direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D5×D4⋊C4, D10.22D8, D10.20SD16, (D4×D5)⋊1C4, D4⋊5(C4×D5), C2.2(D5×D8), C4⋊C4⋊18D10, (C2×C8)⋊25D10, D20⋊12(C2×C4), (C4×D5).93D4, C4.154(D4×D5), C10.21(C2×D8), D20⋊6C4⋊4C2, C2.2(D5×SD16), (C2×C40)⋊21C22, C20.103(C2×D4), D4⋊Dic5⋊3C2, D20⋊5C4⋊17C2, C22.68(D4×D5), (C2×D4).130D10, C20.39(C22×C4), C4⋊Dic5⋊17C22, C10.22(C2×SD16), (C2×C20).209C23, (C2×Dic5).133D4, (C22×D5).152D4, (D4×C10).30C22, (C2×D20).51C22, D10.55(C22⋊C4), Dic5.23(C22⋊C4), C4.4(C2×C4×D5), (D5×C4⋊C4)⋊1C2, (D5×C2×C8)⋊16C2, (C2×D4×D5).3C2, C5⋊2(C2×D4⋊C4), (C5×D4)⋊12(C2×C4), (C5×C4⋊C4)⋊1C22, (C4×D5).48(C2×C4), C2.17(D5×C22⋊C4), (C5×D4⋊C4)⋊18C2, (C2×C5⋊2C8)⋊30C22, (C2×C10).222(C2×D4), C10.57(C2×C22⋊C4), (C2×C4×D5).292C22, (C2×C4).316(C22×D5), SmallGroup(320,396)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D5×D4⋊C4
G = < a,b,c,d,e | a5=b2=c4=d2=e4=1, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd=ece-1=c-1, ede-1=cd >
Subgroups: 1022 in 202 conjugacy classes, 63 normal (37 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, D4, C23, D5, D5, C10, C10, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C2×D4, C2×D4, C24, Dic5, Dic5, C20, C20, D10, D10, C2×C10, C2×C10, D4⋊C4, D4⋊C4, C2×C4⋊C4, C22×C8, C22×D4, C5⋊2C8, C40, C4×D5, C4×D5, D20, D20, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C5×D4, C5×D4, C22×D5, C22×D5, C22×C10, C2×D4⋊C4, C8×D5, C2×C5⋊2C8, C10.D4, C4⋊Dic5, C5×C4⋊C4, C2×C40, C2×C4×D5, C2×C4×D5, C2×D20, D4×D5, D4×D5, C2×C5⋊D4, D4×C10, C23×D5, D20⋊6C4, D20⋊5C4, D4⋊Dic5, C5×D4⋊C4, D5×C4⋊C4, D5×C2×C8, C2×D4×D5, D5×D4⋊C4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D5, C22⋊C4, D8, SD16, C22×C4, C2×D4, D10, D4⋊C4, C2×C22⋊C4, C2×D8, C2×SD16, C4×D5, C22×D5, C2×D4⋊C4, C2×C4×D5, D4×D5, D5×C22⋊C4, D5×D8, D5×SD16, D5×D4⋊C4
►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 28)(2 27)(3 26)(4 30)(5 29)(6 21)(7 25)(8 24)(9 23)(10 22)(11 36)(12 40)(13 39)(14 38)(15 37)(16 31)(17 35)(18 34)(19 33)(20 32)(41 66)(42 70)(43 69)(44 68)(45 67)(46 61)(47 65)(48 64)(49 63)(50 62)(51 76)(52 80)(53 79)(54 78)(55 77)(56 71)(57 75)(58 74)(59 73)(60 72)
(1 14 9 19)(2 15 10 20)(3 11 6 16)(4 12 7 17)(5 13 8 18)(21 31 26 36)(22 32 27 37)(23 33 28 38)(24 34 29 39)(25 35 30 40)(41 56 46 51)(42 57 47 52)(43 58 48 53)(44 59 49 54)(45 60 50 55)(61 76 66 71)(62 77 67 72)(63 78 68 73)(64 79 69 74)(65 80 70 75)
(1 19)(2 20)(3 16)(4 17)(5 18)(6 11)(7 12)(8 13)(9 14)(10 15)(21 36)(22 37)(23 38)(24 39)(25 40)(26 31)(27 32)(28 33)(29 34)(30 35)(41 46)(42 47)(43 48)(44 49)(45 50)(61 66)(62 67)(63 68)(64 69)(65 70)
(1 64 24 44)(2 65 25 45)(3 61 21 41)(4 62 22 42)(5 63 23 43)(6 66 26 46)(7 67 27 47)(8 68 28 48)(9 69 29 49)(10 70 30 50)(11 71 31 51)(12 72 32 52)(13 73 33 53)(14 74 34 54)(15 75 35 55)(16 76 36 56)(17 77 37 57)(18 78 38 58)(19 79 39 59)(20 80 40 60)
