metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.2Dic5, (C2×C20).5D4, (C4×C20).19C4, (C2×Q8).3D10, (D4×C10).15C4, (C2×D4).2Dic5, C4.4D4.3D5, C20.10D4⋊2C2, C5⋊4(C42.C4), (Q8×C10).3C22, C10.45(C23⋊C4), C2.9(C23⋊Dic5), C22.15(C23.D5), (C2×C4).7(C5⋊D4), (C2×C4).2(C2×Dic5), (C2×C20).182(C2×C4), (C5×C4.4D4).1C2, (C2×C10).164(C22⋊C4), SmallGroup(320,100)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.Dic5
G = < a,b,c,d | a4=b4=1, c10=b2, d2=c5, ab=ba, cac-1=a-1b2, dad-1=a-1b-1, cbc-1=b-1, dbd-1=a2b-1, dcd-1=c9 >
Subgroups: 206 in 64 conjugacy classes, 23 normal (17 characteristic)
C1, C2, C2, C4, C22, C22, C5, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C10, C10, C42, C22⋊C4, M4(2), C2×D4, C2×Q8, C20, C2×C10, C2×C10, C4.10D4, C4.4D4, C5⋊2C8, C2×C20, C2×C20, C2×C20, C5×D4, C5×Q8, C22×C10, C42.C4, C4.Dic5, C4×C20, C5×C22⋊C4, D4×C10, Q8×C10, C20.10D4, 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.C4, C23.D5, C23⋊Dic5, C42.Dic5
►On 80 points
Generators in S80
(1 6 11 16)(2 7 12 17)(3 8 13 18)(4 9 14 19)(5 10 15 20)(21 54)(22 45)(23 56)(24 47)(25 58)(26 49)(27 60)(28 51)(29 42)(30 53)(31 44)(32 55)(33 46)(34 57)(35 48)(36 59)(37 50)(38 41)(39 52)(40 43)(61 76 71 66)(62 77 72 67)(63 78 73 68)(64 79 74 69)(65 80 75 70)
(1 64 11 74)(2 75 12 65)(3 66 13 76)(4 77 14 67)(5 68 15 78)(6 79 16 69)(7 70 17 80)(8 61 18 71)(9 72 19 62)(10 63 20 73)(21 49 31 59)(22 60 32 50)(23 51 33 41)(24 42 34 52)(25 53 35 43)(26 44 36 54)(27 55 37 45)(28 46 38 56)(29 57 39 47)(30 48 40 58)
(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 44 6 49 11 54 16 59)(2 53 7 58 12 43 17 48)(3 42 8 47 13 52 18 57)(4 51 9 56 14 41 19 46)(5 60 10 45 15 50 20 55)(21 74 26 79 31 64 36 69)(22 63 27 68 32 73 37 78)(23 72 28 77 33 62 38 67)(24 61 29 66 34 71 39 76)(25 70 30 75 35 80 40 65)
G:=sub<Sym(80)| (1,6,11,16)(2,7,12,17)(3,8,13,18)(4,9,14,19)(5,10,15,20)(21,54)(22,45)(23,56)(24,47)(25,58)(26,49)(27,60)(28,51)(29,42)(30,53)(31,44)(32,55)(33,46)(34,57)(35,48)(36,59)(37,50)(38,41)(39,52)(40,43)(61,76,71,66)(62,77,72,67)(63,78,73,68)(64,79,74,69)(65,80,75,70), (1,64,11,74)(2,75,12,65)(3,66,13,76)(4,77,14,67)(5,68,15,78)(6,79,16,69)(7,70,17,80)(8,61,18,71)(9,72,19,62)(10,63,20,73)(21,49,31,59)(22,60,32,50)(23,51,33,41)(24,42,34,52)(25,53,35,43)(26,44,36,54)(27,55,37,45)(28,46,38,56)(29,57,39,47)(30,48,40,58), (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), (1,44,6,49,11,54,16,59)(2,53,7,58,12,43,17,48)(3,42,8,47,13,52,18,57)(4,51,9,56,14,41,19,46)(5,60,10,45,15,50,20,55)(21,74,26,79,31,64,36,69)(22,63,27,68,32,73,37,78)(23,72,28,77,33,62,38,67)(24,61,29,66,34,71,39,76)(25,70,30,75,35,80,40,65)>;
G:=Group( (1,6,11,16)(2,7,12,17)(3,8,13,18)(4,9,14,19)(5,10,15,20)(21,54)(22,45)(23,56)(24,47)(25,58)(26,49)(27,60)(28,51)(29,42)(30,53)(31,44)(32,55)(33,46)(34,57)(35,48)(36,59)(37,50)(38,41)(39,52)(40,43)(61,76,71,66)(62,77,72,67)(63,78,73,68)(64,79,74,69)(65,80,75,70), (1,64,11,74)(2,75,12,65)(3,66,13,76)(4,77,14,67)(5,68,15,78)(6,79,16,69)(7,70,17,80)(8,61,18,71)(9,72,19,62)(10,63,20,73)(21,49,31,59)(22,60,32,50)(23,51,33,41)(24,42,34,52)(25,53,35,43)(26,44,36,54)(27,55,37,45)(28,46,38,56)(29,57,39,47)(30,48,40,58), (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), (1,44,6,49,11,54,16,59)(2,53,7,58,12,43,17,48)(3,42,8,47,13,52,18,57)(4,51,9,56,14,41,19,46)(5,60,10,45,15,50,20,55)(21,74,26,79,31,64,36,69)(22,63,27,68,32,73,37,78)(23,72,28,77,33,62,38,67)(24,61,29,66,34,71,39,76)(25,70,30,75,35,80,40,65) );
G=PermutationGroup([[(1,6,11,16),(2,7,12,17),(3,8,13,18),(4,9,14,19),(5,10,15,20),(21,54),(22,45),(23,56),(24,47),(25,58),(26,49),(27,60),(28,51),(29,42),(30,53),(31,44),(32,55),(33,46),(34,57),(35,48),(36,59),(37,50),(38,41),(39,52),(40,43),(61,76,71,66),(62,77,72,67),(63,78,73,68),(64,79,74,69),(65,80,75,70)], [(1,64,11,74),(2,75,12,65),(3,66,13,76),(4,77,14,67),(5,68,15,78),(6,79,16,69),(7,70,17,80),(8,61,18,71),(9,72,19,62),(10,63,20,73),(21,49,31,59),(22,60,32,50),(23,51,33,41),(24,42,34,52),(25,53,35,43),(26,44,36,54),(27,55,37,45),(28,46,38,56),(29,57,39,47),(30,48,40,58)], [(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)], [(1,44,6,49,11,54,16,59),(2,53,7,58,12,43,17,48),(3,42,8,47,13,52,18,57),(4,51,9,56,14,41,19,46),(5,60,10,45,15,50,20,55),(21,74,26,79,31,64,36,69),(22,63,27,68,32,73,37,78),(23,72,28,77,33,62,38,67),(24,61,29,66,34,71,39,76),(25,70,30,75,35,80,40,65)]])
41 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | ··· | 4E | 5A | 5B | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | ··· | 20L | 20M | 20N | 20O | 20P |
| order | 1 | 2 | 2 | 2 | 4 | ··· | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | 20 | 20 | 20 |
| size | 1 | 1 | 2 | 8 | 4 | ··· | 4 | 2 | 2 | 40 | 40 | 40 | 40 | 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 | Dic5 | D10 | C5⋊D4 | C23⋊C4 | C42.C4 | C23⋊Dic5 | C42.Dic5 |
| kernel | C42.Dic5 | C20.10D4 | C5×C4.4D4 | C4×C20 | D4×C10 | C2×C20 | C4.4D4 | C42 | C2×D4 | C2×Q8 | C2×C4 | 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
| 9 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 |
| 0 | 0 | 32 | 17 |
| 0 | 0 | 17 | 9 |
| 1 | 30 | 0 | 0 |
| 30 | 40 | 0 | 0 |
| 0 | 0 | 40 | 11 |
| 0 | 0 | 11 | 1 |
| 5 | 27 | 0 | 0 |
| 0 | 36 | 0 | 0 |
| 0 | 0 | 8 | 35 |
| 0 | 0 | 0 | 33 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 9 | 24 | 0 | 0 |
| 0 | 32 | 0 | 0 |
G:=sub<GL(4,GF(41))| [9,0,0,0,0,9,0,0,0,0,32,17,0,0,17,9],[1,30,0,0,30,40,0,0,0,0,40,11,0,0,11,1],[5,0,0,0,27,36,0,0,0,0,8,0,0,0,35,33],[0,0,9,0,0,0,24,32,1,0,0,0,0,1,0,0] >;
C42.Dic5 in GAP, Magma, Sage, TeX
C_4^2.{\rm Dic}_5 % in TeX
G:=Group("C4^2.Dic5"); // GroupNames label
G:=SmallGroup(320,100);
// by ID
G=gap.SmallGroup(320,100);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,219,184,1571,570,297,136,1684,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^10=b^2,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^9>;
// generators/relations