direct product, metabelian, soluble, monomial
Aliases: C5×C42⋊C6, C42⋊1C30, (C4×C20)⋊2C6, C42⋊2C2⋊C15, C42⋊C3⋊1C10, C23.1(C5×A4), (C22×C10).1A4, C22.3(C10×A4), (C5×C42⋊C3)⋊2C2, (C5×C42⋊2C2)⋊C3, (C2×C10).7(C2×A4), SmallGroup(480,657)
Series: Derived ►Chief ►Lower central ►Upper central
| C42 — C5×C42⋊C6 |
Generators and relations for C5×C42⋊C6
G = < a,b,c,d | a5=b4=c4=d6=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=c-1, dcd-1=b-1c >
Smallest permutation representation of C5×C42⋊C6
►On 80 points
Generators in S80
(1 2 4 5 3)(6 13 17 9 18)(7 14 15 10 19)(8 12 16 11 20)(21 61 28 66 75)(22 62 29 67 76)(23 57 30 68 77)(24 58 31 63 78)(25 59 32 64 79)(26 60 27 65 80)(33 49 71 54 40)(34 50 72 55 41)(35 45 73 56 42)(36 46 74 51 43)(37 47 69 52 44)(38 48 70 53 39)
(1 50 7 57)(2 72 14 30)(3 34 19 23)(4 55 15 68)(5 41 10 77)(6 47 8 60)(9 44 11 80)(12 27 13 69)(16 65 17 52)(18 37 20 26)(21 22 24 36)(25 35 33 38)(28 29 31 74)(32 73 71 70)(39 79 42 40)(43 75 76 78)(45 49 48 59)(46 61 62 58)(51 66 67 63)(53 64 56 54)
(1 62 6 49)(2 29 13 71)(3 22 18 33)(4 67 17 54)(5 76 9 40)(7 46 8 59)(10 43 11 79)(12 32 14 74)(15 51 16 64)(19 36 20 25)(21 26 35 23)(24 37 38 34)(27 73 30 28)(31 69 70 72)(39 41 78 44)(42 77 75 80)(45 57 61 60)(47 48 50 58)(52 53 55 63)(56 68 66 65)
(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)
G:=sub<Sym(80)| (1,2,4,5,3)(6,13,17,9,18)(7,14,15,10,19)(8,12,16,11,20)(21,61,28,66,75)(22,62,29,67,76)(23,57,30,68,77)(24,58,31,63,78)(25,59,32,64,79)(26,60,27,65,80)(33,49,71,54,40)(34,50,72,55,41)(35,45,73,56,42)(36,46,74,51,43)(37,47,69,52,44)(38,48,70,53,39), (1,50,7,57)(2,72,14,30)(3,34,19,23)(4,55,15,68)(5,41,10,77)(6,47,8,60)(9,44,11,80)(12,27,13,69)(16,65,17,52)(18,37,20,26)(21,22,24,36)(25,35,33,38)(28,29,31,74)(32,73,71,70)(39,79,42,40)(43,75,76,78)(45,49,48,59)(46,61,62,58)(51,66,67,63)(53,64,56,54), (1,62,6,49)(2,29,13,71)(3,22,18,33)(4,67,17,54)(5,76,9,40)(7,46,8,59)(10,43,11,79)(12,32,14,74)(15,51,16,64)(19,36,20,25)(21,26,35,23)(24,37,38,34)(27,73,30,28)(31,69,70,72)(39,41,78,44)(42,77,75,80)(45,57,61,60)(47,48,50,58)(52,53,55,63)(56,68,66,65), (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)>;
G:=Group( (1,2,4,5,3)(6,13,17,9,18)(7,14,15,10,19)(8,12,16,11,20)(21,61,28,66,75)(22,62,29,67,76)(23,57,30,68,77)(24,58,31,63,78)(25,59,32,64,79)(26,60,27,65,80)(33,49,71,54,40)(34,50,72,55,41)(35,45,73,56,42)(36,46,74,51,43)(37,47,69,52,44)(38,48,70,53,39), (1,50,7,57)(2,72,14,30)(3,34,19,23)(4,55,15,68)(5,41,10,77)(6,47,8,60)(9,44,11,80)(12,27,13,69)(16,65,17,52)(18,37,20,26)(21,22,24,36)(25,35,33,38)(28,29,31,74)(32,73,71,70)(39,79,42,40)(43,75,76,78)(45,49,48,59)(46,61,62,58)(51,66,67,63)(53,64,56,54), (1,62,6,49)(2,29,13,71)(3,22,18,33)(4,67,17,54)(5,76,9,40)(7,46,8,59)(10,43,11,79)(12,32,14,74)(15,51,16,64)(19,36,20,25)(21,26,35,23)(24,37,38,34)(27,73,30,28)(31,69,70,72)(39,41,78,44)(42,77,75,80)(45,57,61,60)(47,48,50,58)(52,53,55,63)(56,68,66,65), (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) );
