?
G = D10⋊6(C4⋊C4) order 320 = 26·5
5th semidirect product of D10 and C4⋊C4 acting via C4⋊C4/C2×C4=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D10⋊6(C4⋊C4), (C22×C4)⋊8F5, C22⋊1(C4⋊F5), C4⋊2(C22⋊F5), C20⋊2(C22⋊C4), (C22×C20)⋊12C4, C5⋊(C23.7Q8), D10.33(C2×D4), (C4×D5).113D4, D10.17(C2×Q8), C23.50(C2×F5), D5.3(C4⋊D4), D10.3Q8⋊5C2, D5.4(C22⋊Q8), D10.27(C4○D4), (C22×D5).22Q8, Dic5⋊6(C22⋊C4), (C22×Dic5)⋊19C4, (C22×D5).100D4, (C22×F5).7C22, C22.90(C22×F5), C10.19(C42⋊C2), (C22×D5).277C23, (C23×D5).135C22, C2.19(D10.C23), (C2×C4×D5)⋊20C4, (C2×C4⋊F5)⋊8C2, (C2×C10)⋊2(C4⋊C4), C2.23(C2×C4⋊F5), C10.23(C2×C4⋊C4), (C2×C4).145(C2×F5), (D5×C22×C4).29C2, (C2×C22⋊F5).7C2, (C2×C20).113(C2×C4), C2.13(C2×C22⋊F5), C10.12(C2×C22⋊C4), (C2×C4×D5).368C22, (C2×C10).72(C22×C4), (C22×C10).72(C2×C4), (C2×Dic5).191(C2×C4), (C22×D5).128(C2×C4), SmallGroup(320,1103)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1002 in 234 conjugacy classes, 70 normal (32 characteristic)
C1, C2 [×3], C2 [×8], C4 [×2], C4 [×8], C22, C22 [×2], C22 [×20], C5, C2×C4 [×2], C2×C4 [×28], C23, C23 [×10], D5 [×4], D5 [×2], C10 [×3], C10 [×2], C22⋊C4 [×4], C4⋊C4 [×4], C22×C4, C22×C4 [×13], C24, Dic5 [×2], Dic5, C20 [×2], C20, F5 [×4], D10 [×2], D10 [×6], D10 [×10], C2×C10, C2×C10 [×2], C2×C10 [×2], C2.C42 [×2], C2×C22⋊C4 [×2], C2×C4⋊C4 [×2], C23×C4, C4×D5 [×4], C4×D5 [×6], C2×Dic5 [×2], C2×Dic5 [×2], C2×C20 [×2], C2×C20 [×2], C2×F5 [×12], C22×D5 [×2], C22×D5 [×4], C22×D5 [×4], C22×C10, C23.7Q8, C4⋊F5 [×4], C22⋊F5 [×4], C2×C4×D5 [×4], C2×C4×D5 [×4], C22×Dic5, C22×C20, C22×F5 [×4], C23×D5, D10.3Q8 [×2], C2×C4⋊F5 [×2], C2×C22⋊F5 [×2], D5×C22×C4, D10⋊6(C4⋊C4)
Quotients:
C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×6], Q8 [×2], C23, C22⋊C4 [×4], C4⋊C4 [×4], C22×C4, C2×D4 [×3], C2×Q8, C4○D4 [×2], F5, C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C4⋊D4 [×2], C22⋊Q8 [×2], C2×F5 [×3], C23.7Q8, C4⋊F5 [×2], C22⋊F5 [×2], C22×F5, C2×C4⋊F5, D10.C23, C2×C22⋊F5, D10⋊6(C4⋊C4)
Generators and relations
G = < a,b,c,d | a10=b2=c4=d4=1, bab=cac-1=a-1, dad-1=a3, cbc-1=a8b, dbd-1=a7b, dcd-1=c-1 >
►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 32)(2 31)(3 40)(4 39)(5 38)(6 37)(7 36)(8 35)(9 34)(10 33)(11 69)(12 68)(13 67)(14 66)(15 65)(16 64)(17 63)(18 62)(19 61)(20 70)(21 43)(22 42)(23 41)(24 50)(25 49)(26 48)(27 47)(28 46)(29 45)(30 44)(51 79)(52 78)(53 77)(54 76)(55 75)(56 74)(57 73)(58 72)(59 71)(60 80)
(1 65 30 53)(2 64 21 52)(3 63 22 51)(4 62 23 60)(5 61 24 59)(6 70 25 58)(7 69 26 57)(8 68 27 56)(9 67 28 55)(10 66 29 54)(11 50 73 38)(12 49 74 37)(13 48 75 36)(14 47 76 35)(15 46 77 34)(16 45 78 33)(17 44 79 32)(18 43 80 31)(19 42 71 40)(20 41 72 39)
