metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42⋊2Dic5, (C4×C20)⋊12C4, (Q8×C10)⋊12C4, (C2×Q8)⋊2Dic5, (C2×D4).9D10, C5⋊4(C42⋊3C4), C4.4D4.2D5, (C22×C10).16D4, C23.7(C5⋊D4), C23⋊Dic5.4C2, C10.44(C23⋊C4), C2.8(C23⋊Dic5), (D4×C10).172C22, C22.14(C23.D5), (C2×C4).1(C2×Dic5), (C2×C20).181(C2×C4), (C5×C4.4D4).9C2, (C2×C10).163(C22⋊C4), SmallGroup(320,99)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42⋊Dic5
G = < a,b,c,d | a4=b4=c10=1, d2=c5, ab=ba, cac-1=a-1b2, dad-1=a-1b-1, cbc-1=b-1, dbd-1=a2b-1, dcd-1=c-1 >
Subgroups: 302 in 70 conjugacy classes, 23 normal (17 characteristic)
C1, C2, C2 [×3], C4 [×5], C22, C22 [×4], C5, C2×C4, C2×C4 [×4], D4, Q8, C23 [×2], C10, C10 [×3], C42, C22⋊C4 [×4], C2×D4, C2×Q8, Dic5 [×2], C20 [×3], C2×C10, C2×C10 [×4], C23⋊C4 [×2], C4.4D4, C2×Dic5 [×2], C2×C20, C2×C20 [×2], C5×D4, C5×Q8, C22×C10 [×2], C42⋊3C4, C23.D5 [×2], C4×C20, C5×C22⋊C4 [×2], D4×C10, Q8×C10, C23⋊Dic5 [×2], C5×C4.4D4, C42⋊Dic5
Quotients: C1, C2 [×3], C4 [×2], C22, C2×C4, D4 [×2], D5, C22⋊C4, Dic5 [×2], D10, C23⋊C4, C2×Dic5, C5⋊D4 [×2], C42⋊3C4, C23.D5, C23⋊Dic5, C42⋊Dic5
►On 80 points
Generators in S80
(1 54 19 45)(2 60 20 41)(3 56 16 47)(4 52 17 43)(5 58 18 49)(6 42 11 51)(7 48 12 57)(8 44 13 53)(9 50 14 59)(10 46 15 55)(21 63 36 68)(22 80 37 75)(23 65 38 70)(24 72 39 77)(25 67 40 62)(26 74 31 79)(27 69 32 64)(28 76 33 71)(29 61 34 66)(30 78 35 73)
(1 26 9 21)(2 22 10 27)(3 28 6 23)(4 24 7 29)(5 30 8 25)(11 38 16 33)(12 34 17 39)(13 40 18 35)(14 36 19 31)(15 32 20 37)(41 75 55 64)(42 65 56 76)(43 77 57 66)(44 67 58 78)(45 79 59 68)(46 69 60 80)(47 71 51 70)(48 61 52 72)(49 73 53 62)(50 63 54 74)
(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)
(2 5)(3 4)(6 7)(8 10)(11 17)(12 16)(13 20)(14 19)(15 18)(21 36 26 31)(22 35 27 40)(23 34 28 39)(24 33 29 38)(25 32 30 37)(41 73 46 78)(42 72 47 77)(43 71 48 76)(44 80 49 75)(45 79 50 74)(51 66 56 61)(52 65 57 70)(53 64 58 69)(54 63 59 68)(55 62 60 67)
G:=sub<Sym(80)| (1,54,19,45)(2,60,20,41)(3,56,16,47)(4,52,17,43)(5,58,18,49)(6,42,11,51)(7,48,12,57)(8,44,13,53)(9,50,14,59)(10,46,15,55)(21,63,36,68)(22,80,37,75)(23,65,38,70)(24,72,39,77)(25,67,40,62)(26,74,31,79)(27,69,32,64)(28,76,33,71)(29,61,34,66)(30,78,35,73), (1,26,9,21)(2,22,10,27)(3,28,6,23)(4,24,7,29)(5,30,8,25)(11,38,16,33)(12,34,17,39)(13,40,18,35)(14,36,19,31)(15,32,20,37)(41,75,55,64)(42,65,56,76)(43,77,57,66)(44,67,58,78)(45,79,59,68)(46,69,60,80)(47,71,51,70)(48,61,52,72)(49,73,53,62)(50,63,54,74), (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), (2,5)(3,4)(6,7)(8,10)(11,17)(12,16)(13,20)(14,19)(15,18)(21,36,26,31)(22,35,27,40)(23,34,28,39)(24,33,29,38)(25,32,30,37)(41,73,46,78)(42,72,47,77)(43,71,48,76)(44,80,49,75)(45,79,50,74)(51,66,56,61)(52,65,57,70)(53,64,58,69)(54,63,59,68)(55,62,60,67)>;
