direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C5xC4.4D4, C42:5C10, C20.39D4, C4.4(C5xD4), (C4xC20):12C2, (Q8xC10):9C2, (C2xQ8):2C10, C2.8(D4xC10), C22:C4:5C10, (C2xD4).5C10, C10.71(C2xD4), (D4xC10).12C2, C23.2(C2xC10), C10.44(C4oD4), (C2xC20).66C22, (C2xC10).79C23, (C22xC10).2C22, C22.14(C22xC10), C2.7(C5xC4oD4), (C2xC4).6(C2xC10), (C5xC22:C4):13C2, SmallGroup(160,185)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5xC4.4D4
G = < a,b,c,d | a5=b4=c4=1, d2=b2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=b2c-1 >
Subgroups: 116 in 76 conjugacy classes, 44 normal (16 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C2xC4, C2xC4, D4, Q8, C23, C10, C10, C10, C42, C22:C4, C2xD4, C2xQ8, C20, C20, C2xC10, C2xC10, C4.4D4, C2xC20, C2xC20, C5xD4, C5xQ8, C22xC10, C4xC20, C5xC22:C4, D4xC10, Q8xC10, C5xC4.4D4
Quotients: C1, C2, C22, C5, D4, C23, C10, C2xD4, C4oD4, C2xC10, C4.4D4, C5xD4, C22xC10, D4xC10, C5xC4oD4, C5xC4.4D4
►On 80 points
(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 16 31 76)(2 17 32 77)(3 18 33 78)(4 19 34 79)(5 20 35 80)(6 21 11 26)(7 22 12 27)(8 23 13 28)(9 24 14 29)(10 25 15 30)(36 61 51 66)(37 62 52 67)(38 63 53 68)(39 64 54 69)(40 65 55 70)(41 56 46 71)(42 57 47 72)(43 58 48 73)(44 59 49 74)(45 60 50 75)
(1 56 21 61)(2 57 22 62)(3 58 23 63)(4 59 24 64)(5 60 25 65)(6 36 76 41)(7 37 77 42)(8 38 78 43)(9 39 79 44)(10 40 80 45)(11 51 16 46)(12 52 17 47)(13 53 18 48)(14 54 19 49)(15 55 20 50)(26 66 31 71)(27 67 32 72)(28 68 33 73)(29 69 34 74)(30 70 35 75)
(1 51 31 36)(2 52 32 37)(3 53 33 38)(4 54 34 39)(5 55 35 40)(6 71 11 56)(7 72 12 57)(8 73 13 58)(9 74 14 59)(10 75 15 60)(16 61 76 66)(17 62 77 67)(18 63 78 68)(19 64 79 69)(20 65 80 70)(21 46 26 41)(22 47 27 42)(23 48 28 43)(24 49 29 44)(25 50 30 45)
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), (1,16,31,76)(2,17,32,77)(3,18,33,78)(4,19,34,79)(5,20,35,80)(6,21,11,26)(7,22,12,27)(8,23,13,28)(9,24,14,29)(10,25,15,30)(36,61,51,66)(37,62,52,67)(38,63,53,68)(39,64,54,69)(40,65,55,70)(41,56,46,71)(42,57,47,72)(43,58,48,73)(44,59,49,74)(45,60,50,75), (1,56,21,61)(2,57,22,62)(3,58,23,63)(4,59,24,64)(5,60,25,65)(6,36,76,41)(7,37,77,42)(8,38,78,43)(9,39,79,44)(10,40,80,45)(11,51,16,46)(12,52,17,47)(13,53,18,48)(14,54,19,49)(15,55,20,50)(26,66,31,71)(27,67,32,72)(28,68,33,73)(29,69,34,74)(30,70,35,75), (1,51,31,36)(2,52,32,37)(3,53,33,38)(4,54,34,39)(5,55,35,40)(6,71,11,56)(7,72,12,57)(8,73,13,58)(9,74,14,59)(10,75,15,60)(16,61,76,66)(17,62,77,67)(18,63,78,68)(19,64,79,69)(20,65,80,70)(21,46,26,41)(22,47,27,42)(23,48,28,43)(24,49,29,44)(25,50,30,45)>;
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), (1,16,31,76)(2,17,32,77)(3,18,33,78)(4,19,34,79)(5,20,35,80)(6,21,11,26)(7,22,12,27)(8,23,13,28)(9,24,14,29)(10,25,15,30)(36,61,51,66)(37,62,52,67)(38,63,53,68)(39,64,54,69)(40,65,55,70)(41,56,46,71)(42,57,47,72)(43,58,48,73)(44,59,49,74)(45,60,50,75), (1,56,21,61)(2,57,22,62)(3,58,23,63)(4,59,24,64)(5,60,25,65)(6,36,76,41)(7,37,77,42)(8,38,78,43)(9,39,79,44)(10,40,80,45)(11,51,16,46)(12,52,17,47)(13,53,18,48)(14,54,19,49)(15,55,20,50)(26,66,31,71)(27,67,32,72)(28,68,33,73)(29,69,34,74)(30,70,35,75), (1,51,31,36)(2,52,32,37)(3,53,33,38)(4,54,34,39)(5,55,35,40)(6,71,11,56)(7,72,12,57)(8,73,13,58)(9,74,14,59)(10,75,15,60)(16,61,76,66)(17,62,77,67)(18,63,78,68)(19,64,79,69)(20,65,80,70)(21,46,26,41)(22,47,27,42)(23,48,28,43)(24,49,29,44)(25,50,30,45) );
