direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D5×C4⋊C4, D10.5Q8, D10.22D4, C4⋊3(C4×D5), C20⋊4(C2×C4), (C4×D5)⋊1C4, C2.3(D4×D5), C2.2(Q8×D5), C4⋊Dic5⋊11C2, Dic5⋊4(C2×C4), (C2×C4).30D10, C10.23(C2×D4), C10.12(C2×Q8), D10.19(C2×C4), C10.D4⋊11C2, C10.22(C22×C4), (C2×C20).23C22, (C2×C10).32C23, C22.16(C22×D5), (C2×Dic5).32C22, (C22×D5).43C22, C5⋊2(C2×C4⋊C4), (C5×C4⋊C4)⋊2C2, (C2×C4×D5).1C2, C2.11(C2×C4×D5), SmallGroup(160,112)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D5×C4⋊C4
G = < a,b,c,d | a5=b2=c4=d4=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >
Subgroups: 264 in 92 conjugacy classes, 49 normal (19 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C2×C4, C2×C4, C2×C4, C23, D5, C10, C4⋊C4, C4⋊C4, C22×C4, Dic5, Dic5, C20, C20, D10, C2×C10, C2×C4⋊C4, C4×D5, C4×D5, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×D5, C10.D4, C4⋊Dic5, C5×C4⋊C4, C2×C4×D5, C2×C4×D5, D5×C4⋊C4
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, D5, C4⋊C4, C22×C4, C2×D4, C2×Q8, D10, C2×C4⋊C4, C4×D5, C22×D5, C2×C4×D5, D4×D5, Q8×D5, D5×C4⋊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 8)(2 7)(3 6)(4 10)(5 9)(11 16)(12 20)(13 19)(14 18)(15 17)(21 26)(22 30)(23 29)(24 28)(25 27)(31 36)(32 40)(33 39)(34 38)(35 37)(41 46)(42 50)(43 49)(44 48)(45 47)(51 56)(52 60)(53 59)(54 58)(55 57)(61 66)(62 70)(63 69)(64 68)(65 67)(71 76)(72 80)(73 79)(74 78)(75 77)
(1 59 9 54)(2 60 10 55)(3 56 6 51)(4 57 7 52)(5 58 8 53)(11 46 16 41)(12 47 17 42)(13 48 18 43)(14 49 19 44)(15 50 20 45)(21 76 26 71)(22 77 27 72)(23 78 28 73)(24 79 29 74)(25 80 30 75)(31 66 36 61)(32 67 37 62)(33 68 38 63)(34 69 39 64)(35 70 40 65)
(1 34 14 24)(2 35 15 25)(3 31 11 21)(4 32 12 22)(5 33 13 23)(6 36 16 26)(7 37 17 27)(8 38 18 28)(9 39 19 29)(10 40 20 30)(41 76 51 66)(42 77 52 67)(43 78 53 68)(44 79 54 69)(45 80 55 70)(46 71 56 61)(47 72 57 62)(48 73 58 63)(49 74 59 64)(50 75 60 65)
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), (1,8)(2,7)(3,6)(4,10)(5,9)(11,16)(12,20)(13,19)(14,18)(15,17)(21,26)(22,30)(23,29)(24,28)(25,27)(31,36)(32,40)(33,39)(34,38)(35,37)(41,46)(42,50)(43,49)(44,48)(45,47)(51,56)(52,60)(53,59)(54,58)(55,57)(61,66)(62,70)(63,69)(64,68)(65,67)(71,76)(72,80)(73,79)(74,78)(75,77), (1,59,9,54)(2,60,10,55)(3,56,6,51)(4,57,7,52)(5,58,8,53)(11,46,16,41)(12,47,17,42)(13,48,18,43)(14,49,19,44)(15,50,20,45)(21,76,26,71)(22,77,27,72)(23,78,28,73)(24,79,29,74)(25,80,30,75)(31,66,36,61)(32,67,37,62)(33,68,38,63)(34,69,39,64)(35,70,40,65), (1,34,14,24)(2,35,15,25)(3,31,11,21)(4,32,12,22)(5,33,13,23)(6,36,16,26)(7,37,17,27)(8,38,18,28)(9,39,19,29)(10,40,20,30)(41,76,51,66)(42,77,52,67)(43,78,53,68)(44,79,54,69)(45,80,55,70)(46,71,56,61)(47,72,57,62)(48,73,58,63)(49,74,59,64)(50,75,60,65)>;
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), (1,8)(2,7)(3,6)(4,10)(5,9)(11,16)(12,20)(13,19)(14,18)(15,17)(21,26)(22,30)(23,29)(24,28)(25,27)(31,36)(32,40)(33,39)(34,38)(35,37)(41,46)(42,50)(43,49)(44,48)(45,47)(51,56)(52,60)(53,59)(54,58)(55,57)(61,66)(62,70)(63,69)(64,68)(65,67)(71,76)(72,80)(73,79)(74,78)(75,77), (1,59,9,54)(2,60,10,55)(3,56,6,51)(4,57,7,52)(5,58,8,53)(11,46,16,41)(12,47,17,42)(13,48,18,43)(14,49,19,44)(15,50,20,45)(21,76,26,71)(22,77,27,72)(23,78,28,73)(24,79,29,74)(25,80,30,75)(31,66,36,61)(32,67,37,62)(33,68,38,63)(34,69,39,64)(35,70,40,65), (1,34,14,24)(2,35,15,25)(3,31,11,21)(4,32,12,22)(5,33,13,23)(6,36,16,26)(7,37,17,27)(8,38,18,28)(9,39,19,29)(10,40,20,30)(41,76,51,66)(42,77,52,67)(43,78,53,68)(44,79,54,69)(45,80,55,70)(46,71,56,61)(47,72,57,62)(48,73,58,63)(49,74,59,64)(50,75,60,65) );