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), (1,28)(2,27)(3,26)(4,30)(5,29)(6,21)(7,25)(8,24)(9,23)(10,22)(11,36)(12,40)(13,39)(14,38)(15,37)(16,31)(17,35)(18,34)(19,33)(20,32)(41,66)(42,70)(43,69)(44,68)(45,67)(46,61)(47,65)(48,64)(49,63)(50,62)(51,76)(52,80)(53,79)(54,78)(55,77)(56,71)(57,75)(58,74)(59,73)(60,72), (1,14,9,19)(2,15,10,20)(3,11,6,16)(4,12,7,17)(5,13,8,18)(21,31,26,36)(22,32,27,37)(23,33,28,38)(24,34,29,39)(25,35,30,40)(41,56,46,51)(42,57,47,52)(43,58,48,53)(44,59,49,54)(45,60,50,55)(61,76,66,71)(62,77,67,72)(63,78,68,73)(64,79,69,74)(65,80,70,75), (1,19)(2,20)(3,16)(4,17)(5,18)(6,11)(7,12)(8,13)(9,14)(10,15)(21,36)(22,37)(23,38)(24,39)(25,40)(26,31)(27,32)(28,33)(29,34)(30,35)(41,46)(42,47)(43,48)(44,49)(45,50)(61,66)(62,67)(63,68)(64,69)(65,70), (1,64,24,44)(2,65,25,45)(3,61,21,41)(4,62,22,42)(5,63,23,43)(6,66,26,46)(7,67,27,47)(8,68,28,48)(9,69,29,49)(10,70,30,50)(11,71,31,51)(12,72,32,52)(13,73,33,53)(14,74,34,54)(15,75,35,55)(16,76,36,56)(17,77,37,57)(18,78,38,58)(19,79,39,59)(20,80,40,60)>;
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), (1,28)(2,27)(3,26)(4,30)(5,29)(6,21)(7,25)(8,24)(9,23)(10,22)(11,36)(12,40)(13,39)(14,38)(15,37)(16,31)(17,35)(18,34)(19,33)(20,32)(41,66)(42,70)(43,69)(44,68)(45,67)(46,61)(47,65)(48,64)(49,63)(50,62)(51,76)(52,80)(53,79)(54,78)(55,77)(56,71)(57,75)(58,74)(59,73)(60,72), (1,14,9,19)(2,15,10,20)(3,11,6,16)(4,12,7,17)(5,13,8,18)(21,31,26,36)(22,32,27,37)(23,33,28,38)(24,34,29,39)(25,35,30,40)(41,56,46,51)(42,57,47,52)(43,58,48,53)(44,59,49,54)(45,60,50,55)(61,76,66,71)(62,77,67,72)(63,78,68,73)(64,79,69,74)(65,80,70,75), (1,19)(2,20)(3,16)(4,17)(5,18)(6,11)(7,12)(8,13)(9,14)(10,15)(21,36)(22,37)(23,38)(24,39)(25,40)(26,31)(27,32)(28,33)(29,34)(30,35)(41,46)(42,47)(43,48)(44,49)(45,50)(61,66)(62,67)(63,68)(64,69)(65,70), (1,64,24,44)(2,65,25,45)(3,61,21,41)(4,62,22,42)(5,63,23,43)(6,66,26,46)(7,67,27,47)(8,68,28,48)(9,69,29,49)(10,70,30,50)(11,71,31,51)(12,72,32,52)(13,73,33,53)(14,74,34,54)(15,75,35,55)(16,76,36,56)(17,77,37,57)(18,78,38,58)(19,79,39,59)(20,80,40,60) );
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)], [(1,28),(2,27),(3,26),(4,30),(5,29),(6,21),(7,25),(8,24),(9,23),(10,22),(11,36),(12,40),(13,39),(14,38),(15,37),(16,31),(17,35),(18,34),(19,33),(20,32),(41,66),(42,70),(43,69),(44,68),(45,67),(46,61),(47,65),(48,64),(49,63),(50,62),(51,76),(52,80),(53,79),(54,78),(55,77),(56,71),(57,75),(58,74),(59,73),(60,72)], [(1,14,9,19),(2,15,10,20),(3,11,6,16),(4,12,7,17),(5,13,8,18),(21,31,26,36),(22,32,27,37),(23,33,28,38),(24,34,29,39),(25,35,30,40),(41,