(1 38 6 33)(2 35 5 36)(3 32 4 39)(7 40 10 31)(8 37 9 34)(11 58 16 53)(12 55 15 56)(13 52 14 59)(17 60 20 51)(18 57 19 54)(21 47 24 48)(22 44 23 41)(25 45 30 50)(26 42 29 43)(27 49 28 46)(61 75 64 76)(62 72 63 79)(65 73 70 78)(66 80 69 71)(67 77 68 74)
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), (1,32)(2,31)(3,40)(4,39)(5,38)(6,37)(7,36)(8,35)(9,34)(10,33)(11,69)(12,68)(13,67)(14,66)(15,65)(16,64)(17,63)(18,62)(19,61)(20,70)(21,43)(22,42)(23,41)(24,50)(25,49)(26,48)(27,47)(28,46)(29,45)(30,44)(51,79)(52,78)(53,77)(54,76)(55,75)(56,74)(57,73)(58,72)(59,71)(60,80), (1,65,30,53)(2,64,21,52)(3,63,22,51)(4,62,23,60)(5,61,24,59)(6,70,25,58)(7,69,26,57)(8,68,27,56)(9,67,28,55)(10,66,29,54)(11,50,73,38)(12,49,74,37)(13,48,75,36)(14,47,76,35)(15,46,77,34)(16,45,78,33)(17,44,79,32)(18,43,80,31)(19,42,71,40)(20,41,72,39), (1,38,6,33)(2,35,5,36)(3,32,4,39)(7,40,10,31)(8,37,9,34)(11,58,16,53)(12,55,15,56)(13,52,14,59)(17,60,20,51)(18,57,19,54)(21,47,24,48)(22,44,23,41)(25,45,30,50)(26,42,29,43)(27,49,28,46)(61,75,64,76)(62,72,63,79)(65,73,70,78)(66,80,69,71)(67,77,68,74)>;
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), (1,32)(2,31)(3,40)(4,39)(5,38)(6,37)(7,36)(8,35)(9,34)(10,33)(11,69)(12,68)(13,67)(14,66)(15,65)(16,64)(17,63)(18,62)(19,61)(20,70)(21,43)(22,42)(23,41)(24,50)(25,49)(26,48)(27,47)(28,46)(29,45)(30,44)(51,79)(52,78)(53,77)(54,76)(55,75)(56,74)(57,73)(58,72)(59,71)(60,80), (1,65,30,53)(2,64,21,52)(3,63,22,51)(4,62,23,60)(5,61,24,59)(6,70,25,58)(7,69,26,57)(8,68,27,56)(9,67,28,55)(10,66,29,54)(11,50,73,38)(12,49,74,37)(13,48,75,36)(14,47,76,35)(15,46,77,34)(16,45,78,33)(17,44,79,32)(18,43,80,31)(19,42,71,40)(20,41,72,39), (1,38,6,33)(2,35,5,36)(3,32,4,39)(7,40,10,31)(8,37,9,34)(11,58,16,53)(12,55,15,56)(13,52,14,59)(17,60,20,51)(18,57,19,54)(21,47,24,48)(22,44,23,41)(25,45,30,50)(26,42,29,43)(27,49,28,46)(61,75,64,76)(62,72,63,79)(65,73,70,78)(66,80,69,71)(67,77,68,74) );
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)], [(1,32),(2,31),(3,40),(4,39),(5,38),(6,37),(7,36),(8,35),(9,34),(10,33),(11,69),(12,68),(13,67),(14,66),(15,65),(16,64),(17,63),(18,62),(19,61),(20,70),(21,43),(22,42),(23,41),(24,50),(25,49),(26,48),(27,47),(28,46),(29,45),(30,44),(51,79),(52,78),(53,77),(54,76),(55,75),(56,74),(57,73),(58,72),(59,71),(60,80)], [(1,65,30,53),(2,64,21,52),(3,63,22,51),(4,62,23,60),(5,61,24,59),(6,70,25,58),(7,69,26,57),(8,68,27,56),(9,67,28,55),(10,66,29,54),(11,50,73,38),(12,49,74,37