direct product, metabelian, supersoluble, monomial
Aliases: C5×Q8⋊2D5, D20⋊4C10, C20.39D10, (C5×Q8)⋊5D5, Q8⋊2(C5×D5), (C5×D20)⋊9C2, (C4×D5)⋊3C10, (D5×C20)⋊7C2, (C5×Q8)⋊3C10, C4.7(D5×C10), C20.7(C2×C10), (Q8×C52)⋊4C2, D10.3(C2×C10), C52⋊12(C4○D4), C10.8(C22×C10), (C5×C20).23C22, (C5×C10).26C23, Dic5.9(C2×C10), C10.47(C22×D5), (D5×C10).16C22, (C5×Dic5).26C22, C5⋊3(C5×C4○D4), C2.9(D5×C2×C10), SmallGroup(400,188)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×Q8⋊2D5
G = < a,b,c,d,e | a5=b4=d5=e2=1, c2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=ebe=b-1, bd=db, cd=dc, ce=ec, ede=d-1 >
Subgroups: 252 in 92 conjugacy classes, 46 normal (16 characteristic)
C1, C2, C2, C4, C4, C22, C5, C5, C2×C4, D4, Q8, D5, C10, C10, C4○D4, Dic5, C20, C20, D10, C2×C10, C52, C4×D5, D20, C2×C20, C5×D4, C5×Q8, C5×Q8, C5×D5, C5×C10, Q8⋊2D5, C5×C4○D4, C5×Dic5, C5×C20, D5×C10, D5×C20, C5×D20, Q8×C52, C5×Q8⋊2D5
Quotients: C1, C2, C22, C5, C23, D5, C10, C4○D4, D10, C2×C10, C22×D5, C22×C10, C5×D5, Q8⋊2D5, C5×C4○D4, D5×C10, D5×C2×C10, C5×Q8⋊2D5
►On 80 points
Generators in S80
(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)
(1 18 8 13)(2 19 9 14)(3 20 10 15)(4 16 6 11)(5 17 7 12)(21 31 26 36)(22 32 27 37)(23 33 28 38)(24 34 29 39)(25 35 30 40)(41 51 46 56)(42 52 47 57)(43 53 48 58)(44 54 49 59)(45 55 50 60)(61 76 66 71)(62 77 67 72)(63 78 68 73)(64 79 69 74)(65 80 70 75)
(1 28 8 23)(2 29 9 24)(3 30 10 25)(4 26 6 21)(5 27 7 22)(11 36 16 31)(12 37 17 32)(13 38 18 33)(14 39 19 34)(15 40 20 35)(41 66 46 61)(42 67 47 62)(43 68 48 63)(44 69 49 64)(45 70 50 65)(51 76 56 71)(52 77 57 72)(53 78 58 73)(54 79 59 74)(55 80 60 75)
(1 4 2 5 3)(6 9 7 10 8)(11 14 12 15 13)(16 19 17 20 18)(21 24 22 25 23)(26 29 27 30 28)(31 34 32 35 33)(36 39 37 40 38)(41 43 45 42 44)(46 48 50 47 49)(51 53 55 52 54)(56 58 60 57 59)(61 63 65 62 64)(66 68 70 67 69)(71 73 75 72 74)(76 78 80 77 79)
(1 41)(2 42)(3 43)(4 44)(5 45)(6 49)(7 50)(8 46)(9 47)(10 48)(11 54)(12 55)(13 51)(14 52)(15 53)(16 59)(17 60)(18 56)(19 57)(20 58)(21 64)(22 65)(23 61)(24 62)(25 63)(26 69)(27 70)(28 66)(29 67)(30 68)(31 74)(32 75)(33 71)(34 72)(35 73)(36 79)(37 80)(38 76)(39 77)(40 78)
