metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: (D4×C10)⋊21C4, (Q8×C10)⋊19C4, C4○D4⋊4Dic5, (C2×Q8)⋊7Dic5, (C2×D4)⋊9Dic5, C4○D4.37D10, (C2×C20).198D4, C20.453(C2×D4), Q8.7(C2×Dic5), D4.7(C2×Dic5), (C4×Dic5)⋊7C22, D4⋊2Dic5⋊10C2, C20.85(C22⋊C4), C20.145(C22×C4), (C2×C20).482C23, C5⋊7(C42⋊C22), (C22×C10).114D4, (C22×C4).162D10, C23.31(C5⋊D4), C4.Dic5⋊24C22, C4.23(C23.D5), C4.16(C22×Dic5), C22.6(C23.D5), C23.21D10⋊20C2, (C22×C20).208C22, (C5×C4○D4)⋊10C4, (C2×C4○D4).5D5, (C10×C4○D4).5C2, (C5×D4).38(C2×C4), (C2×C10).40(C2×D4), C4.144(C2×C5⋊D4), (C5×Q8).40(C2×C4), (C2×C20).299(C2×C4), (C2×C4).90(C5⋊D4), (C2×C4.Dic5)⋊22C2, (C2×C4).29(C2×Dic5), C22.12(C2×C5⋊D4), C2.21(C2×C23.D5), C10.126(C2×C22⋊C4), (C5×C4○D4).42C22, (C2×C4).567(C22×D5), (C2×C10).91(C22⋊C4), SmallGroup(320,863)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for (D4×C10)⋊21C4
G = < a,b,c,d | a10=b4=c2=d4=1, ab=ba, ac=ca, dad-1=a-1b2, cbc=b-1, bd=db, dcd-1=bc >
Subgroups: 398 in 154 conjugacy classes, 67 normal (43 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, C10, C10, C42, C22⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C4○D4, Dic5, C20, C20, C2×C10, C2×C10, C4≀C2, C42⋊C2, C2×M4(2), C2×C4○D4, C5⋊2C8, C2×Dic5, C2×C20, C2×C20, C5×D4, C5×D4, C5×Q8, C5×Q8, C22×C10, C22×C10, C42⋊C22, C2×C5⋊2C8, C4.Dic5, C4.Dic5, C4×Dic5, C4⋊Dic5, C23.D5, C22×C20, C22×C20, D4×C10, D4×C10, Q8×C10, C5×C4○D4, C5×C4○D4, D4⋊2Dic5, C2×C4.Dic5, C23.21D10, C10×C4○D4, (D4×C10)⋊21C4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D5, C22⋊C4, C22×C4, C2×D4, Dic5, D10, C2×C22⋊C4, C2×Dic5, C5⋊D4, C22×D5, C42⋊C22, C23.D5, C22×Dic5, C2×C5⋊D4, C2×C23.D5, (D4×C10)⋊21C4
►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 40 67 12)(2 31 68 13)(3 32 69 14)(4 33 70 15)(5 34 61 16)(6 35 62 17)(7 36 63 18)(8 37 64 19)(9 38 65 20)(10 39 66 11)(21 73 47 58)(22 74 48 59)(23 75 49 60)(24 76 50 51)(25 77 41 52)(26 78 42 53)(27 79 43 54)(28 80 44 55)(29 71 45 56)(30 72 46 57)
(1 46)(2 47)(3 48)(4 49)(5 50)(6 41)(7 42)(8 43)(9 44)(10 45)(11 56)(12 57)(13 58)(14 59)(15 60)(16 51)(17 52)(18 53)(19 54)(20 55)(21 68)(22 69)(23 70)(24 61)(25 62)(26 63)(27 64)(28 65)(29 66)(30 67)(31 73)(32 74)(33 75)(34 76)(35 77)(36 78)(37 79)(38 80)(39 71)(40 72)
(2 66)(3 9)(4 64)(5 7)(6 62)(8 70)(10 68)(11 31)(13 39)(14 20)(15 37)(16 18)(17 35)(19 33)(21 56 47 71)(22 80 48 55)(23 54 49 79)(24 78 50 53)(25 52 41 77)(26 76 42 51)(27 60 43 75)(28 74 44 59)(29 58 45 73)(30 72 46 57)(32 38)(34 36)(61 63)(65 69)
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), (1,40,67,12)(2,31,68,13)(3,32,69,14)(4,33,70,15)(5,34,61,16)(6,35,62,17)(7,36,63,18)(8,37,64,19)(9,38,65,20)(10,39,66,11)(21,73,47,58)(22,74,48,59)(23,75,49,60)(24,76,50,51)(25,77,41,52)(26,78,42,53)(27,79,43,54)(28,80,44,55)(29,71,45,56)(30,72,46,57), (1,46)(2,47)(3,48)(4,49)(5,50)(6,41)(7,42)(8,43)(9,44)(10,45)(11,56)(12,57)(13,58)(14,59)(15,60)(16,51)(17,52)(18,53)(19,54)(20,55)(21,68)(22,69)(23,70)(24,61)(25,62)(26,63)(27,64)(28,65)(29,66)(30,67)(31,73)(32,74)(33,75)(34,76)(35,77)(36,78)(37,79)(38,80)(39,71)(40,72), (2,66)(3,9)(4,64)(5,7)(6,62)(8,70)(10,68)(11,31)(13,39)(14,20)(15,37)(16,18)(17,35)(19,33)(21,56,47,71)(22,80,48,55)(23,54,49,79)(24,78,50,53)(25,52,41,77)(26,76,42,51)(27,60,43,75)(28,74,44,59)(29,58,45,73)(30,72,46,57)(32,38)(34,36)(61,63)(65,69)>;
