direct product, metacyclic, supersoluble, monomial, A-group
Aliases: C5×D5⋊C8, D5⋊C40, C20.3C20, C20.17F5, D10.2C20, C5⋊C8⋊3C10, C5⋊1(C2×C40), (C5×D5)⋊3C8, C52⋊5(C2×C8), C4.3(C5×F5), (C5×C20).9C4, C2.1(C10×F5), C10.1(C2×C20), (C4×D5).5C10, (D5×C10).8C4, C10.42(C2×F5), (D5×C20).10C2, Dic5.5(C2×C10), (C5×Dic5).10C22, (C5×C5⋊C8)⋊7C2, (C5×C10).13(C2×C4), SmallGroup(400,135)
Series: Derived ►Chief ►Lower central ►Upper central
| C5 — C5×D5⋊C8 |
Generators and relations for C5×D5⋊C8
G = < a,b,c,d | a5=b5=c2=d8=1, ab=ba, ac=ca, ad=da, cbc=b-1, dbd-1=b3, dcd-1=b2c >
Smallest permutation representation of C5×D5⋊C8
►On 80 points
Generators in S80
(1 21 75 63 33)(2 22 76 64 34)(3 23 77 57 35)(4 24 78 58 36)(5 17 79 59 37)(6 18 80 60 38)(7 19 73 61 39)(8 20 74 62 40)(9 55 25 41 71)(10 56 26 42 72)(11 49 27 43 65)(12 50 28 44 66)(13 51 29 45 67)(14 52 30 46 68)(15 53 31 47 69)(16 54 32 48 70)
(1 21 75 63 33)(2 64 22 34 76)(3 35 57 77 23)(4 78 36 24 58)(5 17 79 59 37)(6 60 18 38 80)(7 39 61 73 19)(8 74 40 20 62)(9 55 25 41 71)(10 42 56 72 26)(11 65 43 27 49)(12 28 66 50 44)(13 51 29 45 67)(14 46 52 68 30)(15 69 47 31 53)(16 32 70 54 48)
(1 47)(2 16)(3 55)(4 72)(5 43)(6 12)(7 51)(8 68)(9 35)(10 24)(11 79)(13 39)(14 20)(15 75)(17 65)(18 50)(19 29)(21 69)(22 54)(23 25)(26 58)(27 37)(28 80)(30 62)(31 33)(32 76)(34 70)(36 42)(38 66)(40 46)(41 77)(44 60)(45 73)(48 64)(49 59)(52 74)(53 63)(56 78)(57 71)(61 67)
(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)
G:=sub<Sym(80)| (1,21,75,63,33)(2,22,76,64,34)(3,23,77,57,35)(4,24,78,58,36)(5,17,79,59,37)(6,18,80,60,38)(7,19,73,61,39)(8,20,74,62,40)(9,55,25,41,71)(10,56,26,42,72)(11,49,27,43,65)(12,50,28,44,66)(13,51,29,45,67)(14,52,30,46,68)(15,53,31,47,69)(16,54,32,48,70), (1,21,75,63,33)(2,64,22,34,76)(3,35,57,77,23)(4,78,36,24,58)(5,17,79,59,37)(6,60,18,38,80)(7,39,61,73,19)(8,74,40,20,62)(9,55,25,41,71)(10,42,56,72,26)(11,65,43,27,49)(12,28,66,50,44)(13,51,29,45,67)(14,46,52,68,30)(15,69,47,31,53)(16,32,70,54,48), (1,47)(2,16)(3,55)(4,72)(5,43)(6,12)(7,51)(8,68)(9,35)(10,24)(11,79)(13,39)(14,20)(15,75)(17,65)(18,50)(19,29)(21,69)(22,54)(23,25)(26,58)(27,37)(28,80)(30,62)(31,33)(32,76)(34,70)(36,42)(38,66)(40,46)(41,77)(44,60)(45,73)(48,64)(49,59)(52,74)(53,63)(56,78)(57,71)(61,67), (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)>;
G:=Group( (1,21,75,63,33)(2,22,76,64,34)(3,23,77,57,35)(4,24,78,58,36)(5,17,79,59,37)(6,18,80,60,38)(7,19,73,61,39)(8,20,74,62,40)(9,55,25,41,71)(10,56,26,42,72)(11,49,27,43,65)(12,50,28,44,66)(13,51,29,45,67)(14,52,30,46,68)(15,53,31,47,69)(16,54,32,48,70), (1,21,75,63,33)(2,64,22,34,76)(3,35,57,77,23)(4,78,36,24,58)(5,17,79,59,37)(6,60,18,38,80)(7,39,61,73,19)(8,74,40,20,62)(9,55,25,41,71)(10,42,56,72,26)(11,65,43,27,49)(12,28,66,50,44)(13,51,29,45,67)(14,46,52,68,30)(15,69,47,31,53)(16,32,70,54,48), (1,47)(2,16)(3,55)(4,72)(5,43)(6,12)(7,51)(8,68)(9,35)(10,24)(11,79)(13,39)(14,20)(15,75)(17,65)(18,50)(19,29)(21,69)(22,54)(23,25)(26,58)(27,37)(28,80)(30,62)(31,33)(32,76)(34,70)(36,42)(38,66)(40,46)(41,77)(44,60)(45,73)(48,64)(49,59)(52,74)(53,63)(56,78)(57,71)(61,67), (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) );