G:=sub<Sym(80)| (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), (1,18,8,13)(2,19,9,14)(3,20,10,15)(4,16,6,11)(5,17,7,12)(21,31,26,36)(22,32,27,37)(23,33,28,38)(24,34,29,39)(25,35,30,40)(41,51,46,56)(42,52,47,57)(43,53,48,58)(44,54,49,59)(45,55,50,60)(61,76,66,71)(62,77,67,72)(63,78,68,73)(64,79,69,74)(65,80,70,75), (1,28,8,23)(2,29,9,24)(3,30,10,25)(4,26,6,21)(5,27,7,22)(11,36,16,31)(12,37,17,32)(13,38,18,33)(14,39,19,34)(15,40,20,35)(41,66,46,61)(42,67,47,62)(43,68,48,63)(44,69,49,64)(45,70,50,65)(51,76,56,71)(52,77,57,72)(53,78,58,73)(54,79,59,74)(55,80,60,75), (1,4,2,5,3)(6,9,7,10,8)(11,14,12,15,13)(16,19,17,20,18)(21,24,22,25,23)(26,29,27,30,28)(31,34,32,35,33)(36,39,37,40,38)(41,43,45,42,44)(46,48,50,47,49)(51,53,55,52,54)(56,58,60,57,59)(61,63,65,62,64)(66,68,70,67,69)(71,73,75,72,74)(76,78,80,77,79), (1,41)(2,42)(3,43)(4,44)(5,45)(6,49)(7,50)(8,46)(9,47)(10,48)(11,54)(12,55)(13,51)(14,52)(15,53)(16,59)(17,60)(18,56)(19,57)(20,58)(21,64)(22,65)(23,61)(24,62)(25,63)(26,69)(27,70)(28,66)(29,67)(30,68)(31,74)(32,75)(33,71)(34,72)(35,73)(36,79)(37,80)(38,76)(39,77)(40,78)>;
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), (1,18,8,13)(2,19,9,14)(3,20,10,15)(4,16,6,11)(5,17,7,12)(21,31,26,36)(22,32,27,37)(23,33,28,38)(24,34,29,39)(25,35,30,40)(41,51,46,56)(42,52,47,57)(43,53,48,58)(44,54,49,59)(45,55,50,60)(61,76,66,71)(62,77,67,72)(63,78,68,73)(64,79,69,74)(65,80,70,75), (1,28,8,23)(2,29,9,24)(3,30,10,25)(4,26,6,21)(5,27,7,22)(11,36,16,31)(12,37,17,32)(13,38,18,33)(14,39,19,34)(15,40,20,35)(41,66,46,61)(42,67,47,62)(43,68,48,63)(44,69,49,64)(45,70,50,65)(51,76,56,71)(52,77,57,72)(53,78,58,73)(54,79,59,74)(55,80,60,75), (1,4,2,5,3)(6,9,7,10,8)(11,14,12,15,13)(16,19,17,20,18)(21,24,22,25,23)(26,29,27,30,28)(31,34,32,35,33)(36,39,37,40,38)(41,43,45,42,44)(46,48,50,47,49)(51,53,55,52,54)(56,58,60,57,59)(61,63,65,62,64)(66,68,70,67,69)(71,73,75,72,74)(76,78,80,77,79), (1,41)(2,42)(3,43)(4,44)(5,45)(6,49)(7,50)(8,46)(9,47)(10,48)(11,54)(12,55)(13,51)(14,52)(15,53)(16,59)(17,60)(18,56)(19,57)(20,58)(21,64)(22,65)(23,61)(24,62)(25,63)(26,69)(27,70)(28,66)(29,67)(30,68)(31,74)(32,75)(33,71)(34,72)(35,73)(36,79)(37,80)(38,76)(39,77)(40,78) );
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)], [(1,18,8,13),(2,19,9,14),(3,20,10,15),(4,16,6,11),(5,17,7,12),(21,31,26,36),(22,32,27,37),(23,33,28,38),(24,34,29,39),(25,35,30,40),(41,51,46,56),(42,52,47,57),(43,53,48,58),(44,54,49,59),(45,55,50,60),(61,76,66,71),(62,77,67,72),(63,78,68,73),(64,79,69,74),(65,80,70,75)], [(1,28,8,23),(2,29,9,24),(3,30,10,25),(4,26,6,21),(5,27,7,22),(11,36,16,31),(12,37,17,32),(13,38,18,33),(14,39,19,34),(15,40,20,35),(41,66,46,61),(42,67,47,62),(43,68,48,63),(44,69,49,64),(45,70,50,65),(51,76,56,71),(52,77,57,72),(53,78,58,73),(54,79,59,74),(55,80,60,75)], [(1,4,2