direct product, metabelian, soluble, monomial, A-group
Aliases: A4×C23, C22⋊C69, (C2×C46)⋊C3, SmallGroup(276,6)
Series: Derived ►Chief ►Lower central ►Upper central
| C22 — A4×C23 |
Generators and relations for A4×C23
G = < a,b,c,d | a23=b2=c2=d3=1, ab=ba, ac=ca, ad=da, dbd-1=bc=cb, dcd-1=b >
Smallest permutation representation of A4×C23
►On 92 points
Generators in S92
(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)
(1 60)(2 61)(3 62)(4 63)(5 64)(6 65)(7 66)(8 67)(9 68)(10 69)(11 47)(12 48)(13 49)(14 50)(15 51)(16 52)(17 53)(18 54)(19 55)(20 56)(21 57)(22 58)(23 59)(24 76)(25 77)(26 78)(27 79)(28 80)(29 81)(30 82)(31 83)(32 84)(33 85)(34 86)(35 87)(36 88)(37 89)(38 90)(39 91)(40 92)(41 70)(42 71)(43 72)(44 73)(45 74)(46 75)
(1 30)(2 31)(3 32)(4 33)(5 34)(6 35)(7 36)(8 37)(9 38)(10 39)(11 40)(12 41)(13 42)(14 43)(15 44)(16 45)(17 46)(18 24)(19 25)(20 26)(21 27)(22 28)(23 29)(47 92)(48 70)(49 71)(50 72)(51 73)(52 74)(53 75)(54 76)(55 77)(56 78)(57 79)(58 80)(59 81)(60 82)(61 83)(62 84)(63 85)(64 86)(65 87)(66 88)(67 89)(68 90)(69 91)
(24 76 54)(25 77 55)(26 78 56)(27 79 57)(28 80 58)(29 81 59)(30 82 60)(31 83 61)(32 84 62)(33 85 63)(34 86 64)(35 87 65)(36 88 66)(37 89 67)(38 90 68)(39 91 69)(40 92 47)(41 70 48)(42 71 49)(43 72 50)(44 73 51)(45 74 52)(46 75 53)
G:=sub<Sym(92)| (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), (1,60)(2,61)(3,62)(4,63)(5,64)(6,65)(7,66)(8,67)(9,68)(10,69)(11,47)(12,48)(13,49)(14,50)(15,51)(16,52)(17,53)(18,54)(19,55)(20,56)(21,57)(22,58)(23,59)(24,76)(25,77)(26,78)(27,79)(28,80)(29,81)(30,82)(31,83)(32,84)(33,85)(34,86)(35,87)(36,88)(37,89)(38,90)(39,91)(40,92)(41,70)(42,71)(43,72)(44,73)(45,74)(46,75), (1,30)(2,31)(3,32)(4,33)(5,34)(6,35)(7,36)(8,37)(9,38)(10,39)(11,40)(12,41)(13,42)(14,43)(15,44)(16,45)(17,46)(18,24)(19,25)(20,26)(21,27)(22,28)(23,29)(47,92)(48,70)(49,71)(50,72)(51,73)(52,74)(53,75)(54,76)(55,77)(56,78)(57,79)(58,80)(59,81)(60,82)(61,83)(62,84)(63,85)(64,86)(65,87)(66,88)(67,89)(68,90)(69,91), (24,76,54)(25,77,55)(26,78,56)(27,79,57)(28,80,58)(29,81,59)(30,82,60)(31,83,61)(32,84,62)(33,85,63)(34,86,64)(35,87,65)(36,88,66)(37,89,67)(38,90,68)(39,91,69)(40,92,47)(41,70,48)(42,71,49)(43,72,50)(44,73,51)(45,74,52)(46,75,53)>;
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), (1,60)(2,61)(3,62)(4,63)(5,64)(6,65)(7,66)(8,67)(9,68)(10,69)(11,47)(12,48)(13,49)(14,50)(15,51)(16,52)(17,53)(18,54)(19,55)(20,56)(21,57)(22,58)(23,59)(24,76)(25,77)(26,78)(27,79)(28,80)(29,81)(30,82)(31,83)(32,84)(33,85)(34,86)(35,87)(36,88)(37,89)(38,90)(39,91)(40,92)(41,70)(42,71)(43,72)(44,73)(45,74)(46,75), (1,30)(2,31)(3,32)(4,33)(5,34)(6,35)(7,36)(8,37)(9,38)(10,39)(11,40)(12,41)(13,42)(14,43