direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C5×D4⋊5D4, C10.1612+ 1+4, D4⋊5(C5×D4), (C5×D4)⋊23D4, (D4×C20)⋊44C2, (C4×D4)⋊15C10, C42⋊9(C2×C10), C4.40(D4×C10), C22≀C2⋊6C10, C4⋊D4⋊11C10, (C4×C20)⋊43C22, C20.401(C2×D4), (C22×D4)⋊9C10, C22⋊Q8⋊11C10, C22.5(D4×C10), C4.4D4⋊10C10, (D4×C10)⋊38C22, C24.19(C2×C10), (Q8×C10)⋊52C22, (C2×C10).366C24, (C2×C20).713C23, (C22×C20)⋊51C22, C10.194(C22×D4), C22.D4⋊8C10, (C22×C10).98C23, C22.40(C23×C10), C23.16(C22×C10), (C23×C10).19C22, C2.13(C5×2+ 1+4), C4⋊C4⋊5(C2×C10), (D4×C2×C10)⋊24C2, C2.18(D4×C2×C10), (C2×C4○D4)⋊6C10, C22⋊2(C5×C4○D4), (C10×C4○D4)⋊22C2, (C2×D4)⋊13(C2×C10), (C5×C4⋊D4)⋊38C2, (C5×C4⋊C4)⋊73C22, (C2×Q8)⋊12(C2×C10), C2.20(C10×C4○D4), (C5×C22⋊Q8)⋊38C2, (C5×C22≀C2)⋊16C2, (C2×C10)⋊14(C4○D4), (C10×C22⋊C4)⋊34C2, (C2×C22⋊C4)⋊14C10, C22⋊C4⋊16(C2×C10), (C22×C4)⋊11(C2×C10), C10.239(C2×C4○D4), (C2×C10).182(C2×D4), (C5×C4.4D4)⋊30C2, (C5×C22⋊C4)⋊70C22, (C2×C4).59(C22×C10), (C5×C22.D4)⋊27C2, SmallGroup(320,1548)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×D4⋊5D4
G = < a,b,c,d,e | a5=b4=c2=d4=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, dcd-1=ece=b2c, ede=d-1 >
Subgroups: 570 in 334 conjugacy classes, 166 normal (62 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C5, C2×C4, C2×C4, C2×C4, D4, D4, Q8, C23, C23, C23, C10, C10, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×D4, C2×Q8, C4○D4, C24, C20, C20, C2×C10, C2×C10, C2×C10, C2×C22⋊C4, C4×D4, C22≀C2, C4⋊D4, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C22×D4, C2×C4○D4, C2×C20, C2×C20, C2×C20, C5×D4, C5×D4, C5×Q8, C22×C10, C22×C10, C22×C10, D4⋊5D4, C4×C20, C5×C22⋊C4, C5×C22⋊C4, C5×C4⋊C4, C5×C4⋊C4, C22×C20, C22×C20, D4×C10, D4×C10, D4×C10, Q8×C10, C5×C4○D4, C23×C10, C10×C22⋊C4, D4×C20, C5×C22≀C2, C5×C4⋊D4, C5×C4⋊D4, C5×C22⋊Q8, C5×C22.D4, C5×C4.4D4, D4×C2×C10, C10×C4○D4, C5×D4⋊5D4
Quotients: C1, C2, C22, C5, D4, C23, C10, C2×D4, C4○D4, C24, C2×C10, C22×D4, C2×C4○D4, 2+ 1+4, C5×D4, C22×C10, D4⋊5D4, D4×C10, C5×C4○D4, C23×C10, D4×C2×C10, C10×C4○D4, C5×2+ 1+4, C5×D4⋊5D4
►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 55 35 36)(2 51 31 37)(3 52 32 38)(4 53 33 39)(5 54 34 40)(6 63 16 70)(7 64 17 66)(8 65 18 67)(9 61 19 68)(10 62 20 69)(11 56 76 75)(12 57 77 71)(13 58 78 72)(14 59 79 73)(15 60 80 74)(21 48 28 41)(22 49 29 42)(23 50 30 43)(24 46 26 44)(25 47 27 45)
(1 36)(2 37)(3 38)(4 39)(5 40)(6 63)(7 64)(8 65)(9 61)(10 62)(11 56)(12 57)(13 58)(14 59)(15 60)(16 70)(17 66)(18 67)(19 68)(20 69)(21 41)(22 42)(23 43)(24 44)(25 45)(26 46)(27 47)(28 48)(29 49)(30 50)(31 51)(32 52)(33 53)(34 54)(35 55)(71 77)(72 78)(73 79)(74 80)(75 76)
(1 16 30 11)(2 17 26 12)(3 18 27 13)(4 19 28 14)(5 20 29 15)(6 23 76 35)(7 24 77 31)(8 25 78 32)(9 21 79 33)(10 22 80 34)(36 63 50 75)(37 64 46 71)(38 65 47 72)(39 61 48 73)(40 62 49 74)(41 59 53 68)(42 60 54 69)(43 56 55 70)(44 57 51 66)(45 58 52 67)
