direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C10×C4≀C2, C4○D4⋊3C20, (C2×D4)⋊9C20, D4⋊5(C2×C20), (C2×Q8)⋊7C20, Q8⋊5(C2×C20), (D4×C10)⋊33C4, (C2×C42)⋊6C10, (Q8×C10)⋊27C4, C4.72(D4×C10), (C4×C20)⋊56C22, C42⋊15(C2×C10), (C2×C20).519D4, C20.477(C2×D4), C4.7(C22×C20), C23.39(C5×D4), M4(2)⋊9(C2×C10), C22.12(D4×C10), (C10×M4(2))⋊30C2, (C2×M4(2))⋊12C10, C20.211(C22×C4), (C2×C20).896C23, (C22×C10).161D4, C20.164(C22⋊C4), (C5×M4(2))⋊38C22, (C22×C20).584C22, (C2×C4×C20)⋊19C2, (C5×C4○D4)⋊15C4, (C5×D4)⋊35(C2×C4), (C5×Q8)⋊32(C2×C4), (C2×C4).70(C5×D4), (C2×C4).50(C2×C20), C4○D4.6(C2×C10), (C2×C4○D4).6C10, C4.33(C5×C22⋊C4), (C2×C20).444(C2×C4), (C10×C4○D4).20C2, (C2×C10).407(C2×D4), C2.23(C10×C22⋊C4), C22.6(C5×C22⋊C4), C10.152(C2×C22⋊C4), (C2×C4).71(C22×C10), (C5×C4○D4).51C22, (C22×C4).113(C2×C10), (C2×C10).204(C22⋊C4), SmallGroup(320,921)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C10×C4≀C2
G = < a,b,c,d | a10=b4=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=b-1, bd=db, dcd-1=b-1c >
Subgroups: 274 in 170 conjugacy classes, 82 normal (46 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, C10, C10, C10, C42, C42, C2×C8, M4(2), M4(2), C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C4○D4, C20, C20, C2×C10, C2×C10, C4≀C2, C2×C42, C2×M4(2), C2×C4○D4, C40, C2×C20, C2×C20, C5×D4, C5×D4, C5×Q8, C5×Q8, C22×C10, C22×C10, C2×C4≀C2, C4×C20, C4×C20, C2×C40, C5×M4(2), C5×M4(2), C22×C20, C22×C20, D4×C10, D4×C10, Q8×C10, C5×C4○D4, C5×C4○D4, C5×C4≀C2, C2×C4×C20, C10×M4(2), C10×C4○D4, C10×C4≀C2
Quotients: C1, C2, C4, C22, C5, C2×C4, D4, C23, C10, C22⋊C4, C22×C4, C2×D4, C20, C2×C10, C4≀C2, C2×C22⋊C4, C2×C20, C5×D4, C22×C10, C2×C4≀C2, C5×C22⋊C4, C22×C20, D4×C10, C5×C4≀C2, C10×C22⋊C4, C10×C4≀C2
►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 60 64 50)(2 51 65 41)(3 52 66 42)(4 53 67 43)(5 54 68 44)(6 55 69 45)(7 56 70 46)(8 57 61 47)(9 58 62 48)(10 59 63 49)(11 39 29 80)(12 40 30 71)(13 31 21 72)(14 32 22 73)(15 33 23 74)(16 34 24 75)(17 35 25 76)(18 36 26 77)(19 37 27 78)(20 38 28 79)
(1 28)(2 29)(3 30)(4 21)(5 22)(6 23)(7 24)(8 25)(9 26)(10 27)(11 65)(12 66)(13 67)(14 68)(15 69)(16 70)(17 61)(18 62)(19 63)(20 64)(31 53)(32 54)(33 55)(34 56)(35 57)(36 58)(37 59)(38 60)(39 51)(40 52)(41 80)(42 71)(43 72)(44 73)(45 74)(46 75)(47 76)(48 77)(49 78)(50 79)
(1 69)(2 70)(3 61)(4 62)(5 63)(6 64)(7 65)(8 66)(9 67)(10 68)(11 34 29 75)(12 35 30 76)(13 36 21 77)(14 37 22 78)(15 38 23 79)(16 39 24 80)(17 40 25 71)(18 31 26 72)(19 32 27 73)(20 33 28 74)(41 56)(42 57)(43 58)(44 59)(45 60)(46 51)(47 52)(48 53)(49 54)(50 55)
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), (1,60,64,50)(2,51,65,41)(3,52,66,42)(4,53,67,43)(5,54,68,44)(6,55,69,45)(7,56,70,46)(8,57,61,47)(9,58,62,48)(10,59,63,49)(11,39,29,80)(12,40,30,71)(13,31,21,72)(14,32,22,73)(15,33,23,74)(16,34,24,75)(17,35,25,76)(18,36,26,77)(19,37,27,78)(20,38,28,79), (1,28)(2,29)(3,30)(4,21)(5,22)(6,23)(7,24)(8,25)(9,26)(10,27)(11,65)(12,66)(13,67)(14,68)(15,69)(16,70)(17,61)(18,62)(19,63)(20,64)(31,53)(32,54)(33,55)(34,56)(35,57)(36,58)(37,59)(38,60)(39,51)(40,52)(41,80)(42,71)(43,72)(44,73)(45,74)(46,75)(47,76)(48,77)(49,78)(50,79), (1,69)(2,70)(3,61)(4,62)(5,63)(6,64)(7,65)(8,66)(9,67)(10,68)(11,34,29,75)(12,35,30,76)(13,36,21,77)(14,37,22,78)(15,38,23,79)(16,39,24,80)(17,40,25,71)(18,31,26,72)(19,32,27,73)(20,33,28,74)(41,56)(42,57)(43,58)(44,59)(45,60)(46,51)(47,52)(48,53)(49,54)(50,55)>;
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), (1,60,64,50)(2,51,65,41)(3,52,66,42)(4,53,67,43)(5,54,68,44)(6,55,69,45)(7,56,70,46)(8,57,61,47)(9,58,62,48)(10,59,63,49)(11,39,29,80)(12,40,30,71)(13,31,21,72)(14,32,22,73)(15,33,23,74)(16,34,24,75)(17,35,25,76)(18,36,26,77)(19,37,27,78)(20,38,28,79), (1,28)(2,29)(3,30)(4,21)(5,22)(6,23)(7,24)(8,25)(9,26)(10,27)(11,65)(12,66)(13,67)(14,68)(15,69)(16,70)(17,61)(18,62)(19,63)(20,64)(31,53)(32,54)(33,55)(34,56)(35,57)(36,58)(37,59)(38,60)(39,51)(40,52)(41,80)(42,71)(43,72)(44,73)(45,74)(46,75)(47,76)(48,77)(49,78)(50,79), (1,69)(2,70)(3,61)(4,62)(5,63)(6,64)(7,65)(8,66)(9,67)(10,68)(11,34,29,75)(12,35,30,76)(13,36,21,77)(14,37,22,78)(15,38,23,79)(16,39,24,80)(17,40,25,71)(18,31,26,72)(19,32,27,73)(20,33,28,74)(41,56)(42,57)(43,58)(44,59)(45,60)(46,51)(47,52)(48,53)(49,54)(50,55) );
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)], [(1,60,64,50),(2,51,65,41),(3,52,66,42),(4,53,67,43),(5,54,68,44),(6,55,69,45),(7,56,70,46),(8,57,61,47),(9,58,62,48),(10,59,63,49),(11,39,29,80),(12,40,30,71),(13,31,21,72),(14,32,22,73),(15,33,23,74),(16,34,24,75),(17,35,25,76),(18,36,26,77),(19,37,27,78),(20,38,28,79)], [(1,28),(2,29),(3,30),(4,21),(5,22),(6,23),(7,24),(8,25),(9,26),(10,27),(11,65),(12,66),(13,67),(14,68),(15,69),(16,70),(17,61),(18,62),(19,63),(20,64),(31,53),(32,54),(33,55),(34,56),(35,57),(36,58),(37,59),(38,60),(39,51),(40,52),(41,80),(42,71),(43,72),(44,73),(45,74),(46,75),(47,76),(48,77),(49,78),(50,79)], [(1,69),(2,70),(3,61),(4,62),(5,63),(6,64),(7,65),(8,66),(9,67),(10,68),(11,34,29,75),(12,35,30,76),(13,36,21,77),(14,37,22,78),(15,38,23,79),(16,39,24,80),(17,40,25,71),(18,31,26,72),(19,32,27,73),(20,33,28,74),(41,56),(42,57),(43,58),(44,59),(45,60),(46,51),(47,52),(48,53),(49,54),(50,55)]])
140 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | ··· | 4N | 4O | 4P | 5A | 5B | 5C | 5D | 8A | 8B | 8C | 8D | 10A | ··· | 10L | 10M | ··· | 10T | 10U | ··· | 10AB | 20A | ··· | 20P | 20Q | ··· | 20BD | 20BE | ··· | 20BL | 40A | ··· | 40P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | 4 | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
140 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | |||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C5 | C10 | C10 | C10 | C10 | C20 | C20 | C20 | D4 | D4 | C4≀C2 | C5×D4 | C5×D4 | C5×C4≀C2 |
| kernel | C10×C4≀C2 | C5×C4≀C2 | C2×C4×C20 | C10×M4(2) | C10×C4○D4 | D4×C10 | Q8×C10 | C5×C4○D4 | C2×C4≀C2 | C4≀C2 | C2×C42 | C2×M4(2) | C2×C4○D4 | C2×D4 | C2×Q8 | C4○D4 | C2×C20 | C22×C10 | C10 | C2×C4 | C23 | C2 |
| # reps | 1 | 4 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 16 | 4 | 4 | 4 | 8 | 8 | 16 | 3 | 1 | 8 | 12 | 4 | 32 |
Matrix representation of C10×C4≀C2 ►in GL4(𝔽41) generated by
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 32 |
| 4 | 39 | 0 | 0 |
| 28 | 37 | 0 | 0 |
| 0 | 0 | 0 | 32 |
| 0 | 0 | 9 | 0 |
| 40 | 0 | 0 | 0 |
| 37 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 9 |
G:=sub<GL(4,GF(41))| [4,0,0,0,0,4,0,0,0,0,40,0,0,0,0,40],[40,0,0,0,0,40,0,0,0,0,9,0,0,0,0,32],[4,28,0,0,39,37,0,0,0,0,0,9,0,0,32,0],[40,37,0,0,0,1,0,0,0,0,1,0,0,0,0,9] >;
C10×C4≀C2 in GAP, Magma, Sage, TeX
C_{10}\times C_4\wr C_2 % in TeX
G:=Group("C10xC4wrC2"); // GroupNames label
G:=SmallGroup(320,921);
// by ID
G=gap.SmallGroup(320,921);
# by ID
G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,560,589,7004,3511,172,124]);
// Polycyclic
G:=Group<a,b,c,d|a^10=b^4=c^2=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,b*d=d*b,d*c*d^-1=b^-1*c>;
// generators/relations