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)], [(1,8),(2,7),(3,6),(4,10),(5,9),(11,16),(12,20),(13,19),(14,18),(15,17),(21,26),(22,30),(23,29),(24,28),(25,27),(31,36),(32,40),(33,39),(34,38),(35,37),(41,46),(42,50),(43,49),(44,48),(45,47),(51,56),(52,60),(53,59),(54,58),(55,57),(61,66),(62,70),(63,69),(64,68),(65,67),(71,76),(72,80),(73,79),(74,78),(75,77)], [(1,59,9,54),(2,60,10,55),(3,56,6,51),(4,57,7,52),(5,58,8,53),(11,46,16,41),(12,47,17,42),(13,48,18,43),(14,49,19,44),(15,50,20,45),(21,76,26,71),(22,77,27,72),(23,78,28,73),(24,79,29,74),(25,80,30,75),(31,66,36,61),(32,67,37,62),(33,68,38,63),(34,69,39,64),(35,70,40,65)], [(1,34,14,24),(2,35,15,25),(3,31,11,21),(4,32,12,22),(5,33,13,23),(6,36,16,26),(7,37,17,27),(8,38,18,28),(9,39,19,29),(10,40,20,30),(41,76,51,66),(42,77,52,67),(43,78,53,68),(44,79,54,69),(45,80,55,70),(46,71,56,61),(47,72,57,62),(48,73,58,63),(49,74,59,64),(50,75,60,65)]])
D5×C4⋊C4 is a maximal subgroup of
D10.18D8 D4⋊(C4×D5) D10.12D8 D10.16SD16 Q8⋊(C4×D5) D10.11SD16 D10.7Q16 C8⋊(C4×D5) D10.12SD16 D10.17SD16 C40⋊20(C2×C4) D10.13D8 D10.8Q16 C4⋊C4⋊5F5 C20⋊(C4⋊C4) C10.82+ 1+4 C10.2- 1+4 C10.102+ 1+4 C42.91D10 C42.94D10 C42.95D10 C4×D4×D5 C42.108D10 D20⋊24D4 C42.113D10 C4×Q8×D5 C42.126D10 D20⋊10Q8 C42.132D10 C10.392+ 1+4 C10.732- 1+4 C10.432+ 1+4 C10.172- 1+4 D20⋊22D4 C10.1182+ 1+4 C10.522+ 1+4 C10.202- 1+4 C10.212- 1+4 C10.822- 1+4 C10.632+ 1+4 C10.642+ 1+4 C42.148D10 D20⋊7Q8 C42.150D10 C42.151D10 C42.152D10 C42.153D10 C42.161D10 C42.162D10 C42.163D10 D20⋊12D4 C42.174D10 D20⋊9Q8 D30.Q8 D30.2Q8
D5×C4⋊C4 is a maximal quotient of
C10.49(C4×D4) Dic5⋊2C42 C5⋊2(C42⋊8C4) D10⋊2(C4⋊C4) D10⋊3(C4⋊C4) C20⋊5M4(2) C42.30D10 (C8×D5)⋊C4 C8⋊(C4×D5) C8.27(C4×D5) C40⋊20(C2×C4) M4(2).25D10 C10.96(C4×D4) C20⋊5(C4⋊C4) D10⋊4(C4⋊C4) D10⋊5(C4⋊C4) D30.Q8 D30.2Q8
40 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | ··· | 4F | 4G | ··· | 4L | 5A | 5B | 10A | ··· | 10F | 20A | ··· | 20L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 5 | 5 | 5 | 5 | 2 | ··· | 2 | 10 | ··· | 10 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
40 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | - | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C4 | D4 | Q8 | D5 | D10 | C4×D5 | D4×D5 | Q8×D5 |
| kernel | D5×C4⋊C4 | C10.D4 | C4⋊Dic5 | C5×C4⋊C4 | C2×C4×D5 | C4×D5 | D10 | D10 | C4⋊C4 | C2×C4 | C4 | C2 | C2 |
| # reps | 1 | 2 | 1 | 1 | 3 | 8 | 2 | 2 | 2 | 6 | 8 | 2 | 2 |
Matrix representation of D5×C4⋊C4 ►in GL4(𝔽41) generated by
| 0 | 1 | 0 | 0 |
| 40 | 6 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 7 | 7 |
| 0 | 0 | 28 | 34 |
| 32 | 0 | 0 | 0 |
| 0 | 32 | 0 | 0 |
| 0 | 0 | 40 | 21 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(41))| [0,40,0,0,1,6,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,40,0,0,0,0,40],[1,0,0,0,0,1,0,0,0,0,7,28,0,0,7,34],[32,0,0,0,0,32,0,0,0,0,40,0,0,0,21,1] >;
D5×C4⋊C4 in GAP, Magma, Sage, TeX
D_5\times C_4\rtimes C_4
% in TeX
G:=Group("D5xC4:C4"); // GroupNames label
G:=SmallGroup(160,112);
// by ID
G=gap.SmallGroup(160,112);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-5,103,188,50,4613]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^2=c^4=d^4=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations