metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: (D4×D5)⋊2C4, C4⋊C4⋊19D10, (C2×C8)⋊19D10, (C4×D5).3D4, D4.9(C4×D5), C4.155(D4×D5), D4⋊C4⋊16D5, D20⋊6C4⋊5C2, (C2×C40)⋊22C22, D20.15(C2×C4), D4⋊Dic5⋊4C2, C20.104(C2×D4), D20⋊5C4⋊18C2, C22.69(D4×D5), (C2×D4).131D10, C2.2(D8⋊D5), C2.1(D40⋊C2), C20.40(C22×C4), C4⋊Dic5⋊18C22, C10.29(C8⋊C22), (C2×C20).210C23, (C2×Dic5).195D4, (C22×D5).105D4, (C2×D20).52C22, (D4×C10).31C22, C5⋊2(C23.37D4), D10.36(C22⋊C4), Dic5.24(C22⋊C4), C4.5(C2×C4×D5), (C2×D4×D5).4C2, C4⋊C4⋊7D5⋊1C2, (C5×C4⋊C4)⋊2C22, (C4×D5).1(C2×C4), (C2×C8⋊D5)⋊15C2, (C2×C4×D5).9C22, (C2×C5⋊2C8)⋊1C22, (C5×D4).17(C2×C4), C2.18(D5×C22⋊C4), (C5×D4⋊C4)⋊19C2, (C2×C10).223(C2×D4), C10.58(C2×C22⋊C4), (C2×C4).317(C22×D5), SmallGroup(320,397)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for (D4×D5)⋊C4
G = < a,b,c,d,e | a4=b2=c5=d2=e4=1, bab=eae-1=a-1, ac=ca, ad=da, bc=cb, bd=db, ebe-1=ab, dcd=c-1, ce=ec, ede-1=a2d >
Subgroups: 974 in 190 conjugacy classes, 55 normal (37 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, D4, C23, D5, C10, C10, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), C22×C4, C2×D4, C2×D4, C24, Dic5, Dic5, C20, C20, D10, D10, C2×C10, C2×C10, D4⋊C4, D4⋊C4, C42⋊C2, C2×M4(2), C22×D4, C5⋊2C8, C40, C4×D5, D20, D20, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C5×D4, C5×D4, C22×D5, C22×D5, C22×C10, C23.37D4, C8⋊D5, C2×C5⋊2C8, C4×Dic5, C4⋊Dic5, D10⋊C4, C5×C4⋊C4, C2×C40, 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, C4⋊C4⋊7D5, C2×C8⋊D5, C2×D4×D5, (D4×D5)⋊C4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D5, C22⋊C4, C22×C4, C2×D4, D10, C2×C22⋊C4, C8⋊C22, C4×D5, C22×D5, C23.37D4, C2×C4×D5, D4×D5, D5×C22⋊C4, D8⋊D5, D40⋊C2, (D4×D5)⋊C4
►On 80 points
(1 16 6 11)(2 17 7 12)(3 18 8 13)(4 19 9 14)(5 20 10 15)(21 36 26 31)(22 37 27 32)(23 38 28 33)(24 39 29 34)(25 40 30 35)(41 51 46 56)(42 52 47 57)(43 53 48 58)(44 54 49 59)(45 55 50 60)(61 71 66 76)(62 72 67 77)(63 73 68 78)(64 74 69 79)(65 75 70 80)
(1 21)(2 22)(3 23)(4 24)(5 25)(6 26)(7 27)(8 28)(9 29)(10 30)(11 36)(12 37)(13 38)(14 39)(15 40)(16 31)(17 32)(18 33)(19 34)(20 35)(41 71)(42 72)(43 73)(44 74)(45 75)(46 76)(47 77)(48 78)(49 79)(50 80)(51 61)(52 62)(53 63)(54 64)(55 65)(56 66)(57 67)(58 68)(59 69)(60 70)
(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 10)(2 9)(3 8)(4 7)(5 6)(11 20)(12 19)(13 18)(14 17)(15 16)(21 30)(22 29)(23 28)(24 27)(25 26)(31 40)(32 39)(33 38)(34 37)(35 36)(41 45)(42 44)(46 50)(47 49)(51 55)(52 54)(56 60)(57 59)(61 65)(62 64)(66 70)(67 69)(71 75)(72 74)(76 80)(77 79)
