direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D5×C4⋊D4, C4⋊5(D4×D5), C20⋊5(C2×D4), C4⋊C4⋊20D10, D10⋊3(C2×D4), (C4×D5)⋊11D4, C22⋊1(D4×D5), (C2×D4)⋊20D10, Dic5⋊7(C2×D4), C20⋊2D4⋊16C2, C4⋊D20⋊20C2, C20⋊7D4⋊32C2, C22⋊C4⋊25D10, (C22×D5)⋊11D4, (C22×C4)⋊38D10, D10⋊D4⋊18C2, (C2×D20)⋊22C22, (D4×C10)⋊10C22, C4⋊Dic5⋊29C22, C10.63(C22×D4), Dic5⋊D4⋊11C2, D10.62(C4○D4), (C2×C10).148C24, (C2×C20).593C23, (C22×C20)⋊19C22, C23.D5⋊20C22, D10⋊C4⋊25C22, C23.14(C22×D5), C10.D4⋊14C22, (C2×Dic5).69C23, (C23×D5).45C22, C22.169(C23×D5), (C22×C10).186C23, (C22×Dic5)⋊44C22, (C22×D5).193C23, (C2×D4×D5)⋊8C2, C5⋊3(C2×C4⋊D4), C2.36(C2×D4×D5), (D5×C4⋊C4)⋊20C2, (C2×C10)⋊2(C2×D4), (D5×C22×C4)⋊3C2, (C5×C4⋊C4)⋊7C22, (D5×C22⋊C4)⋊4C2, C2.37(D5×C4○D4), (C2×C4×D5)⋊56C22, (C5×C4⋊D4)⋊10C2, C10.150(C2×C4○D4), (C2×C5⋊D4)⋊12C22, (C5×C22⋊C4)⋊9C22, (C2×C4).37(C22×D5), SmallGroup(320,1276)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D5×C4⋊D4
G = < a,b,c,d,e | a5=b2=c4=d4=e2=1, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=ece=c-1, ede=d-1 >
Subgroups: 1822 in 426 conjugacy classes, 121 normal (43 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C5, C2×C4, C2×C4, C2×C4, D4, C23, C23, C23, D5, D5, C10, C10, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×D4, C24, Dic5, Dic5, C20, C20, D10, D10, C2×C10, C2×C10, C2×C10, C2×C22⋊C4, C2×C4⋊C4, C4⋊D4, C4⋊D4, C23×C4, C22×D4, C4×D5, C4×D5, D20, C2×Dic5, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C2×C20, C5×D4, C22×D5, C22×D5, C22×D5, C22×C10, C22×C10, C2×C4⋊D4, C10.D4, C4⋊Dic5, D10⋊C4, C23.D5, C5×C22⋊C4, C5×C4⋊C4, C2×C4×D5, C2×C4×D5, C2×C4×D5, C2×D20, C2×D20, D4×D5, C22×Dic5, C2×C5⋊D4, C22×C20, D4×C10, D4×C10, C23×D5, C23×D5, D5×C22⋊C4, D10⋊D4, D5×C4⋊C4, C4⋊D20, C20⋊7D4, C20⋊2D4, Dic5⋊D4, C5×C4⋊D4, D5×C22×C4, C2×D4×D5, C2×D4×D5, D5×C4⋊D4
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, C24, D10, C4⋊D4, C22×D4, C2×C4○D4, C22×D5, C2×C4⋊D4, D4×D5, C23×D5, C2×D4×D5, D5×C4○D4, D5×C4⋊D4
►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 5)(2 4)(7 10)(8 9)(12 15)(13 14)(17 20)(18 19)(22 25)(23 24)(27 30)(28 29)(32 35)(33 34)(37 40)(38 39)(42 45)(43 44)(47 50)(48 49)(52 55)(53 54)(57 60)(58 59)(62 65)(63 64)(67 70)(68 69)(72 75)(73 74)(77 80)(78 79)
(1 24 14 34)(2 25 15 35)(3 21 11 31)(4 22 12 32)(5 23 13 33)(6 26 16 36)(7 27 17 37)(8 28 18 38)(9 29 19 39)(10 30 20 40)(41 61 51 71)(42 62 52 72)(43 63 53 73)(44 64 54 74)(45 65 55 75)(46 66 56 76)(47 67 57 77)(48 68 58 78)(49 69 59 79)(50 70 60 80)
(1 64 19 79)(2 65 20 80)(3 61 16 76)(4 62 17 77)(5 63 18 78)(6 66 11 71)(7 67 12 72)(8 68 13 73)(9 69 14 74)(10 70 15 75)(21 41 36 56)(22 42 37 57)(23 43 38 58)(24 44 39 59)(25 45 40 60)(26 46 31 51)(27 47 32 52)(28 48 33 53)(29 49 34 54)(30 50 35 55)
(1 14)(2 15)(3 11)(4 12)(5 13)(6 16)(7 17)(8 18)(9 19)(10 20)(41 56)(42 57)(43 58)(44 59)(45 60)(46 51)(47 52)(48 53)(49 54)(50 55)(61 66)(62 67)(63 68)(64 69)(65 70)(71 76)(72 77)(73 78)(74 79)(75 80)
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), (1,5)(2,4)(7,10)(8,9)(12,15)(13,14)(17,20)(18,19)(22,25)(23,24)(27,30)(28,29)(32,35)(33,34)(37,40)(38,39)(42,45)(43,44)(47,50)(48,49)(52,55)(53,54)(57,60)(58,59)(62,65)(63,64)(67,70)(68,69)(72,75)(73,74)(77,80)(78,79), (1,24,14,34)(2,25,15,35)(3,21,11,31)(4,22,12,32)(5,23,13,33)(6,26,16,36)(7,27,17,37)(8,28,18,38)(9,29,19,39)(10,30,20,40)(41,61,51,71)(42,62,52,72)(43,63,53,73)(44,64,54,74)(45,65,55,75)(46,66,56,76)(47,67,57,77)(48,68,58,78)(49,69,59,79)(50,70,60,80), (1,64,19,79)(2,65,20,80)(3,61,16,76)(4,62,17,77)(5,63,18,78)(6,66,11,71)(7,67,12,72)(8,68,13,73)(9,69,14,74)(10,70,15,75)(21,41,36,56)(22,42,37,57)(23,43,38,58)(24,44,39,59)(25,45,40,60)(26,46,31,51)(27,47,32,52)(28,48,33,53)(29,49,34,54)(30,50,35,55), (1,14)(2,15)(3,11)(4,12)(5,13)(6,16)(7,17)(8,18)(9,19)(10,20)(41,56)(42,57)(43,58)(44,59)(45,60)(46,51)(47,52)(48,53)(49,54)(50,55)(61,66)(62,67)(63,68)(64,69)(65,70)(71,76)(72,77)(73,78)(74,79)(75,80)>;
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), (1,5)(2,4)(7,10)(8,9)(12,15)(13,14)(17,20)(18,19)(22,25)(23,24)(27,30)(28,29)(32,35)(33,34)(37,40)(38,39)(42,45)(43,44)(47,50)(48,49)(52,55)(53,54)(57,60)(58,59)(62,65)(63,64)(67,70)(68,69)(72,75)(73,74)(77,80)(78,79), (1,24,14,34)(2,25,15,35)(3,21,11,31)(4,22,12,32)(5,23,13,33)(6,26,16,36)(7,27,17,37)(8,28,18,38)(9,29,19,39)(10,30,20,40)(41,61,51,71)(42,62,52,72)(43,63,53,73)(44,64,54,74)(45,65,55,75)(46,66,56,76)(47,67,57,77)(48,68,58,78)(49,69,59,79)(50,70,60,80), (1,64,19,79)(2,65,20,80)(3,61,16,76)(4,62,17,77)(5,63,18,78)(6,66,11,71)(7,67,12,72)(8,68,13,73)(9,69,14,74)(10,70,15,75)(21,41,36,56)(22,42,37,57)(23,43,38,58)(24,44,39,59)(25,45,40,60)(26,46,31,51)(27,47,32,52)(28,48,33,53)(29,49,34,54)(30,50,35,55), (1,14)(2,15)(3,11)(4,12)(5,13)(6,16)(7,17)(8,18)(9,19)(10,20)(41,56)(42,57)(43,58)(44,59)(45,60)(46,51)(47,52)(48,53)(49,54)(50,55)(61,66)(62,67)(63,68)(64,69)(65,70)(71,76)(72,77)(73,78)(74,79)(75,80) );
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)], [(1,5),(2,4),(7,10),(8,9),(12,15),(13,14),(17,20),(18,19),(22,25),(23,24),(27,30),(28,29),(32,35),(33,34),(37,40),(38,39),(42,45),(43,44),(47,50),(48,49),(52,55),(53,54),(57,60),(58,59),(62,65),(63,64),(67,70),(68,69),(72,75),(73,74),(77,80),(78,79)], [(1,24,14,34),(2,25,15,35),(3,21,11,31),(4,22,12,32),(5,23,13,33),(6,26,16,36),(7,27,17,37),(8,28,18,38),(9,29,19,39),(10,30,20,40),(41,61,51,71),(42,62,52,72),(43,63,53,73),(44,64,54,74),(45,65,55,75),(46,66,56,76),(47,67,57,77),(48,68,58,78),(49,69,59,79),(50,70,60,80)], [(1,64,19,79),(2,65,20,80),(3,61,16,76),(4,62,17,77),(5,63,18,78),(6,66,11,71),(7,67,12,72),(8,68,13,73),(9,69,14,74),(10,70,15,75),(21,41,36,56),(22,42,37,57),(23,43,38,58),(24,44,39,59),(25,45,40,60),(26,46,31,51),(27,47,32,52),(28,48,33,53),(29,49,34,54),(30,50,35,55)], [(1,14),(2,15),(3,11),(4,12),(5,13),(6,16),(7,17),(8,18),(9,19),(10,20),(41,56),(42,57),(43,58),(44,59),(45,60),(46,51),(47,52),(48,53),(49,54),(50,55),(61,66),(62,67),(63,68),(64,69),(65,70),(71,76),(72,77),(73,78),(74,79),(75,80)]])
56 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 2L | 2M | 2N | 2O | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 5A | 5B | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 10K | 10L | 10M | 10N | 20A | ··· | 20H | 20I | 20J | 20K | 20L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | 20 | 20 | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 5 | 5 | 5 | 5 | 10 | 10 | 20 | 20 | 2 | 2 | 2 | 2 | 4 | 4 | 10 | 10 | 10 | 10 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D5 | C4○D4 | D10 | D10 | D10 | D10 | D4×D5 | D4×D5 | D5×C4○D4 |
| kernel | D5×C4⋊D4 | D5×C22⋊C4 | D10⋊D4 | D5×C4⋊C4 | C4⋊D20 | C20⋊7D4 | C20⋊2D4 | Dic5⋊D4 | C5×C4⋊D4 | D5×C22×C4 | C2×D4×D5 | C4×D5 | C22×D5 | C4⋊D4 | D10 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C4 | C22 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 3 | 4 | 4 | 2 | 4 | 4 | 2 | 2 | 6 | 4 | 4 | 4 |
Matrix representation of D5×C4⋊D4 ►in GL6(𝔽41)
| 0 | 1 | 0 | 0 | 0 | 0 |
| 40 | 6 | 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 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 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 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 4 | 0 | 0 |
| 0 | 0 | 20 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 4 |
| 0 | 0 | 0 | 0 | 20 | 40 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 20 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 32 | 5 |
| 0 | 0 | 0 | 0 | 0 | 9 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 21 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 21 | 1 |
G:=sub<GL(6,GF(41))| [0,40,0,0,0,0,1,6,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,20,0,0,0,0,4,40,0,0,0,0,0,0,1,20,0,0,0,0,4,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,20,0,0,0,0,0,40,0,0,0,0,0,0,32,0,0,0,0,0,5,9],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,21,0,0,0,0,0,1,0,0,0,0,0,0,40,21,0,0,0,0,0,1] >;
D5×C4⋊D4 in GAP, Magma, Sage, TeX
D_5\times C_4\rtimes D_4
% in TeX
G:=Group("D5xC4:D4"); // GroupNames label
G:=SmallGroup(320,1276);
// by ID
G=gap.SmallGroup(320,1276);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,100,794,297,12550]);
// Polycyclic
G:=Group<a,b,c,d,e|a^5=b^2=c^4=d^4=e^2=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^-1=e*c*e=c^-1,e*d*e=d^-1>;
// generators/relations