direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: D4×C20, C42⋊4C10, C4⋊C4⋊7C10, C4⋊1(C2×C20), C20⋊9(C2×C4), (C4×C20)⋊11C2, C2.3(D4×C10), C22⋊C4⋊6C10, (C22×C20)⋊4C2, C22⋊1(C2×C20), (C22×C4)⋊2C10, (C2×D4).7C10, C10.66(C2×D4), (D4×C10).14C2, C2.4(C22×C20), C23.7(C2×C10), C10.39(C4○D4), (C2×C10).73C23, C10.45(C22×C4), (C2×C20).121C22, C22.7(C22×C10), (C22×C10).26C22, (C5×C4⋊C4)⋊16C2, (C2×C10)⋊7(C2×C4), (C2×C20)○(D4×C10), C2.2(C5×C4○D4), (C5×C22⋊C4)⋊14C2, (C2×C4).15(C2×C10), (C2×C20)○(C5×C4⋊C4), (C2×C20)○(C5×C22⋊C4), SmallGroup(160,179)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4×C20
G = < a,b,c | a20=b4=c2=1, ab=ba, ac=ca, cbc=b-1 >
Subgroups: 124 in 94 conjugacy classes, 64 normal (24 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C5, C2×C4, C2×C4, C2×C4, D4, C23, C10, C10, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C20, C20, C2×C10, C2×C10, C2×C10, C4×D4, C2×C20, C2×C20, C2×C20, C5×D4, C22×C10, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C22×C20, D4×C10, D4×C20
Quotients: C1, C2, C4, C22, C5, C2×C4, D4, C23, C10, C22×C4, C2×D4, C4○D4, C20, C2×C10, C4×D4, C2×C20, C5×D4, C22×C10, C22×C20, D4×C10, C5×C4○D4, D4×C20
►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 65 24 52)(2 66 25 53)(3 67 26 54)(4 68 27 55)(5 69 28 56)(6 70 29 57)(7 71 30 58)(8 72 31 59)(9 73 32 60)(10 74 33 41)(11 75 34 42)(12 76 35 43)(13 77 36 44)(14 78 37 45)(15 79 38 46)(16 80 39 47)(17 61 40 48)(18 62 21 49)(19 63 22 50)(20 64 23 51)
(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 17)(8 18)(9 19)(10 20)(21 31)(22 32)(23 33)(24 34)(25 35)(26 36)(27 37)(28 38)(29 39)(30 40)(41 64)(42 65)(43 66)(44 67)(45 68)(46 69)(47 70)(48 71)(49 72)(50 73)(51 74)(52 75)(53 76)(54 77)(55 78)(56 79)(57 80)(58 61)(59 62)(60 63)
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), (1,65,24,52)(2,66,25,53)(3,67,26,54)(4,68,27,55)(5,69,28,56)(6,70,29,57)(7,71,30,58)(8,72,31,59)(9,73,32,60)(10,74,33,41)(11,75,34,42)(12,76,35,43)(13,77,36,44)(14,78,37,45)(15,79,38,46)(16,80,39,47)(17,61,40,48)(18,62,21,49)(19,63,22,50)(20,64,23,51), (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,19)(10,20)(21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40)(41,64)(42,65)(43,66)(44,67)(45,68)(46,69)(47,70)(48,71)(49,72)(50,73)(51,74)(52,75)(53,76)(54,77)(55,78)(56,79)(57,80)(58,61)(59,62)(60,63)>;
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), (1,65,24,52)(2,66,25,53)(3,67,26,54)(4,68,27,55)(5,69,28,56)(6,70,29,57)(7,71,30,58)(8,72,31,59)(9,73,32,60)(10,74,33,41)(11,75,34,42)(12,76,35,43)(13,77,36,44)(14,78,37,45)(15,79,38,46)(16,80,39,47)(17,61,40,48)(18,62,21,49)(19,63,22,50)(20,64,23,51), (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,19)(10,20)(21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40)(41,64)(42,65)(43,66)(44,67)(45,68)(46,69)(47,70)(48,71)(49,72)(50,73)(51,74)(52,75)(53,76)(54,77)(55,78)(56,79)(57,80)(58,61)(59,62)(60,63) );
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)], [(1,65,24,52),(2,66,25,53),(3,67,26,54),(4,68,27,55),(5,69,28,56),(6,70,29,57),(7,71,30,58),(8,72,31,59),(9,73,32,60),(10,74,33,41),(11,75,34,42),(12,76,35,43),(13,77,36,44),(14,78,37,45),(15,79,38,46),(16,80,39,47),(17,61,40,48),(18,62,21,49),(19,63,22,50),(20,64,23,51)], [(1,11),(2,12),(3,13),(4,14),(5,15),(6,16),(7,17),(8,18),(9,19),(10,20),(21,31),(22,32),(23,33),(24,34),(25,35),(26,36),(27,37),(28,38),(29,39),(30,40),(41,64),(42,65),(43,66),(44,67),(45,68),(46,69),(47,70),(48,71),(49,72),(50,73),(51,74),(52,75),(53,76),(54,77),(55,78),(56,79),(57,80),(58,61),(59,62),(60,63)]])
D4×C20 is a maximal subgroup of
C20.57D8 C20.50D8 C20.38SD16 D4.3Dic10 C42.47D10 C20⋊7M4(2) C42.48D10 C20⋊7D8 D4.1D20 C42.51D10 D4.2D20 C42.102D10 D4⋊5Dic10 C42.104D10 C42.105D10 C42.106D10 D4⋊6Dic10 C42⋊11D10 C42.108D10 C42⋊12D10 C42.228D10 D20⋊23D4 D20⋊24D4 Dic10⋊23D4 Dic10⋊24D4 D4⋊5D20 D4⋊6D20 C42⋊16D10 C42.229D10 C42.113D10 C42.114D10 C42⋊17D10 C42.115D10 C42.116D10 C42.117D10 C42.118D10 C42.119D10
100 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 5A | 5B | 5C | 5D | 10A | ··· | 10L | 10M | ··· | 10AB | 20A | ··· | 20P | 20Q | ··· | 20AV |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 5 | 5 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 |
100 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | |||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C5 | C10 | C10 | C10 | C10 | C10 | C20 | D4 | C4○D4 | C5×D4 | C5×C4○D4 |
| kernel | D4×C20 | C4×C20 | C5×C22⋊C4 | C5×C4⋊C4 | C22×C20 | D4×C10 | C5×D4 | C4×D4 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | D4 | C20 | C10 | C4 | C2 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 8 | 4 | 4 | 8 | 4 | 8 | 4 | 32 | 2 | 2 | 8 | 8 |
Matrix representation of D4×C20 ►in GL3(𝔽41) generated by
| 39 | 0 | 0 |
| 0 | 40 | 0 |
| 0 | 0 | 40 |
| 40 | 0 | 0 |
| 0 | 0 | 40 |
| 0 | 1 | 0 |
| 1 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 40 |
G:=sub<GL(3,GF(41))| [39,0,0,0,40,0,0,0,40],[40,0,0,0,0,1,0,40,0],[1,0,0,0,1,0,0,0,40] >;
D4×C20 in GAP, Magma, Sage, TeX
D_4\times C_{20} % in TeX
G:=Group("D4xC20"); // GroupNames label
G:=SmallGroup(160,179);
// by ID
G=gap.SmallGroup(160,179);
# by ID
G:=PCGroup([6,-2,-2,-2,-5,-2,-2,480,505,374]);
// Polycyclic
G:=Group<a,b,c|a^20=b^4=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations