metabelian, soluble, monomial, A-group
Aliases: C42⋊(C7⋊C3), C7⋊(C42⋊C3), (C4×C28)⋊2C3, C22.(C7⋊A4), (C2×C14).2A4, SmallGroup(336,57)
Series: Derived ►Chief ►Lower central ►Upper central
| C4×C28 — C42⋊(C7⋊C3) |
Generators and relations for C42⋊(C7⋊C3)
G = < a,b,c,d | a4=b4=c7=d3=1, ab=ba, ac=ca, dad-1=ab-1, bc=cb, dbd-1=a-1b2, dcd-1=c4 >
Smallest permutation representation of C42⋊(C7⋊C3)
►On 84 points
Generators in S84
(8 68 82 71)(9 69 83 72)(10 70 84 73)(11 64 78 74)(12 65 79 75)(13 66 80 76)(14 67 81 77)(36 61 49 50)(37 62 43 51)(38 63 44 52)(39 57 45 53)(40 58 46 54)(41 59 47 55)(42 60 48 56)
(1 33 17 25)(2 34 18 26)(3 35 19 27)(4 29 20 28)(5 30 21 22)(6 31 15 23)(7 32 16 24)(8 82)(9 83)(10 84)(11 78)(12 79)(13 80)(14 81)(36 61 49 50)(37 62 43 51)(38 63 44 52)(39 57 45 53)(40 58 46 54)(41 59 47 55)(42 60 48 56)(64 74)(65 75)(66 76)(67 77)(68 71)(69 72)(70 73)
(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)
(1 64 37)(2 66 41)(3 68 38)(4 70 42)(5 65 39)(6 67 36)(7 69 40)(8 63 35)(9 58 32)(10 60 29)(11 62 33)(12 57 30)(13 59 34)(14 61 31)(15 77 49)(16 72 46)(17 74 43)(18 76 47)(19 71 44)(20 73 48)(21 75 45)(22 79 53)(23 81 50)(24 83 54)(25 78 51)(26 80 55)(27 82 52)(28 84 56)
G:=sub<Sym(84)| (8,68,82,71)(9,69,83,72)(10,70,84,73)(11,64,78,74)(12,65,79,75)(13,66,80,76)(14,67,81,77)(36,61,49,50)(37,62,43,51)(38,63,44,52)(39,57,45,53)(40,58,46,54)(41,59,47,55)(42,60,48,56), (1,33,17,25)(2,34,18,26)(3,35,19,27)(4,29,20,28)(5,30,21,22)(6,31,15,23)(7,32,16,24)(8,82)(9,83)(10,84)(11,78)(12,79)(13,80)(14,81)(36,61,49,50)(37,62,43,51)(38,63,44,52)(39,57,45,53)(40,58,46,54)(41,59,47,55)(42,60,48,56)(64,74)(65,75)(66,76)(67,77)(68,71)(69,72)(70,73), (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), (1,64,37)(2,66,41)(3,68,38)(4,70,42)(5,65,39)(6,67,36)(7,69,40)(8,63,35)(9,58,32)(10,60,29)(11,62,33)(12,57,30)(13,59,34)(14,61,31)(15,77,49)(16,72,46)(17,74,43)(18,76,47)(19,71,44)(20,73,48)(21,75,45)(22,79,53)(23,81,50)(24,83,54)(25,78,51)(26,80,55)(27,82,52)(28,84,56)>;
G:=Group( (8,68,82,71)(9,69,83,72)(10,70,84,73)(11,64,78,74)(12,65,79,75)(13,66,80,76)(14,67,81,77)(36,61,49,50)(37,62,43,51)(38,63,44,52)(39,57,45,53)(40,58,46,54)(41,59,47,55)(42,60,48,56), (1,33,17,25)(2,34,18,26)(3,35,19,27)(4,29,20,28)(5,30,21,22)(6,31,15,23)(7,32,16,24)(8,82)(9,83)(10,84)(11,78)(12,79)(13,80)(14,81)(36,61,49,50)(37,62,43,51)(38,63,44,52)(39,57,45,53)(40,58,46,54)(41,59,47,55)(42,60,48,56)(64,74)(65,75)(66,76)(67,77)(68,71)(69,72)(70,73), (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), (1,64,37)(2,66,41)(3,68,38)(4,70,42)(5,65,39)(6,67,36)(7,69,40)(8,63,35)(9,58,32)(10,60,29)(11,62,33)(12,57,30)(13,59,34)(14,61,31)(15,77,49)(16,72,46)(17,74,43)(18,76,47)(19,71,44)(20,73,48)(21,75,45)(22,79,53)(23,81,50)(24,83,54)(25,78,51)(26,80,55)(27,82,52)(28,84,56) );
