direct product, metabelian, soluble, monomial, A-group
Aliases: A4×C21, (C2×C6)⋊C21, C22⋊(C3×C21), (C2×C42)⋊1C3, (C2×C14)⋊2C32, SmallGroup(252,39)
Series: Derived ►Chief ►Lower central ►Upper central
| C22 — A4×C21 |
Generators and relations for A4×C21
G = < a,b,c,d | a21=b2=c2=d3=1, ab=ba, ac=ca, ad=da, dbd-1=bc=cb, dcd-1=b >
Smallest permutation representation of A4×C21
►On 84 points
Generators in S84
(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 34)(2 35)(3 36)(4 37)(5 38)(6 39)(7 40)(8 41)(9 42)(10 22)(11 23)(12 24)(13 25)(14 26)(15 27)(16 28)(17 29)(18 30)(19 31)(20 32)(21 33)(43 83)(44 84)(45 64)(46 65)(47 66)(48 67)(49 68)(50 69)(51 70)(52 71)(53 72)(54 73)(55 74)(56 75)(57 76)(58 77)(59 78)(60 79)(61 80)(62 81)(63 82)
(1 83)(2 84)(3 64)(4 65)(5 66)(6 67)(7 68)(8 69)(9 70)(10 71)(11 72)(12 73)(13 74)(14 75)(15 76)(16 77)(17 78)(18 79)(19 80)(20 81)(21 82)(22 52)(23 53)(24 54)(25 55)(26 56)(27 57)(28 58)(29 59)(30 60)(31 61)(32 62)(33 63)(34 43)(35 44)(36 45)(37 46)(38 47)(39 48)(40 49)(41 50)(42 51)
(22 71 52)(23 72 53)(24 73 54)(25 74 55)(26 75 56)(27 76 57)(28 77 58)(29 78 59)(30 79 60)(31 80 61)(32 81 62)(33 82 63)(34 83 43)(35 84 44)(36 64 45)(37 65 46)(38 66 47)(39 67 48)(40 68 49)(41 69 50)(42 70 51)
G:=sub<Sym(84)| (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), (1,34)(2,35)(3,36)(4,37)(5,38)(6,39)(7,40)(8,41)(9,42)(10,22)(11,23)(12,24)(13,25)(14,26)(15,27)(16,28)(17,29)(18,30)(19,31)(20,32)(21,33)(43,83)(44,84)(45,64)(46,65)(47,66)(48,67)(49,68)(50,69)(51,70)(52,71)(53,72)(54,73)(55,74)(56,75)(57,76)(58,77)(59,78)(60,79)(61,80)(62,81)(63,82), (1,83)(2,84)(3,64)(4,65)(5,66)(6,67)(7,68)(8,69)(9,70)(10,71)(11,72)(12,73)(13,74)(14,75)(15,76)(16,77)(17,78)(18,79)(19,80)(20,81)(21,82)(22,52)(23,53)(24,54)(25,55)(26,56)(27,57)(28,58)(29,59)(30,60)(31,61)(32,62)(33,63)(34,43)(35,44)(36,45)(37,46)(38,47)(39,48)(40,49)(41,50)(42,51), (22,71,52)(23,72,53)(24,73,54)(25,74,55)(26,75,56)(27,76,57)(28,77,58)(29,78,59)(30,79,60)(31,80,61)(32,81,62)(33,82,63)(34,83,43)(35,84,44)(36,64,45)(37,65,46)(38,66,47)(39,67,48)(40,68,49)(41,69,50)(42,70,51)>;
G:=Group( (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), (1,34)(2,35)(3,36)(4,37)(5,38)(6,39)(7,40)(8,41)(9,42)(10,22)(11,23)(12,24)(13,25)(14,26)(15,27)(16,28)(17,29)(18,30)(19,31)(20,32)(21,33)(43,83)(44,84)(45,64)(46,65)(47,66)(48,67)(49,68)(50,69)(51,70)(52,71)(53,72)(54,73)(55,74)(56,75)(57,76)(58,77)(59,78)(60,79)(61,80)(62,81)(63,82), (1,83)(2,84)(3,64)(4,65)(5,66)(6,67)(7,68)(8,69)(9,70)(10,71)(11,72)(12,73)(13,74)(14,75)(15,76)(16,77)(17,78)(18,79)(19,80)(20,81)(21,82)(22,52)(23,53)(24,54)(25,55)(26,56)(27,57)(28,58)(29,59)(30,60)(31,61)(32,62)(33,63)(34,43)(35,44)(36,45)(37,46)(38,47)(39,48)(40,49)(41,50)(42,51), (22,71,52)(23,72,53)(24,73,54)(25,74,55)(26,75,56)(27,76,57)(28,77,58)(29,78,59)(30,79,60)(31,80,61)(32,81,62)(33,82,63)(34,83,43)(35,84,44)(36,64,45)(37,65,46)(38,66,47)(39,67,48)(40,68,49)(41,69,50)(42,70,51) );