(1 76)(2 77)(3 78)(4 79)(5 80)(6 30)(7 26)(8 27)(9 28)(10 29)(11 35)(12 31)(13 32)(14 33)(15 34)(16 23)(17 24)(18 25)(19 21)(20 22)(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)(51 71)(52 72)(53 73)(54 74)(55 75)
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), (1,55,35,36)(2,51,31,37)(3,52,32,38)(4,53,33,39)(5,54,34,40)(6,63,16,70)(7,64,17,66)(8,65,18,67)(9,61,19,68)(10,62,20,69)(11,56,76,75)(12,57,77,71)(13,58,78,72)(14,59,79,73)(15,60,80,74)(21,48,28,41)(22,49,29,42)(23,50,30,43)(24,46,26,44)(25,47,27,45), (1,36)(2,37)(3,38)(4,39)(5,40)(6,63)(7,64)(8,65)(9,61)(10,62)(11,56)(12,57)(13,58)(14,59)(15,60)(16,70)(17,66)(18,67)(19,68)(20,69)(21,41)(22,42)(23,43)(24,44)(25,45)(26,46)(27,47)(28,48)(29,49)(30,50)(31,51)(32,52)(33,53)(34,54)(35,55)(71,77)(72,78)(73,79)(74,80)(75,76), (1,16,30,11)(2,17,26,12)(3,18,27,13)(4,19,28,14)(5,20,29,15)(6,23,76,35)(7,24,77,31)(8,25,78,32)(9,21,79,33)(10,22,80,34)(36,63,50,75)(37,64,46,71)(38,65,47,72)(39,61,48,73)(40,62,49,74)(41,59,53,68)(42,60,54,69)(43,56,55,70)(44,57,51,66)(45,58,52,67), (1,76)(2,77)(3,78)(4,79)(5,80)(6,30)(7,26)(8,27)(9,28)(10,29)(11,35)(12,31)(13,32)(14,33)(15,34)(16,23)(17,24)(18,25)(19,21)(20,22)(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)(51,71)(52,72)(53,73)(54,74)(55,75)>;
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), (1,55,35,36)(2,51,31,37)(3,52,32,38)(4,53,33,39)(5,54,34,40)(6,63,16,70)(7,64,17,66)(8,65,18,67)(9,61,19,68)(10,62,20,69)(11,56,76,75)(12,57,77,71)(13,58,78,72)(14,59,79,73)(15,60,80,74)(21,48,28,41)(22,49,29,42)(23,50,30,43)(24,46,26,44)(25,47,27,45), (1,36)(2,37)(3,38)(4,39)(5,40)(6,63)(7,64)(8,65)(9,61)(10,62)(11,56)(12,57)(13,58)(14,59)(15,60)(16,70)(17,66)(18,67)(19,68)(20,69)(21,41)(22,42)(23,43)(24,44)(25,45)(26,46)(27,47)(28,48)(29,49)(30,50)(31,51)(32,52)(33,53)(34,54)(35,55)(71,77)(72,78)(73,79)(74,80)(75,76), (1,16,30,11)(2,17,26,12)(3,18,27,13)(4,19,28,14)(5,20,29,15)(6,23,76,35)(7,24,77,31)(8,25,78,32)(9,21,79,33)(10,22,80,34)(36,63,50,75)(37,64,46,71)(38,65,47,72)(39,61,48,73)(40,62,49,74)(41,59,53,68)(42,60,54,69)(43,56,55,70)(44,57,51,66)(45,58,52,67), (1,76)(2,77)(3,78)(4,79)(5,80)(6,30)(7,26)(8,27)(9,28)(10,29)(11,35)(12,31)(13,32)(14,33)(15,34)(16,23)(17,24)(18,25)(19,21)(20,22)(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)(51,71)(52,72)(53,73)(54,74)(55,75) );
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)], [(1,55,35,36),(2,51,31,37),(3,52,32,38),(4,53,33,39),(5,54,34,40),(6,63,16,70),(7,64,17,66),(8,65,18,67),(9,61,19,68),(10,62,20,69),(11,56,76,75),(12,57,77,71),(13,58,78,72),(14,59,79,73),(15,60,80,74),(21,48,28,41),(22,49,29,42),(23,50,30,43),(24,46,26,44),(25,47,27,45)], [(1,36),(2,37),(3,38),(4,39),(5,40),(6,63),(7,64),(8,65),(9,61),(10,62),(11,56),(12,57),(13,58),(14,59),(