direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C5xD4:C4, D4:1C20, C10.13D8, C20.60D4, C10.9SD16, C4:C4:1C10, (C2xC8):2C10, (C2xC40):4C2, (C5xD4):7C4, C2.1(C5xD8), C4.1(C2xC20), C4.11(C5xD4), C20.49(C2xC4), (C2xD4).3C10, (D4xC10).9C2, (C2xC10).46D4, C2.1(C5xSD16), C22.8(C5xD4), C10.35(C22:C4), (C2xC20).114C22, (C5xC4:C4):10C2, C2.6(C5xC22:C4), (C2xC4).17(C2xC10), SmallGroup(160,52)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5xD4:C4
G = < a,b,c,d | a5=b4=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=dbd-1=b-1, dcd-1=bc >
Quotients: C1, C2, C4, C22, C5, C2xC4, D4, C10, C22:C4, D8, SD16, C20, C2xC10, D4:C4, C2xC20, C5xD4, C5xC22:C4, C5xD8, C5xSD16, C5xD4:C4
Smallest permutation representation of C5xD4:C4
►On 80 points
Generators in S80
(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 41 12 30)(2 42 13 26)(3 43 14 27)(4 44 15 28)(5 45 11 29)(6 71 80 35)(7 72 76 31)(8 73 77 32)(9 74 78 33)(10 75 79 34)(16 65 50 24)(17 61 46 25)(18 62 47 21)(19 63 48 22)(20 64 49 23)(36 54 70 59)(37 55 66 60)(38 51 67 56)(39 52 68 57)(40 53 69 58)
(1 55)(2 51)(3 52)(4 53)(5 54)(6 47)(7 48)(8 49)(9 50)(10 46)(11 59)(12 60)(13 56)(14 57)(15 58)(16 78)(17 79)(18 80)(19 76)(20 77)(21 35)(22 31)(23 32)(24 33)(25 34)(26 67)(27 68)(28 69)(29 70)(30 66)(36 45)(37 41)(38 42)(39 43)(40 44)(61 75)(62 71)(63 72)(64 73)(65 74)
(1 8 66 20)(2 9 67 16)(3 10 68 17)(4 6 69 18)(5 7 70 19)(11 76 36 48)(12 77 37 49)(13 78 38 50)(14 79 39 46)(15 80 40 47)(21 44 35 58)(22 45 31 59)(23 41 32 60)(24 42 33 56)(25 43 34 57)(26 74 51 65)(27 75 52 61)(28 71 53 62)(29 72 54 63)(30 73 55 64)
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), (1,41,12,30)(2,42,13,26)(3,43,14,27)(4,44,15,28)(5,45,11,29)(6,71,80,35)(7,72,76,31)(8,73,77,32)(9,74,78,33)(10,75,79,34)(16,65,50,24)(17,61,46,25)(18,62,47,21)(19,63,48,22)(20,64,49,23)(36,54,70,59)(37,55,66,60)(38,51,67,56)(39,52,68,57)(40,53,69,58), (1,55)(2,51)(3,52)(4,53)(5,54)(6,47)(7,48)(8,49)(9,50)(10,46)(11,59)(12,60)(13,56)(14,57)(15,58)(16,78)(17,79)(18,80)(19,76)(20,77)(21,35)(22,31)(23,32)(24,33)(25,34)(26,67)(27,68)(28,69)(29,70)(30,66)(36,45)(37,41)(38,42)(39,43)(40,44)(61,75)(62,71)(63,72)(64,73)(65,74), (1,8,66,20)(2,9,67,16)(3,10,68,17)(4,6,69,18)(5,7,70,19)(11,76,36,48)(12,77,37,49)(13,78,38,50)(14,79,39,46)(15,80,40,47)(21,44,35,58)(22,45,31,59)(23,41,32,60)(24,42,33,56)(25,43,34,57)(26,74,51,65)(27,75,52,61)(28,71,53,62)(29,72,54,63)(30,73,55,64)>;
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), (1,41,12,30)(2,42,13,26)(3,43,14,27)(4,44,15,28)(5,45,11,29)(6,71,80,35)(7,72,76,31)(8,73,77,32)(9,74,78,33)(10,75,79,34)(16,65,50,24)(17,61,46,25)(18,62,47,21)(19,63,48,22)(20,64,49,23)(36,54,70,59)(37,55,66,60)(38,51,67,56)(39,52,68,57)(40,53,69,58), (1,55)(2,51)(3,52)(4,53)(5,54)(6,47)(7,48)(8,49)(9,50)(10,46)(11,59)(12,60)(13,56)(14,57)(15,58)(16,78)(17,79)(18,80)(19,76)(20,77)(21,35)(22,31)(23,32)(24,33)(25,34)(26,67)(27,68)(28,69)(29,70)(30,66)(36,45)(37,41)(38,42)(39,43)(40,44)(61,75)(62,71)(63,72)(64,73)(65,74), (1,8,66,20)(2,9,67,16)(3,10,68,17)(4,6,69,18)(5,7,70,19)(11,76,36,48)(12,77,37,49)(13,78,38,50)(14,79,39,46)(15,80,40,47)(21,44,35,58)(22,45,31,59)(23,41,32,60)(24,42,33,56)(25,43,34,57)(26,74,51,65)(27,75,52,61)(28,71,53,62)(29,72,54,63)(30,73,55,64) );
