direct product, metabelian, soluble, monomial, A-group
Aliases: A4×C52, C102⋊2C3, (C2×C10)⋊C15, C22⋊(C5×C15), SmallGroup(300,42)
Series: Derived ►Chief ►Lower central ►Upper central
| C22 — A4×C52 |
Generators and relations for A4×C52
G = < a,b,c,d,e | a5=b5=c2=d2=e3=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, ece-1=cd=dc, ede-1=c >
Smallest permutation representation of A4×C52
►On 100 points
Generators in S100
(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)
(1 40 93 24 8)(2 36 94 25 9)(3 37 95 21 10)(4 38 91 22 6)(5 39 92 23 7)(11 88 27 49 58)(12 89 28 50 59)(13 90 29 46 60)(14 86 30 47 56)(15 87 26 48 57)(16 79 96 65 43)(17 80 97 61 44)(18 76 98 62 45)(19 77 99 63 41)(20 78 100 64 42)(31 83 70 52 74)(32 84 66 53 75)(33 85 67 54 71)(34 81 68 55 72)(35 82 69 51 73)
(1 68)(2 69)(3 70)(4 66)(5 67)(6 84)(7 85)(8 81)(9 82)(10 83)(11 17)(12 18)(13 19)(14 20)(15 16)(21 31)(22 32)(23 33)(24 34)(25 35)(26 96)(27 97)(28 98)(29 99)(30 100)(36 51)(37 52)(38 53)(39 54)(40 55)(41 60)(42 56)(43 57)(44 58)(45 59)(46 63)(47 64)(48 65)(49 61)(50 62)(71 92)(72 93)(73 94)(74 95)(75 91)(76 89)(77 90)(78 86)(79 87)(80 88)
(1 96)(2 97)(3 98)(4 99)(5 100)(6 77)(7 78)(8 79)(9 80)(10 76)(11 35)(12 31)(13 32)(14 33)(15 34)(16 24)(17 25)(18 21)(19 22)(20 23)(26 68)(27 69)(28 70)(29 66)(30 67)(36 61)(37 62)(38 63)(39 64)(40 65)(41 91)(42 92)(43 93)(44 94)(45 95)(46 53)(47 54)(48 55)(49 51)(50 52)(56 71)(57 72)(58 73)(59 74)(60 75)(81 87)(82 88)(83 89)(84 90)(85 86)
(11 35 17)(12 31 18)(13 32 19)(14 33 20)(15 34 16)(26 68 96)(27 69 97)(28 70 98)(29 66 99)(30 67 100)(41 60 75)(42 56 71)(43 57 72)(44 58 73)(45 59 74)(46 53 63)(47 54 64)(48 55 65)(49 51 61)(50 52 62)(76 89 83)(77 90 84)(78 86 85)(79 87 81)(80 88 82)
G:=sub<Sym(100)| (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), (1,40,93,24,8)(2,36,94,25,9)(3,37,95,21,10)(4,38,91,22,6)(5,39,92,23,7)(11,88,27,49,58)(12,89,28,50,59)(13,90,29,46,60)(14,86,30,47,56)(15,87,26,48,57)(16,79,96,65,43)(17,80,97,61,44)(18,76,98,62,45)(19,77,99,63,41)(20,78,100,64,42)(31,83,70,52,74)(32,84,66,53,75)(33,85,67,54,71)(34,81,68,55,72)(35,82,69,51,73), (1,68)(2,69)(3,70)(4,66)(5,67)(6,84)(7,85)(8,81)(9,82)(10,83)(11,17)(12,18)(13,19)(14,20)(15,16)(21,31)(22,32)(23,33)(24,34)(25,35)(26,96)(27,97)(28,98)(29,99)(30,100)(36,51)(37,52)(38,53)(39,54)(40,55)(41,60)(42,56)(43,57)(44,58)(45,59)(46,63)(47,64)(48,65)(49,61)(50,62)(71,92)(72,93)(73,94)(74,95)(75,91)(76,89)(77,90)(78,86)(79,87)(80,88), (1,96)(2,97)(3,98)(4,99)(5,100)(6,77)(7,78)(8,79)(9,80)(10,76)(11,35)(12,31)(13,32)(14,33)(15,34)(16,24)(17,25)(18,21)(19,22)(20,23)(26,68)(27,69)(28,70)(29,66)(30,67)(36,61)(37,62)(38,63)(39,64)(40,65)(41,91)(42,92)(43,93)(44,94)(45,95)(46,53)(47,54)(48,55)(49,51)(50,52)(56,71)(57,72)(58,73)(59,74)(60,75)(81,87)(82,88)(83,89)(84,90)(85,86), (11,35,17)(12,31,18)(13,32,19)(14,33,20)(15,34,16)(26,68,96)(27,69,97)(28,70,98)(29,66,99)(30,67,100)(41,60,75)(42,56,71)(43,57,72)(44,58,73)(45,59,74)(46,53,63)(47,54,64)(48,55,65)(49,51,61)(50,52,62)(76,89,83)(77,90,84)(78,86,85)(79,87,81)(80,88,82)>;
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)(81,82,83,84,85)(86,87,88,89,90)(91,92,93,94,95)(96,97,98,99,100), (1,40,93,24,8)(2,36,94,25,9)(3,37,95,21,10)(4,38,91,22,6)(5,39,92,23,7)(11,88,27,49,58)(12,89,28,50,59)(13,90,29,46,60)(14,86,30,47,56)(15,87,26,48,57)(16,79,96,65,43)(17,80,97,61,44)(18,76,98,62,45)(19,77,99,63,41)(20,78,100,64,42)(31,83,70,52,74)(32,84,66,53,75)(33,85,67,54,71)(34,81,68,55,72)(35,82,69,51,73), (1,68)(2,69)(3,70)(4,66)(5,67)(6,84)(7,85)(8,81)(9,82)(10,83)(11,17)(12,18)(13,19)(14,20)(15,16)(21,31)(22,32)(23,33)(24,34)(25,35)(26,96)(27,97)(28,98)(29,99)(30,100)(36,51)(37,52)(38,53)(39,54)(40,55)(41,60)(42,56)(43,57)(44,58)(45,59)(46,63)(47,64)(48,65)(49,61)(50,62)(71,92)(72,93)(73,94)(74,95)(75,91)(76,89)(77,90)(78,86)(79,87)(80,88), (1,96)(2,97)(3,98)(4,99)(5,100)(6,77)(7,78)(8,79)(9,80)(10,76)(11,35)(12,31)(13,32)(14,33)(15,34)(16,24)(17,25)(18,21)(19,22)(20,23)(26,68)(27,69)(28,70)(29,66)(30,67)(36,61)(37,62)(38,63)(39,64)(40,65)(41,91)(42,92)(43,93)(44,94)(45,95)(46,53)(47,54)(48,55)(49,51)(50,52)(56,71)(57,72)(58,73)(59,74)(60,75)(81,87)(82,88)(83,89)(84,90)(85,86), (11,35,17)(12,31,18)(13,32,19)(14,33,20)(15,34,16)(26,68,96)(27,69,97)(28,70,98)(29,66,99)(30,67,100)(41,60,75)(42,56,71)(43,57,72)(44,58,73)(45,59,74)(46,53,63)(47,54,64)(48,55,65)(49,51,61)(50,52,62)(76,89,83)(77,90,84)(78,86,85)(79,87,81)(80,88,82) );
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),(81,82,83,84,85),(86,87,88,89,90),(91,92,93,94,95),(96,97,98,99,100)], [(1,40,93,24,8),(2,36,94,25,9),(3,37,95,21,10),(4,38,91,22,6),(5,39,92,23,7),(11,88,27,49,58),(12,89,28,50,59),(13,90,29,46,60),(14,86,30,47,56),(15,87,26,48,57),(16,79,96,65,43),(17,80,97,61,44),(18,76,98,62,45),(19,77,99,63,41),(20,78,100,64,42),(31,83,70,52,74),(32,84,66,53,75),(33,85,67,54,71),(34,81,68,55,72),(35,82,69,51,73)], [(1,68),(2,69),(3,70),(4,66),(5,67),(6,84),(7,85),(8,81),(9,82),(10,83),(11,17),(12,18),(13,19),(14,20),(15,16),(21,31),(22,32),(23,33),(24,34),(25,35),(26,96