G:=Group( (1,54,19,45)(2,60,20,41)(3,56,16,47)(4,52,17,43)(5,58,18,49)(6,42,11,51)(7,48,12,57)(8,44,13,53)(9,50,14,59)(10,46,15,55)(21,63,36,68)(22,80,37,75)(23,65,38,70)(24,72,39,77)(25,67,40,62)(26,74,31,79)(27,69,32,64)(28,76,33,71)(29,61,34,66)(30,78,35,73), (1,26,9,21)(2,22,10,27)(3,28,6,23)(4,24,7,29)(5,30,8,25)(11,38,16,33)(12,34,17,39)(13,40,18,35)(14,36,19,31)(15,32,20,37)(41,75,55,64)(42,65,56,76)(43,77,57,66)(44,67,58,78)(45,79,59,68)(46,69,60,80)(47,71,51,70)(48,61,52,72)(49,73,53,62)(50,63,54,74), (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), (2,5)(3,4)(6,7)(8,10)(11,17)(12,16)(13,20)(14,19)(15,18)(21,36,26,31)(22,35,27,40)(23,34,28,39)(24,33,29,38)(25,32,30,37)(41,73,46,78)(42,72,47,77)(43,71,48,76)(44,80,49,75)(45,79,50,74)(51,66,56,61)(52,65,57,70)(53,64,58,69)(54,63,59,68)(55,62,60,67) );
G=PermutationGroup([(1,54,19,45),(2,60,20,41),(3,56,16,47),(4,52,17,43),(5,58,18,49),(6,42,11,51),(7,48,12,57),(8,44,13,53),(9,50,14,59),(10,46,15,55),(21,63,36,68),(22,80,37,75),(23,65,38,70),(24,72,39,77),(25,67,40,62),(26,74,31,79),(27,69,32,64),(28,76,33,71),(29,61,34,66),(30,78,35,73)], [(1,26,9,21),(2,22,10,27),(3,28,6,23),(4,24,7,29),(5,30,8,25),(11,38,16,33),(12,34,17,39),(13,40,18,35),(14,36,19,31),(15,32,20,37),(41,75,55,64),(42,65,56,76),(43,77,57,66),(44,67,58,78),(45,79,59,68),(46,69,60,80),(47,71,51,70),(48,61,52,72),(49,73,53,62),(50,63,54,74)], [(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)], [(2,5),(3,4),(6,7),(8,10),(11,17),(12,16),(13,20),(14,19),(15,18),(21,36,26,31),(22,35,27,40),(23,34,28,39),(24,33,29,38),(25,32,30,37),(41,73,46,78),(42,72,47,77),(43,71,48,76),(44,80,49,75),(45,79,50,74),(51,66,56,61),(52,65,57,70),(53,64,58,69),(54,63,59,68),(55,62,60,67)])
41 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5A | 5B | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | ··· | 20L | 20M | 20N | 20O | 20P |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | 20 | 20 | 20 |
| size | 1 | 1 | 2 | 4 | 4 | 4 | 4 | 4 | 8 | 40 | 40 | 40 | 40 | 2 | 2 | 2 | ··· | 2 | 8 | 8 | 8 | 8 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
41 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | - | + | - | + | ||||||
| image | C1 | C2 | C2 | C4 | C4 | D4 | D5 | Dic5 | D10 | Dic5 | C5⋊D4 | C23⋊C4 | C42⋊3C4 | C23⋊Dic5 | C42⋊Dic5 |
| kernel | C42⋊Dic5 | C23⋊Dic5 | C5×C4.4D4 | C4×C20 | Q8×C10 | C22×C10 | C4.4D4 | C42 | C2×D4 | C2×Q8 | C23 | C10 | C5 | C2 | C1 |
| # reps | 1 | 2 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 8 | 1 | 2 | 4 | 8 |
Matrix representation of C42⋊Dic5 ►in GL4(𝔽41) generated by
| 20 | 21 | 23 | 10 |
| 9 | 6 | 34 | 13 |
| 19 | 37 | 3 | 7 |
| 18 | 36 | 11 | 35 |
| 18 | 1 | 0 | 28 |
| 5 | 23 | 13 | 13 |
| 3 | 3 | 17 | 40 |
| 38 | 0 | 1 | 24 |
| 38 | 18 | 0 | 0 |
| 20 | 17 | 0 | 0 |
| 26 | 6 | 6 | 23 |
| 38 | 20 | 18 | 21 |
| 5 | 6 | 14 | 12 |
| 6 | 7 | 39 | 25 |
| 21 | 30 | 1 | 2 |
| 10 | 21 | 35 | 28 |
G:=sub<GL(4,GF(41))| [20,9,19,18,21,6,37,36,23,34,3,11,10,13,7,35],[18,5,3,38,1,23,3,0,0,13,17,1,28,13,40,24],[38,20,26,38,18,17,6,20,0,0,6,18,0,0,23,21],[5,6,21,10,6,7,30,21,14,39,1,35,12,25,2,28] >;
C42⋊Dic5 in GAP, Magma, Sage, TeX
C_4^2\rtimes {\rm Dic}_5 % in TeX
G:=Group("C4^2:Dic5"); // GroupNames label
G:=SmallGroup(320,99);
// by ID
G=gap.SmallGroup(320,99);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,232,219,1571,570,297,136,1684,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^10=1,d^2=c^5,a*b=b*a,c*a*c^-1=a^-1*b^2,d*a*d^-1=a^-1*b^-1,c*b*c^-1=b^-1,d*b*d^-1=a^2*b^-1,d*c*d^-1=c^-1>;
// generators/relations