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), (1,40,67,12)(2,31,68,13)(3,32,69,14)(4,33,70,15)(5,34,61,16)(6,35,62,17)(7,36,63,18)(8,37,64,19)(9,38,65,20)(10,39,66,11)(21,73,47,58)(22,74,48,59)(23,75,49,60)(24,76,50,51)(25,77,41,52)(26,78,42,53)(27,79,43,54)(28,80,44,55)(29,71,45,56)(30,72,46,57), (1,46)(2,47)(3,48)(4,49)(5,50)(6,41)(7,42)(8,43)(9,44)(10,45)(11,56)(12,57)(13,58)(14,59)(15,60)(16,51)(17,52)(18,53)(19,54)(20,55)(21,68)(22,69)(23,70)(24,61)(25,62)(26,63)(27,64)(28,65)(29,66)(30,67)(31,73)(32,74)(33,75)(34,76)(35,77)(36,78)(37,79)(38,80)(39,71)(40,72), (2,66)(3,9)(4,64)(5,7)(6,62)(8,70)(10,68)(11,31)(13,39)(14,20)(15,37)(16,18)(17,35)(19,33)(21,56,47,71)(22,80,48,55)(23,54,49,79)(24,78,50,53)(25,52,41,77)(26,76,42,51)(27,60,43,75)(28,74,44,59)(29,58,45,73)(30,72,46,57)(32,38)(34,36)(61,63)(65,69) );
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)], [(1,40,67,12),(2,31,68,13),(3,32,69,14),(4,33,70,15),(5,34,61,16),(6,35,62,17),(7,36,63,18),(8,37,64,19),(9,38,65,20),(10,39,66,11),(21,73,47,58),(22,74,48,59),(23,75,49,60),(24,76,50,51),(25,77,41,52),(26,78,42,53),(27,79,43,54),(28,80,44,55),(29,71,45,56),(30,72,46,57)], [(1,46),(2,47),(3,48),(4,49),(5,50),(6,41),(7,42),(8,43),(9,44),(10,45),(11,56),(12,57),(13,58),(14,59),(15,60),(16,51),(17,52),(18,53),(19,54),(20,55),(21,68),(22,69),(23,70),(24,61),(25,62),(26,63),(27,64),(28,65),(29,66),(30,67),(31,73),(32,74),(33,75),(34,76),(35,77),(36,78),(37,79),(38,80),(39,71),(40,72)], [(2,66),(3,9),(4,64),(5,7),(6,62),(8,70),(10,68),(11,31),(13,39),(14,20),(15,37),(16,18),(17,35),(19,33),(21,56,47,71),(22,80,48,55),(23,54,49,79),(24,78,50,53),(25,52,41,77),(26,76,42,51),(27,60,43,75),(28,74,44,59),(29,58,45,73),(30,72,46,57),(32,38),(34,36),(61,63),(65,69)]])
62 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 5A | 5B | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 10G | ··· | 10R | 20A | ··· | 20H | 20I | ··· | 20T |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 20 | 20 | 20 | 20 | 2 | 2 | 20 | 20 | 20 | 20 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
62 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | - | - | + | |||||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | D4 | D4 | D5 | D10 | Dic5 | Dic5 | Dic5 | D10 | C5⋊D4 | C5⋊D4 | C42⋊C22 | (D4×C10)⋊21C4 |
| kernel | (D4×C10)⋊21C4 | D4⋊2Dic5 | C2×C4.Dic5 | C23.21D10 | C10×C4○D4 | D4×C10 | Q8×C10 | C5×C4○D4 | C2×C20 | C22×C10 | C2×C4○D4 | C22×C4 | C2×D4 | C2×Q8 | C4○D4 | C4○D4 | C2×C4 | C23 | C5 | C1 |
| # reps | 1 | 4 | 1 | 1 | 1 | 2 | 2 | 4 | 3 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 12 | 4 | 2 | 8 |
Matrix representation of (D4×C10)⋊21C4 ►in GL4(𝔽41) generated by
| 35 | 15 | 0 | 0 |
| 3 | 20 | 0 | 0 |
| 0 | 0 | 35 | 15 |
| 0 | 0 | 3 | 20 |
| 32 | 0 | 0 | 0 |
| 0 | 32 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 9 |
| 0 | 0 | 39 | 37 |
| 0 | 0 | 32 | 2 |
| 2 | 4 | 0 | 0 |
| 9 | 39 | 0 | 0 |
| 6 | 6 | 0 | 0 |
| 1 | 35 | 0 | 0 |
| 0 | 0 | 28 | 28 |
| 0 | 0 | 32 | 13 |
G:=sub<GL(4,GF(41))| [35,3,0,0,15,20,0,0,0,0,35,3,0,0,15,20],[32,0,0,0,0,32,0,0,0,0,9,0,0,0,0,9],[0,0,2,9,0,0,4,39,39,32,0,0,37,2,0,0],[6,1,0,0,6,35,0,0,0,0,28,32,0,0,28,13] >;
(D4×C10)⋊21C4 in GAP, Magma, Sage, TeX
(D_4\times C_{10})\rtimes_{21}C_4 % in TeX
G:=Group("(D4xC10):21C4"); // GroupNames label
G:=SmallGroup(320,863);
// by ID
G=gap.SmallGroup(320,863);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,422,387,136,1684,438,102,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^10=b^4=c^2=d^4=1,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1*b^2,c*b*c=b^-1,b*d=d*b,d*c*d^-1=b*c>;
// generators/relations