G=PermutationGroup([[(1,21,75,63,33),(2,22,76,64,34),(3,23,77,57,35),(4,24,78,58,36),(5,17,79,59,37),(6,18,80,60,38),(7,19,73,61,39),(8,20,74,62,40),(9,55,25,41,71),(10,56,26,42,72),(11,49,27,43,65),(12,50,28,44,66),(13,51,29,45,67),(14,52,30,46,68),(15,53,31,47,69),(16,54,32,48,70)], [(1,21,75,63,33),(2,64,22,34,76),(3,35,57,77,23),(4,78,36,24,58),(5,17,79,59,37),(6,60,18,38,80),(7,39,61,73,19),(8,74,40,20,62),(9,55,25,41,71),(10,42,56,72,26),(11,65,43,27,49),(12,28,66,50,44),(13,51,29,45,67),(14,46,52,68,30),(15,69,47,31,53),(16,32,70,54,48)], [(1,47),(2,16),(3,55),(4,72),(5,43),(6,12),(7,51),(8,68),(9,35),(10,24),(11,79),(13,39),(14,20),(15,75),(17,65),(18,50),(19,29),(21,69),(22,54),(23,25),(26,58),(27,37),(28,80),(30,62),(31,33),(32,76),(34,70),(36,42),(38,66),(40,46),(41,77),(44,60),(45,73),(48,64),(49,59),(52,74),(53,63),(56,78),(57,71),(61,67)], [(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)]])
100 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 5A | 5B | 5C | 5D | 5E | ··· | 5I | 8A | ··· | 8H | 10A | 10B | 10C | 10D | 10E | ··· | 10I | 10J | ··· | 10Q | 20A | ··· | 20H | 20I | ··· | 20R | 20S | ··· | 20Z | 40A | ··· | 40AF |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 5 | ··· | 5 | 8 | ··· | 8 | 10 | 10 | 10 | 10 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 5 | 5 | 1 | 1 | 5 | 5 | 1 | 1 | 1 | 1 | 4 | ··· | 4 | 5 | ··· | 5 | 1 | 1 | 1 | 1 | 4 | ··· | 4 | 5 | ··· | 5 | 1 | ··· | 1 | 4 | ··· | 4 | 5 | ··· | 5 | 5 | ··· | 5 |
100 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | |||||||||||||
| image | C1 | C2 | C2 | C4 | C4 | C5 | C8 | C10 | C10 | C20 | C20 | C40 | F5 | C2×F5 | D5⋊C8 | C5×F5 | C10×F5 | C5×D5⋊C8 |
| kernel | C5×D5⋊C8 | C5×C5⋊C8 | D5×C20 | C5×C20 | D5×C10 | D5⋊C8 | C5×D5 | C5⋊C8 | C4×D5 | C20 | D10 | D5 | C20 | C10 | C5 | C4 | C2 | C1 |
| # reps | 1 | 2 | 1 | 2 | 2 | 4 | 8 | 8 | 4 | 8 | 8 | 32 | 1 | 1 | 2 | 4 | 4 | 8 |
Matrix representation of C5×D5⋊C8 ►in GL4(𝔽41) generated by
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 16 | 0 | 0 | 0 |
| 30 | 18 | 0 | 0 |
| 11 | 0 | 10 | 0 |
| 2 | 0 | 0 | 37 |
| 23 | 7 | 0 | 0 |
| 30 | 18 | 0 | 0 |
| 28 | 6 | 0 | 5 |
| 32 | 13 | 33 | 0 |
| 3 | 0 | 17 | 0 |
| 0 | 0 | 32 | 1 |
| 0 | 1 | 38 | 0 |
| 0 | 0 | 14 | 0 |
G:=sub<GL(4,GF(41))| [16,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16],[16,30,11,2,0,18,0,0,0,0,10,0,0,0,0,37],[23,30,28,32,7,18,6,13,0,0,0,33,0,0,5,0],[3,0,0,0,0,0,1,0,17,32,38,14,0,1,0,0] >;
C5×D5⋊C8 in GAP, Magma, Sage, TeX
C_5\times D_5\rtimes C_8
% in TeX
G:=Group("C5xD5:C8"); // GroupNames label
G:=SmallGroup(400,135);
// by ID
G=gap.SmallGroup(400,135);
# by ID
G:=PCGroup([6,-2,-2,-5,-2,-2,-5,120,247,69,5765,599]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^5=c^2=d^8=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,d*b*d^-1=b^3,d*c*d^-1=b^2*c>;
// generators/relations
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