(1 61 21 41)(2 62 22 42)(3 63 23 43)(4 64 24 44)(5 65 25 45)(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)| (1,16,6,11)(2,17,7,12)(3,18,8,13)(4,19,9,14)(5,20,10,15)(21,36,26,31)(22,37,27,32)(23,38,28,33)(24,39,29,34)(25,40,30,35)(41,51,46,56)(42,52,47,57)(43,53,48,58)(44,54,49,59)(45,55,50,60)(61,71,66,76)(62,72,67,77)(63,73,68,78)(64,74,69,79)(65,75,70,80), (1,21)(2,22)(3,23)(4,24)(5,25)(6,26)(7,27)(8,28)(9,29)(10,30)(11,36)(12,37)(13,38)(14,39)(15,40)(16,31)(17,32)(18,33)(19,34)(20,35)(41,71)(42,72)(43,73)(44,74)(45,75)(46,76)(47,77)(48,78)(49,79)(50,80)(51,61)(52,62)(53,63)(54,64)(55,65)(56,66)(57,67)(58,68)(59,69)(60,70), (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), (1,10)(2,9)(3,8)(4,7)(5,6)(11,20)(12,19)(13,18)(14,17)(15,16)(21,30)(22,29)(23,28)(24,27)(25,26)(31,40)(32,39)(33,38)(34,37)(35,36)(41,45)(42,44)(46,50)(47,49)(51,55)(52,54)(56,60)(57,59)(61,65)(62,64)(66,70)(67,69)(71,75)(72,74)(76,80)(77,79), (1,61,21,41)(2,62,22,42)(3,63,23,43)(4,64,24,44)(5,65,25,45)(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( (1,16,6,11)(2,17,7,12)(3,18,8,13)(4,19,9,14)(5,20,10,15)(21,36,26,31)(22,37,27,32)(23,38,28,33)(24,39,29,34)(25,40,30,35)(41,51,46,56)(42,52,47,57)(43,53,48,58)(44,54,49,59)(45,55,50,60)(61,71,66,76)(62,72,67,77)(63,73,68,78)(64,74,69,79)(65,75,70,80), (1,21)(2,22)(3,23)(4,24)(5,25)(6,26)(7,27)(8,28)(9,29)(10,30)(11,36)(12,37)(13,38)(14,39)(15,40)(16,31)(17,32)(18,33)(19,34)(20,35)(41,71)(42,72)(43,73)(44,74)(45,75)(46,76)(47,77)(48,78)(49,79)(50,80)(51,61)(52,62)(53,63)(54,64)(55,65)(56,66)(57,67)(58,68)(59,69)(60,70), (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), (1,10)(2,9)(3,8)(4,7)(5,6)(11,20)(12,19)(13,18)(14,17)(15,16)(21,30)(22,29)(23,28)(24,27)(25,26)(31,40)(32,39)(33,38)(34,37)(35,36)(41,45)(42,44)(46,50)(47,49)(51,55)(52,54)(56,60)(57,59)(61,65)(62,64)(66,70)(67,69)(71,75)(72,74)(76,80)(77,79), (1,61,21,41)(2,62,22,42)(3,63,23,43)(4,64,24,44)(5,65,25,45)(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([[(1,16,6,11),(2,17,7,12),(3,18,8,13),(4,19,9,14),(5,20,10,15),(21,36,26,31),(22,37,27,32),(23,38,28,33),(24,39,29,34),(25,40,30,35),(41,51,46,56),(42,52,47,57),(43,53,48,58),(44,54,49,59),(45,55,50,60),(61,71,66,76),(62,72,67,77),(63,73,68,78),(64,74,69,79),(65,75,70,80)], [(1,21),(2,22),(3,23),(4,24),(5,25),(6,26),(7,27),(8,28),(9,29),(10,30),(11,36),(12,37),(13,38),(14,39),(15,40),(16,31),(17,32),(18,33),(19,34),(20,35),(41,71),(42,72),(43,73),(44,74),(45,75),(46,76),(47,77),(48,78),(49,79),(50,80),(51,61),(52,62),(53,63),(54,64),(55,65),(56,66),(57,67),(58,68),(59,69),(60,70)], [(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)], [(1,10),(2,9),(3,8),(4,7),(5,6),(11,20),(12,19),(13,18),(14,17),(15,16),(21,30),(22,29),(23,28),(24,27),(25,26),(31,40),(32,39),(33,38),(34,37),(35,36),(41,45),(42,44),(46,50),(47,49),(51,55),(52,54),(56,60),(57,59),(61,65),(62,64),(66,70),(67,69),(71,75),(72,74),(76,80),(77,79)], [(1,61,21,41),(2,62,22,42),(3,63,23,43),(4,64,24,44),(5,65,25,45),(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)]])