56,46,51),(42,57,47,52),(43,58,48,53),(44,59,49,54),(45,60,50,55),(61,76,66,71),(62,77,67,72),(63,78,68,73),(64,79,69,74),(65,80,70,75)], [(1,19),(2,20),(3,16),(4,17),(5,18),(6,11),(7,12),(8,13),(9,14),(10,15),(21,36),(22,37),(23,38),(24,39),(25,40),(26,31),(27,32),(28,33),(29,34),(30,35),(41,46),(42,47),(43,48),(44,49),(45,50),(61,66),(62,67),(63,68),(64,69),(65,70)], [(1,64,24,44),(2,65,25,45),(3,61,21,41),(4,62,22,42),(5,63,23,43),(6,66,26,46),(7,67,27,47),(8,68,28,48),(9,69,29,49),(10,70,30,50),(11,71,31,51),(12,72,32,52),(13,73,33,53),(14,74,34,54),(15,75,35,55),(16,76,36,56),(17,77,37,57),(18,78,38,58),(19,79,39,59),(20,80,40,60)]])
56 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5A | 5B | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | 20B | 20C | 20D | 20E | 20F | 20G | 20H | 40A | ··· | 40H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 5 | 5 | 5 | 5 | 20 | 20 | 2 | 2 | 4 | 4 | 10 | 10 | 20 | 20 | 2 | 2 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 2 | ··· | 2 | 8 | 8 | 8 | 8 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | ··· | 4 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | D4 | D4 | D5 | D8 | SD16 | D10 | D10 | D10 | C4×D5 | D4×D5 | D4×D5 | D5×D8 | D5×SD16 |
| kernel | D5×D4⋊C4 | D20⋊6C4 | D20⋊5C4 | D4⋊Dic5 | C5×D4⋊C4 | D5×C4⋊C4 | D5×C2×C8 | C2×D4×D5 | D4×D5 | C4×D5 | C2×Dic5 | C22×D5 | D4⋊C4 | D10 | D10 | C4⋊C4 | C2×C8 | C2×D4 | D4 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 8 | 2 | 1 | 1 | 2 | 4 | 4 | 2 | 2 | 2 | 8 | 2 | 2 | 4 | 4 |
Matrix representation of D5×D4⋊C4 ►in GL6(𝔽41)
| 7 | 1 | 0 | 0 | 0 | 0 |
| 33 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 40 | 40 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 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 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 9 | 0 | 0 |
| 0 | 0 | 9 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 29 | 29 |
| 0 | 0 | 0 | 0 | 29 | 12 |
G:=sub<GL(6,GF(41))| [7,33,0,0,0,0,1,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[40,0,0,0,0,0,40,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,1,0,0,0,0,40,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,0,40,0,0,0,0,40,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,9,0,0,0,0,9,0,0,0,0,0,0,0,29,29,0,0,0,0,29,12] >;
D5×D4⋊C4 in GAP, Magma, Sage, TeX
D_5\times D_4\rtimes C_4
% in TeX
G:=Group("D5xD4:C4"); // GroupNames label
G:=SmallGroup(320,396);
// by ID
G=gap.SmallGroup(320,396);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,219,58,851,438,102,12550]);
// Polycyclic
G:=Group<a,b,c,d,e|a^5=b^2=c^4=d^2=e^4=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,d*c*d=e*c*e^-1=c^-1,e*d*e^-1=c*d>;
// generators/relations