),(27,97),(28,98),(29,99),(30,100),(36,51),(37,52),(38,53),(39,54),(40,55),(41,60),(42,56),(43,57),(44,58),(45,59),(46,63),(47,64),(48,65),(49,61),(50,62),(71,92),(72,93),(73,94),(74,95),(75,91),(76,89),(77,90),(78,86),(79,87),(80,88)], [(1,96),(2,97),(3,98),(4,99),(5,100),(6,77),(7,78),(8,79),(9,80),(10,76),(11,35),(12,31),(13,32),(14,33),(15,34),(16,24),(17,25),(18,21),(19,22),(20,23),(26,68),(27,69),(28,70),(29,66),(30,67),(36,61),(37,62),(38,63),(39,64),(40,65),(41,91),(42,92),(43,93),(44,94),(45,95),(46,53),(47,54),(48,55),(49,51),(50,52),(56,71),(57,72),(58,73),(59,74),(60,75),(81,87),(82,88),(83,89),(84,90),(85,86)], [(11,35,17),(12,31,18),(13,32,19),(14,33,20),(15,34,16),(26,68,96),(27,69,97),(28,70,98),(29,66,99),(30,67,100),(41,60,75),(42,56,71),(43,57,72),(44,58,73),(45,59,74),(46,53,63),(47,54,64),(48,55,65),(49,51,61),(50,52,62),(76,89,83),(77,90,84),(78,86,85),(79,87,81),(80,88,82)]])
100 conjugacy classes
| class | 1 | 2 | 3A | 3B | 5A | ··· | 5X | 10A | ··· | 10X | 15A | ··· | 15AV |
| order | 1 | 2 | 3 | 3 | 5 | ··· | 5 | 10 | ··· | 10 | 15 | ··· | 15 |
| size | 1 | 3 | 4 | 4 | 1 | ··· | 1 | 3 | ··· | 3 | 4 | ··· | 4 |
100 irreducible representations
| dim | 1 | 1 | 1 | 1 | 3 | 3 |
| type | + | + | ||||
| image | C1 | C3 | C5 | C15 | A4 | C5×A4 |
| kernel | A4×C52 | C102 | C5×A4 | C2×C10 | C52 | C5 |
| # reps | 1 | 2 | 24 | 48 | 1 | 24 |
Matrix representation of A4×C52 ►in GL4(𝔽31) generated by
| 16 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 |
| 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 8 |
| 4 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 |
| 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 8 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 30 | 30 | 30 |
| 1 | 0 | 0 | 0 |
| 0 | 30 | 30 | 30 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
| 25 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 30 | 30 | 30 |
G:=sub<GL(4,GF(31))| [16,0,0,0,0,8,0,0,0,0,8,0,0,0,0,8],[4,0,0,0,0,8,0,0,0,0,8,0,0,0,0,8],[1,0,0,0,0,0,1,30,0,1,0,30,0,0,0,30],[1,0,0,0,0,30,0,0,0,30,0,1,0,30,1,0],[25,0,0,0,0,1,0,30,0,0,0,30,0,0,1,30] >;
A4×C52 in GAP, Magma, Sage, TeX
A_4\times C_5^2
% in TeX
G:=Group("A4xC5^2"); // GroupNames label
G:=SmallGroup(300,42);
// by ID
G=gap.SmallGroup(300,42);
# by ID
G:=PCGroup([5,-3,-5,-5,-2,2,3003,5629]);
// Polycyclic
G:=Group<a,b,c,d,e|a^5=b^5=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,b*e=e*b,e*c*e^-1=c*d=d*c,e*d*e^-1=c>;
// generators/relations
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