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5A | 5B | 8A | 8B | 8C | 8D | 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 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 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 | 10 | 10 | 20 | 20 | 2 | 2 | 4 | 4 | 10 | 10 | 20 | 20 | 2 | 2 | 4 | 4 | 20 | 20 | 2 | ··· | 2 | 8 | 8 | 8 | 8 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | ··· | 4 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | D4 | D4 | D5 | D10 | D10 | D10 | C4×D5 | C8⋊C22 | D4×D5 | D4×D5 | D8⋊D5 | D40⋊C2 |
| kernel | (D4×D5)⋊C4 | D20⋊6C4 | D20⋊5C4 | D4⋊Dic5 | C5×D4⋊C4 | C4⋊C4⋊7D5 | C2×C8⋊D5 | C2×D4×D5 | D4×D5 | C4×D5 | C2×Dic5 | C22×D5 | D4⋊C4 | C4⋊C4 | C2×C8 | C2×D4 | D4 | C10 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 8 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 8 | 2 | 2 | 2 | 4 | 4 |
Matrix representation of (D4×D5)⋊C4 ►in GL6(𝔽41)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 3 | 20 |
| 0 | 0 | 0 | 1 | 1 | 20 |
| 0 | 0 | 40 | 1 | 40 | 0 |
| 0 | 0 | 39 | 6 | 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 | 1 | 40 | 1 | 0 |
| 0 | 0 | 2 | 35 | 0 | 1 |
| 6 | 40 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 40 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 40 |
| 0 | 0 | 0 | 0 | 8 | 40 |
| 6 | 40 | 0 | 0 | 0 | 0 |
| 35 | 35 | 0 | 0 | 0 | 0 |
| 0 | 0 | 35 | 1 | 0 | 0 |
| 0 | 0 | 6 | 6 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 35 |
| 0 | 0 | 0 | 0 | 34 | 0 |
| 32 | 0 | 0 | 0 | 0 | 0 |
| 0 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 34 | 16 | 18 | 8 |
| 0 | 0 | 25 | 7 | 0 | 33 |
| 0 | 0 | 16 | 16 | 23 | 25 |
| 0 | 0 | 0 | 5 | 5 | 18 |
G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,40,39,0,0,0,1,1,6,0,0,3,1,40,0,0,0,20,20,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,1,2,0,0,0,40,40,35,0,0,0,0,1,0,0,0,0,0,0,1],[6,1,0,0,0,0,40,0,0,0,0,0,0,0,6,1,0,0,0,0,40,0,0,0,0,0,0,0,7,8,0,0,0,0,40,40],[6,35,0,0,0,0,40,35,0,0,0,0,0,0,35,6,0,0,0,0,1,6,0,0,0,0,0,0,0,34,0,0,0,0,35,0],[32,0,0,0,0,0,0,32,0,0,0,0,0,0,34,25,16,0,0,0,16,7,16,5,0,0,18,0,23,5,0,0,8,33,25,18] >;
(D4×D5)⋊C4 in GAP, Magma, Sage, TeX
(D_4\times D_5)\rtimes C_4
% in TeX
G:=Group("(D4xD5):C4"); // GroupNames label
G:=SmallGroup(320,397);
// by ID
G=gap.SmallGroup(320,397);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,758,219,58,851,438,102,12550]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^2=c^5=d^2=e^4=1,b*a*b=e*a*e^-1=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,e*b*e^-1=a*b,d*c*d=c^-1,c*e=e*c,e*d*e^-1=a^2*d>;
// generators/relations