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)], [(1,16,31,76),(2,17,32,77),(3,18,33,78),(4,19,34,79),(5,20,35,80),(6,21,11,26),(7,22,12,27),(8,23,13,28),(9,24,14,29),(10,25,15,30),(36,61,51,66),(37,62,52,67),(38,63,53,68),(39,64,54,69),(40,65,55,70),(41,56,46,71),(42,57,47,72),(43,58,48,73),(44,59,49,74),(45,60,50,75)], [(1,56,21,61),(2,57,22,62),(3,58,23,63),(4,59,24,64),(5,60,25,65),(6,36,76,41),(7,37,77,42),(8,38,78,43),(9,39,79,44),(10,40,80,45),(11,51,16,46),(12,52,17,47),(13,53,18,48),(14,54,19,49),(15,55,20,50),(26,66,31,71),(27,67,32,72),(28,68,33,73),(29,69,34,74),(30,70,35,75)], [(1,51,31,36),(2,52,32,37),(3,53,33,38),(4,54,34,39),(5,55,35,40),(6,71,11,56),(7,72,12,57),(8,73,13,58),(9,74,14,59),(10,75,15,60),(16,61,76,66),(17,62,77,67),(18,63,78,68),(19,64,79,69),(20,65,80,70),(21,46,26,41),(22,47,27,42),(23,48,28,43),(24,49,29,44),(25,50,30,45)]])
C5xC4.4D4 is a maximal subgroup of
C42.7D10 C42:Dic5 C42.Dic5 C42.61D10 C42.62D10 C42.213D10 D20.23D4 C42.64D10 C42.214D10 C42.65D10 C42:5D10 D20.14D4 C42.233D10 C42.137D10 C42.138D10 C42.139D10 C42.140D10 C42:18D10 C42.141D10 D20:10D4 Dic10:10D4 C42:20D10 C42:21D10 C42.234D10 C42.143D10 C42.144D10 C42:22D10 C42.145D10
70 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | ··· | 4F | 4G | 4H | 5A | 5B | 5C | 5D | 10A | ··· | 10L | 10M | ··· | 10T | 20A | ··· | 20X | 20Y | ··· | 20AF |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
70 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | ||||||||
| image | C1 | C2 | C2 | C2 | C2 | C5 | C10 | C10 | C10 | C10 | D4 | C4oD4 | C5xD4 | C5xC4oD4 |
| kernel | C5xC4.4D4 | C4xC20 | C5xC22:C4 | D4xC10 | Q8xC10 | C4.4D4 | C42 | C22:C4 | C2xD4 | C2xQ8 | C20 | C10 | C4 | C2 |
| # reps | 1 | 1 | 4 | 1 | 1 | 4 | 4 | 16 | 4 | 4 | 2 | 4 | 8 | 16 |
Matrix representation of C5xC4.4D4 ►in GL4(F41) generated by
| 18 | 0 | 0 | 0 |
| 0 | 18 | 0 | 0 |
| 0 | 0 | 10 | 0 |
| 0 | 0 | 0 | 10 |
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 9 | 23 |
| 0 | 0 | 0 | 32 |
| 0 | 40 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 32 | 0 |
| 0 | 0 | 0 | 32 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 39 |
| 0 | 0 | 1 | 40 |
G:=sub<GL(4,GF(41))| [18,0,0,0,0,18,0,0,0,0,10,0,0,0,0,10],[40,0,0,0,0,40,0,0,0,0,9,0,0,0,23,32],[0,1,0,0,40,0,0,0,0,0,32,0,0,0,0,32],[0,1,0,0,1,0,0,0,0,0,1,1,0,0,39,40] >;
C5xC4.4D4 in GAP, Magma, Sage, TeX
C_5\times C_4._4D_4
% in TeX
G:=Group("C5xC4.4D4"); // GroupNames label
G:=SmallGroup(160,185);
// by ID
G=gap.SmallGroup(160,185);
# by ID
G:=PCGroup([6,-2,-2,-2,-5,-2,-2,505,487,1514,194]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^4=c^4=1,d^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=b^2*c^-1>;
// generators/relations