,5,3),(6,9,7,10,8),(11,14,12,15,13),(16,19,17,20,18),(21,24,22,25,23),(26,29,27,30,28),(31,34,32,35,33),(36,39,37,40,38),(41,43,45,42,44),(46,48,50,47,49),(51,53,55,52,54),(56,58,60,57,59),(61,63,65,62,64),(66,68,70,67,69),(71,73,75,72,74),(76,78,80,77,79)], [(1,41),(2,42),(3,43),(4,44),(5,45),(6,49),(7,50),(8,46),(9,47),(10,48),(11,54),(12,55),(13,51),(14,52),(15,53),(16,59),(17,60),(18,56),(19,57),(20,58),(21,64),(22,65),(23,61),(24,62),(25,63),(26,69),(27,70),(28,66),(29,67),(30,68),(31,74),(32,75),(33,71),(34,72),(35,73),(36,79),(37,80),(38,76),(39,77),(40,78)]])
100 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 5A | 5B | 5C | 5D | 5E | ··· | 5N | 10A | 10B | 10C | 10D | 10E | ··· | 10N | 10O | ··· | 10Z | 20A | ··· | 20L | 20M | ··· | 20AP | 20AQ | ··· | 20AX |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 5 | ··· | 5 | 10 | 10 | 10 | 10 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 10 | 10 | 10 | 2 | 2 | 2 | 5 | 5 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 10 | ··· | 10 | 2 | ··· | 2 | 4 | ··· | 4 | 5 | ··· | 5 |
100 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | |||||||||
| image | C1 | C2 | C2 | C2 | C5 | C10 | C10 | C10 | D5 | C4○D4 | D10 | C5×D5 | C5×C4○D4 | D5×C10 | Q8⋊2D5 | C5×Q8⋊2D5 |
| kernel | C5×Q8⋊2D5 | D5×C20 | C5×D20 | Q8×C52 | Q8⋊2D5 | C4×D5 | D20 | C5×Q8 | C5×Q8 | C52 | C20 | Q8 | C5 | C4 | C5 | C1 |
| # reps | 1 | 3 | 3 | 1 | 4 | 12 | 12 | 4 | 2 | 2 | 6 | 8 | 8 | 24 | 2 | 8 |
Matrix representation of C5×Q8⋊2D5 ►in GL4(𝔽41) generated by
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 1 | 5 |
| 0 | 0 | 16 | 40 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 4 | 30 |
| 0 | 0 | 9 | 37 |
| 37 | 7 | 0 | 0 |
| 0 | 10 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 10 | 34 | 0 | 0 |
| 20 | 31 | 0 | 0 |
| 0 | 0 | 36 | 24 |
| 0 | 0 | 40 | 5 |
G:=sub<GL(4,GF(41))| [16,0,0,0,0,16,0,0,0,0,1,0,0,0,0,1],[40,0,0,0,0,40,0,0,0,0,1,16,0,0,5,40],[1,0,0,0,0,1,0,0,0,0,4,9,0,0,30,37],[37,0,0,0,7,10,0,0,0,0,1,0,0,0,0,1],[10,20,0,0,34,31,0,0,0,0,36,40,0,0,24,5] >;
C5×Q8⋊2D5 in GAP, Magma, Sage, TeX
C_5\times Q_8\rtimes_2D_5
% in TeX
G:=Group("C5xQ8:2D5"); // GroupNames label
G:=SmallGroup(400,188);
// by ID
G=gap.SmallGroup(400,188);
# by ID
G:=PCGroup([6,-2,-2,-2,-5,-2,-5,247,794,404,194,11525]);
// Polycyclic
G:=Group<a,b,c,d,e|a^5=b^4=d^5=e^2=1,c^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=e*b*e=b^-1,b*d=d*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations