direct product, metabelian, soluble, monomial, A-group
Aliases: C3×C7⋊A4, C21⋊A4, C7⋊2(C3×A4), (C2×C42)⋊2C3, (C2×C14)⋊3C32, (C2×C6)⋊(C7⋊C3), C22⋊2(C3×C7⋊C3), SmallGroup(252,40)
Series: Derived ►Chief ►Lower central ►Upper central
| C2×C14 — C3×C7⋊A4 |
Generators and relations for C3×C7⋊A4
G = < a,b,c,d,e | a3=b7=c2=d2=e3=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, ebe-1=b4, ece-1=cd=dc, ede-1=c >
Smallest permutation representation of C3×C7⋊A4
►On 84 points
Generators in S84
(1 57 29)(2 58 30)(3 59 31)(4 60 32)(5 61 33)(6 62 34)(7 63 35)(8 64 36)(9 65 37)(10 66 38)(11 67 39)(12 68 40)(13 69 41)(14 70 42)(15 71 43)(16 72 44)(17 73 45)(18 74 46)(19 75 47)(20 76 48)(21 77 49)(22 78 50)(23 79 51)(24 80 52)(25 81 53)(26 82 54)(27 83 55)(28 84 56)
(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 8)(2 9)(3 10)(4 11)(5 12)(6 13)(7 14)(15 22)(16 23)(17 24)(18 25)(19 26)(20 27)(21 28)(29 36)(30 37)(31 38)(32 39)(33 40)(34 41)(35 42)(43 50)(44 51)(45 52)(46 53)(47 54)(48 55)(49 56)(57 64)(58 65)(59 66)(60 67)(61 68)(62 69)(63 70)(71 78)(72 79)(73 80)(74 81)(75 82)(76 83)(77 84)
(1 22)(2 23)(3 24)(4 25)(5 26)(6 27)(7 28)(8 15)(9 16)(10 17)(11 18)(12 19)(13 20)(14 21)(29 50)(30 51)(31 52)(32 53)(33 54)(34 55)(35 56)(36 43)(37 44)(38 45)(39 46)(40 47)(41 48)(42 49)(57 78)(58 79)(59 80)(60 81)(61 82)(62 83)(63 84)(64 71)(65 72)(66 73)(67 74)(68 75)(69 76)(70 77)
(1 29 57)(2 31 61)(3 33 58)(4 35 62)(5 30 59)(6 32 63)(7 34 60)(8 50 71)(9 52 75)(10 54 72)(11 56 76)(12 51 73)(13 53 77)(14 55 74)(15 36 78)(16 38 82)(17 40 79)(18 42 83)(19 37 80)(20 39 84)(21 41 81)(22 43 64)(23 45 68)(24 47 65)(25 49 69)(26 44 66)(27 46 70)(28 48 67)
G:=sub<Sym(84)| (1,57,29)(2,58,30)(3,59,31)(4,60,32)(5,61,33)(6,62,34)(7,63,35)(8,64,36)(9,65,37)(10,66,38)(11,67,39)(12,68,40)(13,69,41)(14,70,42)(15,71,43)(16,72,44)(17,73,45)(18,74,46)(19,75,47)(20,76,48)(21,77,49)(22,78,50)(23,79,51)(24,80,52)(25,81,53)(26,82,54)(27,83,55)(28,84,56), (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), (1,8)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14)(15,22)(16,23)(17,24)(18,25)(19,26)(20,27)(21,28)(29,36)(30,37)(31,38)(32,39)(33,40)(34,41)(35,42)(43,50)(44,51)(45,52)(46,53)(47,54)(48,55)(49,56)(57,64)(58,65)(59,66)(60,67)(61,68)(62,69)(63,70)(71,78)(72,79)(73,80)(74,81)(75,82)(76,83)(77,84), (1,22)(2,23)(3,24)(4,25)(5,26)(6,27)(7,28)(8,15)(9,16)(10,17)(11,18)(12,19)(13,20)(14,21)(29,50)(30,51)(31,52)(32,53)(33,54)(34,55)(35,56)(36,43)(37,44)(38,45)(39,46)(40,47)(41,48)(42,49)(57,78)(58,79)(59,80)(60,81)(61,82)(62,83)(63,84)(64,71)(65,72)(66,73)(67,74)(68,75)(69,76)(70,77), (1,29,57)(2,31,61)(3,33,58)(4,35,62)(5,30,59)(6,32,63)(7,34,60)(8,50,71)(9,52,75)(10,54,72)(11,56,76)(12,51,73)(13,53,77)(14,55,74)(15,36,78)(16,38,82)(17,40,79)(18,42,83)(19,37,80)(20,39,84)(21,41,81)(22,43,64)(23,45,68)(24,47,65)(25,49,69)(26,44,66)(27,46,70)(28,48,67)>;
G:=Group( (1,57,29)(2,58,30)(3,59,31)(4,60,32)(5,61,33)(6,62,34)(7,63,35)(8,64,36)(9,65,37)(10,66,38)(11,67,39)(12,68,40)(13,69,41)(14,70,42)(15,71,43)(16,72,44)(17,73,45)(18,74,46)(19,75,47)(20,76,48)(21,77,49)(22,78,50)(23,79,51)(24,80,52)(25,81,53)(26,82,54)(27,83,55)(28,84,56), (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), (1,8)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14)(15,22)(16,23)(17,24)(18,25)(19,26)(20,27)(21,28)(29,36)(30,37)(31,38)(32,39)(33,40)(34,41)(35,42)(43,50)(44,51)(45,52)(46,53)(47,54)(48,55)(49,56)(57,64)(58,65)(59,66)(60,67)(61,68)(62,69)(63,70)(71,78)(72,79)(73,80)(74,81)(75,82