metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.3Dic5, C4⋊Q8.3D5, (C2×C20).6D4, (C4×C20).20C4, (C2×Q8).4D10, (Q8×C10).13C4, (C2×Q8).3Dic5, C5⋊4(C42.3C4), (Q8×C10).4C22, C10.47(C23⋊C4), C20.10D4.2C2, C2.11(C23⋊Dic5), C22.17(C23.D5), (C5×C4⋊Q8).3C2, (C2×C4).8(C5⋊D4), (C2×C4).4(C2×Dic5), (C2×C20).184(C2×C4), (C2×C10).170(C22⋊C4), SmallGroup(320,106)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.3Dic5
G = < a,b,c,d | a4=b4=1, c10=b2, d2=c5, ab=ba, cac-1=a-1, dad-1=a-1b, cbc-1=b-1, dbd-1=a2b, dcd-1=c9 >
Subgroups: 174 in 60 conjugacy classes, 23 normal (17 characteristic)
C1, C2, C2, C4, C22, C5, C8, C2×C4, C2×C4, C2×C4, Q8, C10, C10, C42, C4⋊C4, M4(2), C2×Q8, C20, C2×C10, C4.10D4, C4⋊Q8, C5⋊2C8, C2×C20, C2×C20, C2×C20, C5×Q8, C42.3C4, C4.Dic5, C4×C20, C5×C4⋊C4, Q8×C10, C20.10D4, C5×C4⋊Q8, C42.3Dic5
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, C42.3Dic5
►On 80 points
Generators in S80
(41 75 51 65)(42 66 52 76)(43 77 53 67)(44 68 54 78)(45 79 55 69)(46 70 56 80)(47 61 57 71)(48 72 58 62)(49 63 59 73)(50 74 60 64)
(1 35 11 25)(2 26 12 36)(3 37 13 27)(4 28 14 38)(5 39 15 29)(6 30 16 40)(7 21 17 31)(8 32 18 22)(9 23 19 33)(10 34 20 24)(41 75 51 65)(42 66 52 76)(43 77 53 67)(44 68 54 78)(45 79 55 69)(46 70 56 80)(47 61 57 71)(48 72 58 62)(49 63 59 73)(50 74 60 64)
(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 42 6 47 11 52 16 57)(2 51 7 56 12 41 17 46)(3 60 8 45 13 50 18 55)(4 49 9 54 14 59 19 44)(5 58 10 43 15 48 20 53)(21 80 26 65 31 70 36 75)(22 69 27 74 32 79 37 64)(23 78 28 63 33 68 38 73)(24 67 29 72 34 77 39 62)(25 76 30 61 35 66 40 71)
G:=sub<Sym(80)| (41,75,51,65)(42,66,52,76)(43,77,53,67)(44,68,54,78)(45,79,55,69)(46,70,56,80)(47,61,57,71)(48,72,58,62)(49,63,59,73)(50,74,60,64), (1,35,11,25)(2,26,12,36)(3,37,13,27)(4,28,14,38)(5,39,15,29)(6,30,16,40)(7,21,17,31)(8,32,18,22)(9,23,19,33)(10,34,20,24)(41,75,51,65)(42,66,52,76)(43,77,53,67)(44,68,54,78)(45,79,55,69)(46,70,56,80)(47,61,57,71)(48,72,58,62)(49,63,59,73)(50,74,60,64), (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), (1,42,6,47,11,52,16,57)(2,51,7,56,12,41,17,46)(3,60,8,45,13,50,18,55)(4,49,9,54,14,59,19,44)(5,58,10,43,15,48,20,53)(21,80,26,65,31,70,36,75)(22,69,27,74,32,79,37,64)(23,78,28,63,33,68,38,73)(24,67,29,72,34,77,39,62)(25,76,30,61,35,66,40,71)>;
G:=Group( (41,75,51,65)(42,66,52,76)(43,77,53,67)(44,68,54,78)(45,79,55,69)(46,70,56,80)(47,61,57,71)(48,72,58,62)(49,63,59,73)(50,74,60,64), (1,35,11,25)(2,26,12,36)(3,37,13,27)(4,28,14,38)(5,39,15,29)(6,30,16,40)(7,21,17,31)(8,32,18,22)(9,23,19,33)(10,34,20,24)(41,75,51,65)(42,66,52,76)(43,77,53,67)(44,68,54,78)(45,79,55,69)(46,70,56,80)(47,61,57,71)(48,72,58,62)(49,63,59,73)(50,74,60,64), (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), (1,42,6,47,11,52,16,57)(2,51,7,56,12,41,17,46)(3,60,8,45,13,50,18,55)(4,49,9,54,14,59,19,44)(5,58,10,43,15,48,20,53)(21,80,26,65,31,70,36,75)(22,69,27,74,32,79,37,64)(23,78,28,63,33,68,38,73)(24,67,29,72,34,77,39,62)(25,76,30,61,35,66,40,71) );
G=PermutationGroup([[(41,75,51,65),(42,66,52,76),(43,77,53,67),(44,68,54,78),(45,79,55,69),(46,70,56,80),(47,61,57,71),(48,72,58,62),(49,63,59,73),(50,74,60,64)], [(1,35,11,25),(2,26,12,36),(3,37,13,27),(4,28,14,38),(5,39,15,29),(6,30,16,40),(7,21,17,31),(8,32,18,22),(9,23,19,33),(10,34,20,24),(41,75,51,65),(42,66,52,76),(43,77,53,67),(44,68,54,78),(45,79,55,69),(46,70,56,80),(47,61,57,71),(48,72,58,62),(49,63,59,73),(50,74,60,64)], [(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)], [(1,42,6,47,11,52,16,57),(2,51,7,56,12,41,17,46),(3,60,8,45,13,50,18,55),(4,49,9,54,14,59,19,44),(5,58,10,43,15,48,20,53),(21,80,26,65,31,70,36,75),(22,69,27,74,32,79,37,64),(23,78,28,63,33,68,38,73),(24,67,29,72,34,77,39,62),(25,76,30,61,35,66,40,71)]])
41 conjugacy classes
| class | 1 | 2A | 2B | 4A | ··· | 4E | 4F | 5A | 5B | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 20A | ··· | 20L | 20M | ··· | 20T |
| order | 1 | 2 | 2 | 4 | ··· | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 2 | 4 | ··· | 4 | 8 | 2 | 2 | 40 | 40 | 40 | 40 | 2 | ··· | 2 | 4 | ··· | 4 | 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 | C42.3C4 | C23⋊Dic5 | C42.3Dic5 |
| kernel | C42.3Dic5 | C20.10D4 | C5×C4⋊Q8 | C4×C20 | Q8×C10 | C2×C20 | C4⋊Q8 | C42 | C2×Q8 | 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 C42.3Dic5 ►in GL6(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 23 | 0 | 0 | 0 | 0 | 0 |
| 16 | 25 | 0 | 0 | 0 | 0 |
| 0 | 0 | 26 | 15 | 0 | 0 |
| 0 | 0 | 15 | 15 | 0 | 0 |
| 0 | 0 | 0 | 0 | 26 | 15 |
| 0 | 0 | 0 | 0 | 15 | 15 |
| 25 | 39 | 0 | 0 | 0 | 0 |
| 26 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 26 | 15 | 0 | 0 |
| 0 | 0 | 15 | 15 | 0 | 0 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,40,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,40,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,40,0],[23,16,0,0,0,0,0,25,0,0,0,0,0,0,26,15,0,0,0,0,15,15,0,0,0,0,0,0,26,15,0,0,0,0,15,15],[25,26,0,0,0,0,39,16,0,0,0,0,0,0,0,0,26,15,0,0,0,0,15,15,0,0,1,0,0,0,0,0,0,1,0,0] >;
C42.3Dic5 in GAP, Magma, Sage, TeX
C_4^2._3{\rm Dic}_5 % in TeX
G:=Group("C4^2.3Dic5"); // GroupNames label
G:=SmallGroup(320,106);
// by ID
G=gap.SmallGroup(320,106);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,232,219,184,1571,570,297,136,1684,12550]);
// 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,d*a*d^-1=a^-1*b,c*b*c^-1=b^-1,d*b*d^-1=a^2*b,d*c*d^-1=c^9>;
// generators/relations