)(15,44)(16,45)(17,46)(18,24)(19,25)(20,26)(21,27)(22,28)(23,29)(47,92)(48,70)(49,71)(50,72)(51,73)(52,74)(53,75)(54,76)(55,77)(56,78)(57,79)(58,80)(59,81)(60,82)(61,83)(62,84)(63,85)(64,86)(65,87)(66,88)(67,89)(68,90)(69,91), (24,76,54)(25,77,55)(26,78,56)(27,79,57)(28,80,58)(29,81,59)(30,82,60)(31,83,61)(32,84,62)(33,85,63)(34,86,64)(35,87,65)(36,88,66)(37,89,67)(38,90,68)(39,91,69)(40,92,47)(41,70,48)(42,71,49)(43,72,50)(44,73,51)(45,74,52)(46,75,53) );
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)], [(1,60),(2,61),(3,62),(4,63),(5,64),(6,65),(7,66),(8,67),(9,68),(10,69),(11,47),(12,48),(13,49),(14,50),(15,51),(16,52),(17,53),(18,54),(19,55),(20,56),(21,57),(22,58),(23,59),(24,76),(25,77),(26,78),(27,79),(28,80),(29,81),(30,82),(31,83),(32,84),(33,85),(34,86),(35,87),(36,88),(37,89),(38,90),(39,91),(40,92),(41,70),(42,71),(43,72),(44,73),(45,74),(46,75)], [(1,30),(2,31),(3,32),(4,33),(5,34),(6,35),(7,36),(8,37),(9,38),(10,39),(11,40),(12,41),(13,42),(14,43),(15,44),(16,45),(17,46),(18,24),(19,25),(20,26),(21,27),(22,28),(23,29),(47,92),(48,70),(49,71),(50,72),(51,73),(52,74),(53,75),(54,76),(55,77),(56,78),(57,79),(58,80),(59,81),(60,82),(61,83),(62,84),(63,85),(64,86),(65,87),(66,88),(67,89),(68,90),(69,91)], [(24,76,54),(25,77,55),(26,78,56),(27,79,57),(28,80,58),(29,81,59),(30,82,60),(31,83,61),(32,84,62),(33,85,63),(34,86,64),(35,87,65),(36,88,66),(37,89,67),(38,90,68),(39,91,69),(40,92,47),(41,70,48),(42,71,49),(43,72,50),(44,73,51),(45,74,52),(46,75,53)]])
92 conjugacy classes
| class | 1 | 2 | 3A | 3B | 23A | ··· | 23V | 46A | ··· | 46V | 69A | ··· | 69AR |
| order | 1 | 2 | 3 | 3 | 23 | ··· | 23 | 46 | ··· | 46 | 69 | ··· | 69 |
| size | 1 | 3 | 4 | 4 | 1 | ··· | 1 | 3 | ··· | 3 | 4 | ··· | 4 |
92 irreducible representations
| dim | 1 | 1 | 1 | 1 | 3 | 3 |
| type | + | + | ||||
| image | C1 | C3 | C23 | C69 | A4 | A4×C23 |
| kernel | A4×C23 | C2×C46 | A4 | C22 | C23 | C1 |
| # reps | 1 | 2 | 22 | 44 | 1 | 22 |
Matrix representation of A4×C23 ►in GL3(𝔽139) generated by
| 91 | 0 | 0 |
| 0 | 91 | 0 |
| 0 | 0 | 91 |
| 0 | 1 | 138 |
| 1 | 0 | 138 |
| 0 | 0 | 138 |
| 138 | 0 | 0 |
| 138 | 0 | 1 |
| 138 | 1 | 0 |
| 1 | 138 | 0 |
| 0 | 138 | 1 |
| 0 | 138 | 0 |
G:=sub<GL(3,GF(139))| [91,0,0,0,91,0,0,0,91],[0,1,0,1,0,0,138,138,138],[138,138,138,0,0,1,0,1,0],[1,0,0,138,138,138,0,1,0] >;
A4×C23 in GAP, Magma, Sage, TeX
A_4\times C_{23} % in TeX
G:=Group("A4xC23"); // GroupNames label
G:=SmallGroup(276,6);
// by ID
G=gap.SmallGroup(276,6);
# by ID
G:=PCGroup([4,-3,-23,-2,2,1658,3315]);
// Polycyclic
G:=Group<a,b,c,d|a^23=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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