),(13,48,75,36),(14,47,76,35),(15,46,77,34),(16,45,78,33),(17,44,79,32),(18,43,80,31),(19,42,71,40),(20,41,72,39)], [(1,38,6,33),(2,35,5,36),(3,32,4,39),(7,40,10,31),(8,37,9,34),(11,58,16,53),(12,55,15,56),(13,52,14,59),(17,60,20,51),(18,57,19,54),(21,47,24,48),(22,44,23,41),(25,45,30,50),(26,42,29,43),(27,49,28,46),(61,75,64,76),(62,72,63,79),(65,73,70,78),(66,80,69,71),(67,77,68,74)])
Matrix representation ►G ⊆ GL6(𝔽41)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 40 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 40 | 0 | 1 | 0 |
| 0 | 0 | 0 | 40 | 1 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 38 | 22 | 19 | 3 |
| 0 | 0 | 19 | 0 | 22 | 3 |
| 0 | 0 | 38 | 3 | 22 | 0 |
| 0 | 0 | 0 | 3 | 19 | 22 |
| 32 | 2 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 27 | 0 | 14 | 34 |
| 0 | 0 | 27 | 14 | 7 | 0 |
| 0 | 0 | 0 | 7 | 14 | 27 |
| 0 | 0 | 34 | 14 | 0 | 27 |
| 32 | 0 | 0 | 0 | 0 | 0 |
| 1 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 38 | 3 | 22 | 0 |
| 0 | 0 | 19 | 3 | 0 | 38 |
| 0 | 0 | 38 | 0 | 3 | 19 |
| 0 | 0 | 0 | 22 | 3 | 38 |
G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,1,1,1,1,0,0,40,0,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,38,19,38,0,0,0,22,0,3,3,0,0,19,22,22,19,0,0,3,3,0,22],[32,0,0,0,0,0,2,9,0,0,0,0,0,0,27,27,0,34,0,0,0,14,7,14,0,0,14,7,14,0,0,0,34,0,27,27],[32,1,0,0,0,0,0,9,0,0,0,0,0,0,38,19,38,0,0,0,3,3,0,22,0,0,22,0,3,3,0,0,0,38,19,38] >;
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4P | 5 | 10A | ··· | 10G | 20A | ··· | 20H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 5 | 5 | 5 | 5 | 10 | 10 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 4 | 4 | ··· | 4 | 4 | ··· | 4 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | + | + | + | + | ||||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | D4 | D4 | Q8 | C4○D4 | F5 | C2×F5 | C2×F5 | C22⋊F5 | C4⋊F5 | D10.C23 |
| kernel | D10⋊6(C4⋊C4) | D10.3Q8 | C2×C4⋊F5 | C2×C22⋊F5 | D5×C22×C4 | C2×C4×D5 | C22×Dic5 | C22×C20 | C4×D5 | C22×D5 | C22×D5 | D10 | C22×C4 | C2×C4 | C23 | C4 | C22 | C2 |
| # reps | 1 | 2 | 2 | 2 | 1 | 4 | 2 | 2 | 4 | 2 | 2 | 4 | 1 | 2 | 1 | 4 | 4 | 4 |
In GAP, Magma, Sage, TeX
D_{10}\rtimes_6(C_4\rtimes C_4) % in TeX
G:=Group("D10:6(C4:C4)"); // GroupNames label
G:=SmallGroup(320,1103);
// by ID
G=gap.SmallGroup(320,1103);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,477,232,422,6278,1595]);
// Polycyclic
G:=Group<a,b,c,d|a^10=b^2=c^4=d^4=1,b*a*b=c*a*c^-1=a^-1,d*a*d^-1=a^3,c*b*c^-1=a^8*b,d*b*d^-1=a^7*b,d*c*d^-1=c^-1>;
// generators/relations