direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: D4×C2×C10, C20⋊4C23, C24⋊3C10, C10.16C24, C4⋊(C22×C10), (C2×C10)⋊2C23, C23⋊3(C2×C10), (C23×C10)⋊2C2, (C22×C4)⋊5C10, C22⋊(C22×C10), (C2×C20)⋊15C22, (C22×C20)⋊12C2, C2.1(C23×C10), (C22×C10)⋊6C22, (C2×C4)⋊4(C2×C10), SmallGroup(160,229)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4×C2×C10
G = < a,b,c,d | a2=b10=c4=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Subgroups: 316 in 236 conjugacy classes, 156 normal (10 characteristic)
C1, C2, C2, C2, C4, C22, C22, C5, C2×C4, D4, C23, C23, C23, C10, C10, C10, C22×C4, C2×D4, C24, C20, C2×C10, C2×C10, C22×D4, C2×C20, C5×D4, C22×C10, C22×C10, C22×C10, C22×C20, D4×C10, C23×C10, D4×C2×C10
Quotients: C1, C2, C22, C5, D4, C23, C10, C2×D4, C24, C2×C10, C22×D4, C5×D4, C22×C10, D4×C10, C23×C10, D4×C2×C10
►On 80 points
Generators in S80
(1 42)(2 43)(3 44)(4 45)(5 46)(6 47)(7 48)(8 49)(9 50)(10 41)(11 52)(12 53)(13 54)(14 55)(15 56)(16 57)(17 58)(18 59)(19 60)(20 51)(21 33)(22 34)(23 35)(24 36)(25 37)(26 38)(27 39)(28 40)(29 31)(30 32)(61 73)(62 74)(63 75)(64 76)(65 77)(66 78)(67 79)(68 80)(69 71)(70 72)
(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 18 27 74)(2 19 28 75)(3 20 29 76)(4 11 30 77)(5 12 21 78)(6 13 22 79)(7 14 23 80)(8 15 24 71)(9 16 25 72)(10 17 26 73)(31 64 44 51)(32 65 45 52)(33 66 46 53)(34 67 47 54)(35 68 48 55)(36 69 49 56)(37 70 50 57)(38 61 41 58)(39 62 42 59)(40 63 43 60)
(1 74)(2 75)(3 76)(4 77)(5 78)(6 79)(7 80)(8 71)(9 72)(10 73)(11 30)(12 21)(13 22)(14 23)(15 24)(16 25)(17 26)(18 27)(19 28)(20 29)(31 51)(32 52)(33 53)(34 54)(35 55)(36 56)(37 57)(38 58)(39 59)(40 60)(41 61)(42 62)(43 63)(44 64)(45 65)(46 66)(47 67)(48 68)(49 69)(50 70)
G:=sub<Sym(80)| (1,42)(2,43)(3,44)(4,45)(5,46)(6,47)(7,48)(8,49)(9,50)(10,41)(11,52)(12,53)(13,54)(14,55)(15,56)(16,57)(17,58)(18,59)(19,60)(20,51)(21,33)(22,34)(23,35)(24,36)(25,37)(26,38)(27,39)(28,40)(29,31)(30,32)(61,73)(62,74)(63,75)(64,76)(65,77)(66,78)(67,79)(68,80)(69,71)(70,72), (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), (1,18,27,74)(2,19,28,75)(3,20,29,76)(4,11,30,77)(5,12,21,78)(6,13,22,79)(7,14,23,80)(8,15,24,71)(9,16,25,72)(10,17,26,73)(31,64,44,51)(32,65,45,52)(33,66,46,53)(34,67,47,54)(35,68,48,55)(36,69,49,56)(37,70,50,57)(38,61,41,58)(39,62,42,59)(40,63,43,60), (1,74)(2,75)(3,76)(4,77)(5,78)(6,79)(7,80)(8,71)(9,72)(10,73)(11,30)(12,21)(13,22)(14,23)(15,24)(16,25)(17,26)(18,27)(19,28)(20,29)(31,51)(32,52)(33,53)(34,54)(35,55)(36,56)(37,57)(38,58)(39,59)(40,60)(41,61)(42,62)(43,63)(44,64)(45,65)(46,66)(47,67)(48,68)(49,69)(50,70)>;
G:=Group( (1,42)(2,43)(3,44)(4,45)(5,46)(6,47)(7,48)(8,49)(9,50)(10,41)(11,52)(12,53)(13,54)(14,55)(15,56)(16,57)(17,58)(18,59)(19,60)(20,51)(21,33)(22,34)(23,35)(24,36)(25,37)(26,38)(27,39)(28,40)(29,31)(30,32)(61,73)(62,74)(63,75)(64,76)(65,77)(66,78)(67,79)(68,80)(69,71)(70,72), (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), (1,18,27,74)(2,19,28,75)(3,20,29,76)(4,11,30,77)(5,12,21,78)(6,13,22,79)(7,14,23,80)(8,15,24,71)(9,16,25,72)(10,17,26,73)(31,64,44,51)(32,65,45,52)(33,66,46,53)(34,67,47,54)(35,68,48,55)(36,69,49,56)(37,70,50,57)(38,61,41,58)(39,62,42,59)(40,63,43,60), (1,74)(2,75)(3,76)(4,77)(5,78)(6,79)(7,80)(8,71)(9,72)(10,73)(11,30)(12,21)(13,22)(14,23)(15,24)(16,25)(17,26)(18,27)(19,28)(20,29)(31,51)(32,52)(33,53)(34,54)(35,55)(36,56)(37,57)(38,58)(39,59)(40,60)(41,61)(42,62)(43,63)(44,64)(45,65)(46,66)(47,67)(48,68)(49,69)(50,70) );
G=PermutationGroup([[(1,42),(2,43),(3,44),(4,45),(5,46),(6,47),(7,48),(8,49),(9,50),(10,41),(11,52),(12,53),(13,54),(14,55),(15,56),(16,57),(17,58),(18,59),(19,60),(20,51),(21,33),(22,34),(23,35),(24,36),(25,37),(26,38),(27,39),(28,40),(29,31),(30,32),(61,73),(62,74),(63,75),(64,76),(65,77),(66,78),(67,79),(68,80),(69,71),(70,72)], [(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)], [(1,18,27,74),(2,19,28,75),(3,20,29,76),(4,11,30,77),(5,12,21,78),(6,13,22,79),(7,14,23,80),(8,15,24,71),(9,16,25,72),(10,17,26,73),(31,64,44,51),(32,65,45,52),(33,66,46,53),(34,67,47,54),(35,68,48,55),(36,69,49,56),(37,70,50,57),(38,61,41,58),(39,62,42,59),(40,63,43,60)], [(1,74),(2,75),(3,76),(4,77),(5,78),(6,79),(7,80),(8,71),(9,72),(10,73),(11,30),(12,21),(13,22),(14,23),(15,24),(16,25),(17,26),(18,27),(19,28),(20,29),(31,51),(32,52),(33,53),(34,54),(35,55),(36,56),(37,57),(38,58),(39,59),(40,60),(41,61),(42,62),(43,63),(44,64),(45,65),(46,66),(47,67),(48,68),(49,69),(50,70)]])
D4×C2×C10 is a maximal subgroup of
(D4×C10)⋊18C4 (C2×C10)⋊8D8 (C5×D4).31D4 C24.18D10 C24.19D10 C24.20D10 C24.21D10 C24.38D10 C24⋊8D10 C24.41D10 C24.42D10
100 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | ··· | 2O | 4A | 4B | 4C | 4D | 5A | 5B | 5C | 5D | 10A | ··· | 10AB | 10AC | ··· | 10BH | 20A | ··· | 20P |
| order | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
100 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C5 | C10 | C10 | C10 | D4 | C5×D4 |
| kernel | D4×C2×C10 | C22×C20 | D4×C10 | C23×C10 | C22×D4 | C22×C4 | C2×D4 | C24 | C2×C10 | C22 |
| # reps | 1 | 1 | 12 | 2 | 4 | 4 | 48 | 8 | 4 | 16 |
Matrix representation of D4×C2×C10 ►in GL4(𝔽41) generated by
| 1 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 40 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 25 | 0 |
| 0 | 0 | 0 | 25 |
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 40 |
| 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 40 |
| 0 | 0 | 40 | 0 |
G:=sub<GL(4,GF(41))| [1,0,0,0,0,40,0,0,0,0,40,0,0,0,0,40],[40,0,0,0,0,1,0,0,0,0,25,0,0,0,0,25],[40,0,0,0,0,40,0,0,0,0,0,1,0,0,40,0],[1,0,0,0,0,1,0,0,0,0,0,40,0,0,40,0] >;
D4×C2×C10 in GAP, Magma, Sage, TeX
D_4\times C_2\times C_{10} % in TeX
G:=Group("D4xC2xC10"); // GroupNames label
G:=SmallGroup(160,229);
// by ID
G=gap.SmallGroup(160,229);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-5,-2,985]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^10=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations