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, C4, C22, C22, C5, C2×C4, C2×C4, D4, Q8, C23, C10, C10, C42, C22⋊C4, C2×D4, C2×Q8, Dic5, C20, C2×C10, C2×C10, C23⋊C4, C4.4D4, C2×Dic5, C2×C20, C2×C20, C5×D4, C5×Q8, C22×C10, C42⋊3C4, C23.D5, C4×C20, C5×C22⋊C4, D4×C10, Q8×C10, C23⋊Dic5, C5×C4.4D4, C42⋊Dic5
Quotients: C1, C2, C4, C22, C2×C4, D4, D5, C22⋊C4, Dic5, D10, C23⋊C4, C2×Dic5, C5⋊D4, C42⋊3C4, C23.D5, C23⋊Dic5, C42⋊Dic5
►On 80 points
Generators in S80
(1 68 19 41)(2 64 20 47)(3 70 16 43)(4 66 17 49)(5 62 18 45)(6 48 15 65)(7 44 11 61)(8 50 12 67)(9 46 13 63)(10 42 14 69)(21 79 34 74)(22 55 35 60)(23 71 36 76)(24 57 37 52)(25 73 38 78)(26 59 39 54)(27 75 40 80)(28 51 31 56)(29 77 32 72)(30 53 33 58)
(1 26 9 21)(2 22 10 27)(3 28 6 23)(4 24 7 29)(5 30 8 25)(11 32 17 37)(12 38 18 33)(13 34 19 39)(14 40 20 35)(15 36 16 31)(41 54 63 74)(42 75 64 55)(43 56 65 76)(44 77 66 57)(45 58 67 78)(46 79 68 59)(47 60 69 80)(48 71 70 51)(49 52 61 72)(50 73 62 53)
(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 16)(12 20)(13 19)(14 18)(15 17)(21 34 26 39)(22 33 27 38)(23 32 28 37)(24 31 29 36)(25 40 30 35)(41 54 46 59)(42 53 47 58)(43 52 48 57)(44 51 49 56)(45 60 50 55)(61 76 66 71)(62 75 67 80)(63 74 68 79)(64 73 69 78)(65 72 70 77)
G:=sub<Sym(80)| (1,68,19,41)(2,64,20,47)(3,70,16,43)(4,66,17,49)(5,62,18,45)(6,48,15,65)(7,44,11,61)(8,50,12,67)(9,46,13,63)(10,42,14,69)(21,79,34,74)(22,55,35,60)(23,71,36,76)(24,57,37,52)(25,73,38,78)(26,59,39,54)(27,75,40,80)(28,51,31,56)(29,77,32,72)(30,53,33,58), (1,26,9,21)(2,22,10,27)(3,28,6,23)(4,24,7,29)(5,30,8,25)(11,32,17,37)(12,38,18,33)(13,34,19,39)(14,40,20,35)(15,36,16,31)(41,54,63,74)(42,75,64,55)(43,56,65,76)(44,77,66,57)(45,58,67,78)(46,79,68,59)(47,60,69,80)(48,71,70,51)(49,52,61,72)(50,73,62,53), (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)(11,16)(12,20)(13,19)(14,18)(15,17)(21,34,26,39)(22,33,27,38)(23,32,28,37)(24,31,29,36)(25,40,30,35)(41,54,46,59)(42,53,47,58)(43,52,48,57)(44,51,49,56)(45,60,50,55)(61,76,66,71)(62,75,67,80)(63,74,68,79)(64,73,69,78)(65,72,70,77)>;
G:=Group( (1,68,19,41)(2,64,20,47)(3,70,16,43)(4,66,17,49)(5,62,18,45)(6,48,15,65)(7,44,11,61)(8,50,12,67)(9,46,13,63)(10,42,14,69)(21,79,34,74)(22,55,35,60)(23,71,36,76)(24,57,37,52)(25,73,38,78)(26,59,39,54)(27,75,40,80)(28,51,31,56)(29,77,32,72)(30,53,33,58), (1,26,9,21)(2,22,10,27)(3,28,6,23)(4,24,7,29)(5,30,8,25)(11,32,17,37)(12,38,18,33)(13,34,19,39)(14,40,20,35)(15,36,16,31)(41,54,63,74)(42,75,64,55)(43,56,65,76)(44,77,66,57)(45,58,67,78)(46,79,68,59)(47,60,69,80)(48,71,70,51)(49,52,61,72)(50,73,62,53), (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)(11,16)(12,20)(13,19)(14,18)(15,17)(21,34,26,39)(22,33,27,38)(23,32,28,37)(24,31,29,36)(25,40,30,35)(41,54,46,59)(42,53,47,58)(43,52,48,57)(44,51,49,56)(45,60,50,55)(61,76,66,71)(62,75,67,80)(63,74,68,79)(64,73,69,78)(65,72,70,77) );
G=PermutationGroup([[(1,68,19,41),(2,64,20,47),(3,70,16,43),(4,66,17,49),(5,62,18,45),(6,48,15,65),(7,44,11,61),(8,50,12,67),(9,46,13,63),(10,42,14,69),(21,79,34,74),(22,55,35,60),(23,71,36,76),(24,57,37,52),(25,73,38,78),(26,59,39,54),(27,75,40,80),(28,51,31,56),(29,77,32,72),(30,53,33,58)], [(1,26,9,21),(2,22,10,27),(3,28,6,23),(4,24,7,29),(5,30,8,25),(11,32,17,37),(12,38,18,33),(13,34,19,39),(14,40,20,35),(15,36,16,31),(41,54,63,74),(42,75,64,55),(43,56,65,76),(44,77,66,57),(45,58,67,78),(46,79,68,59),(47,60,69,80),(48,71,70,51),(49,52,61,72),(50,73,62,53)], [(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),(11,16),(12,20),(13,19),(14,18),(15,17),(21,34,26,39),(22,33,27,38),(23,32,28,37),(24,31,29,36),(25,40,30,35),(41,54,46,59),(42,53,47,58),(43,52,48,57),(44,51,49,56),(45,60,50,55),(61,76,66,71),(62,75,67,80),(63,74,68,79),(64,73,69,78),(65,72,70,77)]])
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