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)], [(1,41,12,30),(2,42,13,26),(3,43,14,27),(4,44,15,28),(5,45,11,29),(6,71,80,35),(7,72,76,31),(8,73,77,32),(9,74,78,33),(10,75,79,34),(16,65,50,24),(17,61,46,25),(18,62,47,21),(19,63,48,22),(20,64,49,23),(36,54,70,59),(37,55,66,60),(38,51,67,56),(39,52,68,57),(40,53,69,58)], [(1,55),(2,51),(3,52),(4,53),(5,54),(6,47),(7,48),(8,49),(9,50),(10,46),(11,59),(12,60),(13,56),(14,57),(15,58),(16,78),(17,79),(18,80),(19,76),(20,77),(21,35),(22,31),(23,32),(24,33),(25,34),(26,67),(27,68),(28,69),(29,70),(30,66),(36,45),(37,41),(38,42),(39,43),(40,44),(61,75),(62,71),(63,72),(64,73),(65,74)], [(1,8,66,20),(2,9,67,16),(3,10,68,17),(4,6,69,18),(5,7,70,19),(11,76,36,48),(12,77,37,49),(13,78,38,50),(14,79,39,46),(15,80,40,47),(21,44,35,58),(22,45,31,59),(23,41,32,60),(24,42,33,56),(25,43,34,57),(26,74,51,65),(27,75,52,61),(28,71,53,62),(29,72,54,63),(30,73,55,64)]])
C5xD4:C4 is a maximal subgroup of
Dic5:4D8 D4.D5:5C4 Dic5:6SD16 Dic5.14D8 Dic5.5D8 D4:Dic10 Dic10:2D4 D4.Dic10 C4:C4.D10 C20:Q8:C2 D4.2Dic10 Dic10.D4 (C8xDic5):C2 (D4xD5):C4 D4:(C4xD5) D4:2D5:C4 D4:D20 D10.12D8 D10:D8 D20.8D4 D10.16SD16 D10:SD16 C40:6C4:C2 C5:2C8:D4 D4:3D20 C5:(C8:2D4) D4.D20 C40:5C4:C2 D4:D5:6C4 D20:3D4 D20.D4 D8xC20 SD16xC20
70 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 5A | 5B | 5C | 5D | 8A | 8B | 8C | 8D | 10A | ··· | 10L | 10M | ··· | 10T | 20A | ··· | 20H | 20I | ··· | 20P | 40A | ··· | 40P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 2 | 2 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 |
70 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | |||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C5 | C10 | C10 | C10 | C20 | D4 | D4 | D8 | SD16 | C5xD4 | C5xD4 | C5xD8 | C5xSD16 |
| kernel | C5xD4:C4 | C5xC4:C4 | C2xC40 | D4xC10 | C5xD4 | D4:C4 | C4:C4 | C2xC8 | C2xD4 | D4 | C20 | C2xC10 | C10 | C10 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 4 | 16 | 1 | 1 | 2 | 2 | 4 | 4 | 8 | 8 |
Matrix representation of C5xD4:C4 ►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 | 0 | 1 |
| 0 | 0 | 40 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
| 0 | 1 | 0 | 0 |
| 40 | 0 | 0 | 0 |
| 0 | 0 | 29 | 12 |
| 0 | 0 | 12 | 12 |
G:=sub<GL(4,GF(41))| [18,0,0,0,0,18,0,0,0,0,10,0,0,0,0,10],[40,0,0,0,0,40,0,0,0,0,0,40,0,0,1,0],[1,0,0,0,0,40,0,0,0,0,0,1,0,0,1,0],[0,40,0,0,1,0,0,0,0,0,29,12,0,0,12,12] >;
C5xD4:C4 in GAP, Magma, Sage, TeX
C_5\times D_4\rtimes C_4
% in TeX
G:=Group("C5xD4:C4"); // GroupNames label
G:=SmallGroup(160,52);
// by ID
G=gap.SmallGroup(160,52);
# by ID
G:=PCGroup([6,-2,-2,-5,-2,-2,-2,240,265,2403,1209,117]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^4=c^2=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=d*b*d^-1=b^-1,d*c*d^-1=b*c>;
// generators/relations
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