15,60),(16,70),(17,66),(18,67),(19,68),(20,69),(21,41),(22,42),(23,43),(24,44),(25,45),(26,46),(27,47),(28,48),(29,49),(30,50),(31,51),(32,52),(33,53),(34,54),(35,55),(71,77),(72,78),(73,79),(74,80),(75,76)], [(1,16,30,11),(2,17,26,12),(3,18,27,13),(4,19,28,14),(5,20,29,15),(6,23,76,35),(7,24,77,31),(8,25,78,32),(9,21,79,33),(10,22,80,34),(36,63,50,75),(37,64,46,71),(38,65,47,72),(39,61,48,73),(40,62,49,74),(41,59,53,68),(42,60,54,69),(43,56,55,70),(44,57,51,66),(45,58,52,67)], [(1,76),(2,77),(3,78),(4,79),(5,80),(6,30),(7,26),(8,27),(9,28),(10,29),(11,35),(12,31),(13,32),(14,33),(15,34),(16,23),(17,24),(18,25),(19,21),(20,22),(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),(51,71),(52,72),(53,73),(54,74),(55,75)]])
125 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | ··· | 2I | 2J | 2K | 2L | 4A | ··· | 4F | 4G | ··· | 4L | 5A | 5B | 5C | 5D | 10A | ··· | 10L | 10M | ··· | 10AJ | 10AK | ··· | 10AV | 20A | ··· | 20X | 20Y | ··· | 20AV |
| order | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 5 | 5 | 5 | 5 | 10 | ··· | 10 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
125 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | ||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C5 | C10 | C10 | C10 | C10 | C10 | C10 | C10 | C10 | C10 | D4 | C4○D4 | C5×D4 | C5×C4○D4 | 2+ 1+4 | C5×2+ 1+4 |
| kernel | C5×D4⋊5D4 | C10×C22⋊C4 | D4×C20 | C5×C22≀C2 | C5×C4⋊D4 | C5×C22⋊Q8 | C5×C22.D4 | C5×C4.4D4 | D4×C2×C10 | C10×C4○D4 | D4⋊5D4 | C2×C22⋊C4 | C4×D4 | C22≀C2 | C4⋊D4 | C22⋊Q8 | C22.D4 | C4.4D4 | C22×D4 | C2×C4○D4 | C5×D4 | C2×C10 | D4 | C22 | C10 | C2 |
| # reps | 1 | 2 | 2 | 2 | 3 | 1 | 2 | 1 | 1 | 1 | 4 | 8 | 8 | 8 | 12 | 4 | 8 | 4 | 4 | 4 | 4 | 4 | 16 | 16 | 1 | 4 |
Matrix representation of C5×D4⋊5D4 ►in GL4(𝔽41) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 18 | 0 |
| 0 | 0 | 0 | 18 |
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 1 | 2 |
| 0 | 0 | 40 | 40 |
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 1 | 2 |
| 0 | 0 | 0 | 40 |
| 0 | 40 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 9 | 18 |
| 0 | 0 | 32 | 32 |
| 0 | 40 | 0 | 0 |
| 40 | 0 | 0 | 0 |
| 0 | 0 | 32 | 23 |
| 0 | 0 | 9 | 9 |
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,18,0,0,0,0,18],[40,0,0,0,0,40,0,0,0,0,1,40,0,0,2,40],[40,0,0,0,0,40,0,0,0,0,1,0,0,0,2,40],[0,1,0,0,40,0,0,0,0,0,9,32,0,0,18,32],[0,40,0,0,40,0,0,0,0,0,32,9,0,0,23,9] >;
C5×D4⋊5D4 in GAP, Magma, Sage, TeX
C_5\times D_4\rtimes_5D_4
% in TeX
G:=Group("C5xD4:5D4"); // GroupNames label
G:=SmallGroup(320,1548);
// by ID
G=gap.SmallGroup(320,1548);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-5,-2,-2,560,1149,3446,1242]);
// Polycyclic
G:=Group<a,b,c,d,e|a^5=b^4=c^2=d^4=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,d*c*d^-1=e*c*e=b^2*c,e*d*e=d^-1>;
// generators/relations