G=PermutationGroup([[(1,2,4,5,3),(6,13,17,9,18),(7,14,15,10,19),(8,12,16,11,20),(21,61,28,66,75),(22,62,29,67,76),(23,57,30,68,77),(24,58,31,63,78),(25,59,32,64,79),(26,60,27,65,80),(33,49,71,54,40),(34,50,72,55,41),(35,45,73,56,42),(36,46,74,51,43),(37,47,69,52,44),(38,48,70,53,39)], [(1,50,7,57),(2,72,14,30),(3,34,19,23),(4,55,15,68),(5,41,10,77),(6,47,8,60),(9,44,11,80),(12,27,13,69),(16,65,17,52),(18,37,20,26),(21,22,24,36),(25,35,33,38),(28,29,31,74),(32,73,71,70),(39,79,42,40),(43,75,76,78),(45,49,48,59),(46,61,62,58),(51,66,67,63),(53,64,56,54)], [(1,62,6,49),(2,29,13,71),(3,22,18,33),(4,67,17,54),(5,76,9,40),(7,46,8,59),(10,43,11,79),(12,32,14,74),(15,51,16,64),(19,36,20,25),(21,26,35,23),(24,37,38,34),(27,73,30,28),(31,69,70,72),(39,41,78,44),(42,77,75,80),(45,57,61,60),(47,48,50,58),(52,53,55,63),(56,68,66,65)], [(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)]])
50 conjugacy classes
| class | 1 | 2A | 2B | 3A | 3B | 4A | 4B | 4C | 5A | 5B | 5C | 5D | 6A | 6B | 10A | 10B | 10C | 10D | 10E | 10F | 10G | 10H | 15A | ··· | 15H | 20A | ··· | 20H | 20I | 20J | 20K | 20L | 30A | ··· | 30H |
| order | 1 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | 6 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 15 | ··· | 15 | 20 | ··· | 20 | 20 | 20 | 20 | 20 | 30 | ··· | 30 |
| size | 1 | 3 | 4 | 16 | 16 | 6 | 6 | 12 | 1 | 1 | 1 | 1 | 16 | 16 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 16 | ··· | 16 | 6 | ··· | 6 | 12 | 12 | 12 | 12 | 16 | ··· | 16 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 6 | 6 |
| type | + | + | + | + | ||||||||||
| image | C1 | C2 | C3 | C5 | C6 | C10 | C15 | C30 | A4 | C2×A4 | C5×A4 | C10×A4 | C42⋊C6 | C5×C42⋊C6 |
| kernel | C5×C42⋊C6 | C5×C42⋊C3 | C5×C42⋊2C2 | C42⋊C6 | C4×C20 | C42⋊C3 | C42⋊2C2 | C42 | C22×C10 | C2×C10 | C23 | C22 | C5 | C1 |
| # reps | 1 | 1 | 2 | 4 | 2 | 4 | 8 | 8 | 1 | 1 | 4 | 4 | 2 | 8 |
Matrix representation of C5×C42⋊C6 ►in GL6(𝔽61)
| 58 | 0 | 0 | 0 | 0 | 0 |
| 0 | 58 | 0 | 0 | 0 | 0 |
| 0 | 0 | 58 | 0 | 0 | 0 |
| 0 | 0 | 0 | 58 | 0 | 0 |
| 0 | 0 | 0 | 0 | 58 | 0 |
| 0 | 0 | 0 | 0 | 0 | 58 |
| 0 | 50 | 0 | 0 | 1 | 6 |
| 0 | 50 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 56 |
| 1 | 60 | 0 | 0 | 0 | 56 |
| 0 | 0 | 1 | 0 | 0 | 55 |
| 0 | 0 | 0 | 0 | 0 | 50 |
| 55 | 55 | 6 | 10 | 1 | 0 |
| 50 | 0 | 11 | 60 | 1 | 0 |
| 55 | 5 | 6 | 11 | 0 | 0 |
| 5 | 55 | 6 | 11 | 50 | 1 |
| 6 | 56 | 5 | 50 | 11 | 0 |
| 11 | 11 | 50 | 1 | 60 | 0 |
| 1 | 0 | 60 | 50 | 10 | 55 |
| 0 | 0 | 60 | 0 | 60 | 0 |
| 0 | 1 | 60 | 50 | 11 | 5 |
| 0 | 0 | 0 | 0 | 11 | 55 |
| 0 | 0 | 0 | 0 | 50 | 56 |
| 0 | 0 | 0 | 60 | 1 | 11 |
G:=sub<GL(6,GF(61))| [58,0,0,0,0,0,0,58,0,0,0,0,0,0,58,0,0,0,0,0,0,58,0,0,0,0,0,0,58,0,0,0,0,0,0,58],[0,0,0,1,0,0,50,50,0,60,0,0,0,0,0,0,1,0,0,0,1,0,0,0,1,0,0,0,0,0,6,0,56,56,55,50],[55,50,55,5,6,11,55,0,5,55,56,11,6,11,6,6,5,50,10,60,11,11,50,1,1,1,0,50,11,60,0,0,0,1,0,0],[1,0,0,0,0,0,0,0,1,0,0,0,60,60,60,0,0,0,50,0,50,0,0,60,10,60,11,11,50,1,55,0,5,55,56,11] >;
C5×C42⋊C6 in GAP, Magma, Sage, TeX
C_5\times C_4^2\rtimes C_6
% in TeX
G:=Group("C5xC4^2:C6"); // GroupNames label
G:=SmallGroup(480,657);
// by ID
G=gap.SmallGroup(480,657);
# by ID
G:=PCGroup([7,-2,-3,-5,-2,2,-2,2,4203,850,360,10504,5786,102,5052,8833]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^4=c^4=d^6=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=c^-1,d*c*d^-1=b^-1*c>;
// generators/relations
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