G=PermutationGroup([[(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)], [(1,34),(2,35),(3,36),(4,37),(5,38),(6,39),(7,40),(8,41),(9,42),(10,22),(11,23),(12,24),(13,25),(14,26),(15,27),(16,28),(17,29),(18,30),(19,31),(20,32),(21,33),(43,83),(44,84),(45,64),(46,65),(47,66),(48,67),(49,68),(50,69),(51,70),(52,71),(53,72),(54,73),(55,74),(56,75),(57,76),(58,77),(59,78),(60,79),(61,80),(62,81),(63,82)], [(1,83),(2,84),(3,64),(4,65),(5,66),(6,67),(7,68),(8,69),(9,70),(10,71),(11,72),(12,73),(13,74),(14,75),(15,76),(16,77),(17,78),(18,79),(19,80),(20,81),(21,82),(22,52),(23,53),(24,54),(25,55),(26,56),(27,57),(28,58),(29,59),(30,60),(31,61),(32,62),(33,63),(34,43),(35,44),(36,45),(37,46),(38,47),(39,48),(40,49),(41,50),(42,51)], [(22,71,52),(23,72,53),(24,73,54),(25,74,55),(26,75,56),(27,76,57),(28,77,58),(29,78,59),(30,79,60),(31,80,61),(32,81,62),(33,82,63),(34,83,43),(35,84,44),(36,64,45),(37,65,46),(38,66,47),(39,67,48),(40,68,49),(41,69,50),(42,70,51)]])
84 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | ··· | 3H | 6A | 6B | 7A | ··· | 7F | 14A | ··· | 14F | 21A | ··· | 21L | 21M | ··· | 21AV | 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 | 4 | ··· | 4 | 3 | 3 | 1 | ··· | 1 | 3 | ··· | 3 | 1 | ··· | 1 | 4 | ··· | 4 | 3 | ··· | 3 |
84 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 |
| type | + | + | ||||||||
| image | C1 | C3 | C3 | C7 | C21 | C21 | A4 | C3×A4 | C7×A4 | A4×C21 |
| kernel | A4×C21 | C7×A4 | C2×C42 | C3×A4 | A4 | C2×C6 | C21 | C7 | C3 | C1 |
| # reps | 1 | 6 | 2 | 6 | 36 | 12 | 1 | 2 | 6 | 12 |
Matrix representation of A4×C21 ►in GL3(𝔽43) generated by
| 13 | 0 | 0 |
| 0 | 13 | 0 |
| 0 | 0 | 13 |
| 0 | 1 | 42 |
| 1 | 0 | 42 |
| 0 | 0 | 42 |
| 42 | 0 | 0 |
| 42 | 0 | 1 |
| 42 | 1 | 0 |
| 1 | 42 | 0 |
| 0 | 42 | 1 |
| 0 | 42 | 0 |
G:=sub<GL(3,GF(43))| [13,0,0,0,13,0,0,0,13],[0,1,0,1,0,0,42,42,42],[42,42,42,0,0,1,0,1,0],[1,0,0,42,42,42,0,1,0] >;
A4×C21 in GAP, Magma, Sage, TeX
A_4\times C_{21} % in TeX
G:=Group("A4xC21"); // GroupNames label
G:=SmallGroup(252,39);
// by ID
G=gap.SmallGroup(252,39);
# by ID
G:=PCGroup([5,-3,-3,-7,-2,2,2523,4729]);
// Polycyclic
G:=Group<a,b,c,d|a^21=b^2=c^2=d^3=1,a*b=b*a,a*c=c*a,a*d=d*a,d*b*d^-1=b*c=c*b,d*c*d^-1=b>;
// generators/relations
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