)(76,83)(77,84), (1,22)(2,23)(3,24)(4,25)(5,26)(6,27)(7,28)(8,15)(9,16)(10,17)(11,18)(12,19)(13,20)(14,21)(29,50)(30,51)(31,52)(32,53)(33,54)(34,55)(35,56)(36,43)(37,44)(38,45)(39,46)(40,47)(41,48)(42,49)(57,78)(58,79)(59,80)(60,81)(61,82)(62,83)(63,84)(64,71)(65,72)(66,73)(67,74)(68,75)(69,76)(70,77), (1,29,57)(2,31,61)(3,33,58)(4,35,62)(5,30,59)(6,32,63)(7,34,60)(8,50,71)(9,52,75)(10,54,72)(11,56,76)(12,51,73)(13,53,77)(14,55,74)(15,36,78)(16,38,82)(17,40,79)(18,42,83)(19,37,80)(20,39,84)(21,41,81)(22,43,64)(23,45,68)(24,47,65)(25,49,69)(26,44,66)(27,46,70)(28,48,67) );
G=PermutationGroup([[(1,57,29),(2,58,30),(3,59,31),(4,60,32),(5,61,33),(6,62,34),(7,63,35),(8,64,36),(9,65,37),(10,66,38),(11,67,39),(12,68,40),(13,69,41),(14,70,42),(15,71,43),(16,72,44),(17,73,45),(18,74,46),(19,75,47),(20,76,48),(21,77,49),(22,78,50),(23,79,51),(24,80,52),(25,81,53),(26,82,54),(27,83,55),(28,84,56)], [(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)], [(1,8),(2,9),(3,10),(4,11),(5,12),(6,13),(7,14),(15,22),(16,23),(17,24),(18,25),(19,26),(20,27),(21,28),(29,36),(30,37),(31,38),(32,39),(33,40),(34,41),(35,42),(43,50),(44,51),(45,52),(46,53),(47,54),(48,55),(49,56),(57,64),(58,65),(59,66),(60,67),(61,68),(62,69),(63,70),(71,78),(72,79),(73,80),(74,81),(75,82),(76,83),(77,84)], [(1,22),(2,23),(3,24),(4,25),(5,26),(6,27),(7,28),(8,15),(9,16),(10,17),(11,18),(12,19),(13,20),(14,21),(29,50),(30,51),(31,52),(32,53),(33,54),(34,55),(35,56),(36,43),(37,44),(38,45),(39,46),(40,47),(41,48),(42,49),(57,78),(58,79),(59,80),(60,81),(61,82),(62,83),(63,84),(64,71),(65,72),(66,73),(67,74),(68,75),(69,76),(70,77)], [(1,29,57),(2,31,61),(3,33,58),(4,35,62),(5,30,59),(6,32,63),(7,34,60),(8,50,71),(9,52,75),(10,54,72),(11,56,76),(12,51,73),(13,53,77),(14,55,74),(15,36,78),(16,38,82),(17,40,79),(18,42,83),(19,37,80),(20,39,84),(21,41,81),(22,43,64),(23,45,68),(24,47,65),(25,49,69),(26,44,66),(27,46,70),(28,48,67)]])
36 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | ··· | 3H | 6A | 6B | 7A | 7B | 14A | ··· | 14F | 21A | 21B | 21C | 21D | 42A | ··· | 42L |
| order | 1 | 2 | 3 | 3 | 3 | ··· | 3 | 6 | 6 | 7 | 7 | 14 | ··· | 14 | 21 | 21 | 21 | 21 | 42 | ··· | 42 |
| size | 1 | 3 | 1 | 1 | 28 | ··· | 28 | 3 | 3 | 3 | 3 | 3 | ··· | 3 | 3 | 3 | 3 | 3 | 3 | ··· | 3 |
36 irreducible representations
| dim | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 3 | 3 |
| type | + | + | |||||||
| image | C1 | C3 | C3 | A4 | C7⋊C3 | C3×A4 | C3×C7⋊C3 | C7⋊A4 | C3×C7⋊A4 |
| kernel | C3×C7⋊A4 | C7⋊A4 | C2×C42 | C21 | C2×C6 | C7 | C22 | C3 | C1 |
| # reps | 1 | 6 | 2 | 1 | 2 | 2 | 4 | 6 | 12 |
Matrix representation of C3×C7⋊A4 ►in GL3(𝔽43) generated by
| 36 | 0 | 0 |
| 0 | 36 | 0 |
| 0 | 0 | 36 |
| 0 | 0 | 1 |
| 1 | 0 | 25 |
| 0 | 1 | 24 |
| 32 | 19 | 37 |
| 11 | 34 | 41 |
| 19 | 37 | 19 |
| 15 | 35 | 4 |
| 24 | 30 | 6 |
| 35 | 4 | 40 |
| 6 | 0 | 15 |
| 0 | 0 | 37 |
| 0 | 6 | 37 |
G:=sub<GL(3,GF(43))| [36,0,0,0,36,0,0,0,36],[0,1,0,0,0,1,1,25,24],[32,11,19,19,34,37,37,41,19],[15,24,35,35,30,4,4,6,40],[6,0,0,0,0,6,15,37,37] >;
C3×C7⋊A4 in GAP, Magma, Sage, TeX
C_3\times C_7\rtimes A_4
% in TeX
G:=Group("C3xC7:A4"); // GroupNames label
G:=SmallGroup(252,40);
// by ID
G=gap.SmallGroup(252,40);
# by ID
G:=PCGroup([5,-3,-3,-2,2,-7,272,543,1804]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^7=c^2=d^2=e^3=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,e*b*e^-1=b^4,e*c*e^-1=c*d=d*c,e*d*e^-1=c>;
// generators/relations
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