(𝔽43)
| 42 | 0 | 0 | 0 | 0 | 0 |
| 0 | 42 | 0 | 0 | 0 | 0 |
| 0 | 0 | 42 | 0 | 0 | 0 |
| 0 | 0 | 0 | 42 | 0 | 0 |
| 0 | 0 | 0 | 0 | 42 | 0 |
| 0 | 0 | 0 | 0 | 0 | 42 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 1 | 19 | 18 | 0 | 0 | 0 |
| 0 | 0 | 0 | 25 | 0 | 42 |
| 0 | 0 | 0 | 25 | 24 | 17 |
| 0 | 0 | 0 | 24 | 42 | 18 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 42 | 1 | 0 |
| 0 | 0 | 0 | 42 | 0 | 1 |
| 0 | 0 | 0 | 18 | 1 | 25 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 18 | 42 | 42 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 19 | 1 | 25 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 1 | 24 |
G:=sub<GL(6,GF(43))| [42,0,0,0,0,0,0,42,0,0,0,0,0,0,42,0,0,0,0,0,0,42,0,0,0,0,0,0,42,0,0,0,0,0,0,42],[0,0,1,0,0,0,1,0,19,0,0,0,0,1,18,0,0,0,0,0,0,25,25,24,0,0,0,0,24,42,0,0,0,42,17,18],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,42,42,18,0,0,0,1,0,1,0,0,0,0,1,25],[1,18,0,0,0,0,0,42,1,0,0,0,0,42,0,0,0,0,0,0,0,19,1,1,0,0,0,1,0,1,0,0,0,25,0,24] >;
C2×C72⋊C3 in GAP, Magma, Sage, TeX
C_2\times C_7^2\rtimes C_3
% in TeX
G:=Group("C2xC7^2:C3"); // GroupNames label
G:=SmallGroup(294,16);
// by ID
G=gap.SmallGroup(294,16);
# by ID
G:=PCGroup([4,-2,-3,-7,-7,78,679]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^7=c^7=d^3=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^4,d*c*d^-1=c^4>;
// generators/relations
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