G=PermutationGroup([[(8,68,82,71),(9,69,83,72),(10,70,84,73),(11,64,78,74),(12,65,79,75),(13,66,80,76),(14,67,81,77),(36,61,49,50),(37,62,43,51),(38,63,44,52),(39,57,45,53),(40,58,46,54),(41,59,47,55),(42,60,48,56)], [(1,33,17,25),(2,34,18,26),(3,35,19,27),(4,29,20,28),(5,30,21,22),(6,31,15,23),(7,32,16,24),(8,82),(9,83),(10,84),(11,78),(12,79),(13,80),(14,81),(36,61,49,50),(37,62,43,51),(38,63,44,52),(39,57,45,53),(40,58,46,54),(41,59,47,55),(42,60,48,56),(64,74),(65,75),(66,76),(67,77),(68,71),(69,72),(70,73)], [(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)], [(1,64,37),(2,66,41),(3,68,38),(4,70,42),(5,65,39),(6,67,36),(7,69,40),(8,63,35),(9,58,32),(10,60,29),(11,62,33),(12,57,30),(13,59,34),(14,61,31),(15,77,49),(16,72,46),(17,74,43),(18,76,47),(19,71,44),(20,73,48),(21,75,45),(22,79,53),(23,81,50),(24,83,54),(25,78,51),(26,80,55),(27,82,52),(28,84,56)]])
40 conjugacy classes
| class | 1 | 2 | 3A | 3B | 4A | 4B | 4C | 4D | 7A | 7B | 14A | ··· | 14F | 28A | ··· | 28X |
| order | 1 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 7 | 7 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 3 | 112 | 112 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | ··· | 3 | 3 | ··· | 3 |
40 irreducible representations
| dim | 1 | 1 | 3 | 3 | 3 | 3 | 3 |
| type | + | + | |||||
| image | C1 | C3 | A4 | C7⋊C3 | C42⋊C3 | C7⋊A4 | C42⋊(C7⋊C3) |
| kernel | C42⋊(C7⋊C3) | C4×C28 | C2×C14 | C42 | C7 | C22 | C1 |
| # reps | 1 | 2 | 1 | 2 | 4 | 6 | 24 |
Matrix representation of C42⋊(C7⋊C3) ►in GL3(𝔽337) generated by
| 1 | 0 | 0 |
| 0 | 148 | 0 |
| 0 | 0 | 189 |
| 148 | 0 | 0 |
| 0 | 148 | 0 |
| 0 | 0 | 336 |
| 79 | 0 | 0 |
| 0 | 175 | 0 |
| 0 | 0 | 295 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
G:=sub<GL(3,GF(337))| [1,0,0,0,148,0,0,0,189],[148,0,0,0,148,0,0,0,336],[79,0,0,0,175,0,0,0,295],[0,0,1,1,0,0,0,1,0] >;
C42⋊(C7⋊C3) in GAP, Magma, Sage, TeX
C_4^2\rtimes (C_7\rtimes C_3)
% in TeX
G:=Group("C4^2:(C7:C3)"); // GroupNames label
G:=SmallGroup(336,57);
// by ID
G=gap.SmallGroup(336,57);
# by ID
G:=PCGroup([6,-3,-2,2,-7,-2,2,73,1015,3188,266,579,5044,9077]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^7=d^3=1,a*b=b*a,a*c=c*a,d*a*d^-1=a*b^-1,b*c=c*b,d*b*d^-1=a^-1*b^2,d*c*d^-1=c^4>;
// generators/relations
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