direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C10×C8⋊C22, C40⋊8C23, C20.82C24, C8⋊(C22×C10), D8⋊3(C2×C10), (C2×D8)⋊11C10, (C10×D8)⋊25C2, C4.66(D4×C10), (C2×C40)⋊29C22, SD16⋊1(C2×C10), (C2×SD16)⋊4C10, (C2×C20).525D4, C20.329(C2×D4), D4⋊2(C22×C10), (C5×D8)⋊19C22, (C5×D4)⋊13C23, C4.5(C23×C10), (C5×Q8)⋊12C23, Q8⋊2(C22×C10), C23.50(C5×D4), (C10×SD16)⋊15C2, (C22×D4)⋊11C10, (D4×C10)⋊66C22, (C2×M4(2))⋊3C10, M4(2)⋊3(C2×C10), (Q8×C10)⋊54C22, C22.23(D4×C10), (C10×M4(2))⋊13C2, (C2×C20).975C23, (C5×SD16)⋊17C22, C10.203(C22×D4), (C22×C10).172D4, (C5×M4(2))⋊29C22, (C22×C20).465C22, (C2×C8)⋊2(C2×C10), (D4×C2×C10)⋊26C2, C2.27(D4×C2×C10), C4○D4⋊4(C2×C10), (C10×C4○D4)⋊27C2, (C2×C4○D4)⋊11C10, (C2×D4)⋊15(C2×C10), (C2×Q8)⋊14(C2×C10), (C2×C4).136(C5×D4), (C2×C10).419(C2×D4), (C5×C4○D4)⋊24C22, (C2×C4).45(C22×C10), (C22×C4).76(C2×C10), SmallGroup(320,1575)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 530 in 298 conjugacy classes, 162 normal (30 characteristic)
C1, C2, C2 [×2], C2 [×8], C4 [×2], C4 [×2], C4 [×2], C22, C22 [×2], C22 [×22], C5, C8 [×4], C2×C4 [×2], C2×C4 [×4], C2×C4 [×5], D4 [×6], D4 [×11], Q8 [×2], Q8, C23, C23 [×11], C10, C10 [×2], C10 [×8], C2×C8 [×2], M4(2) [×4], D8 [×8], SD16 [×8], C22×C4, C22×C4, C2×D4, C2×D4 [×6], C2×D4 [×4], C2×Q8, C4○D4 [×4], C4○D4 [×2], C24, C20 [×2], C20 [×2], C20 [×2], C2×C10, C2×C10 [×2], C2×C10 [×22], C2×M4(2), C2×D8 [×2], C2×SD16 [×2], C8⋊C22 [×8], C22×D4, C2×C4○D4, C40 [×4], C2×C20 [×2], C2×C20 [×4], C2×C20 [×5], C5×D4 [×6], C5×D4 [×11], C5×Q8 [×2], C5×Q8, C22×C10, C22×C10 [×11], C2×C8⋊C22, C2×C40 [×2], C5×M4(2) [×4], C5×D8 [×8], C5×SD16 [×8], C22×C20, C22×C20, D4×C10, D4×C10 [×6], D4×C10 [×4], Q8×C10, C5×C4○D4 [×4], C5×C4○D4 [×2], C23×C10, C10×M4(2), C10×D8 [×2], C10×SD16 [×2], C5×C8⋊C22 [×8], D4×C2×C10, C10×C4○D4, C10×C8⋊C22
Quotients:
C1, C2 [×15], C22 [×35], C5, D4 [×4], C23 [×15], C10 [×15], C2×D4 [×6], C24, C2×C10 [×35], C8⋊C22 [×2], C22×D4, C5×D4 [×4], C22×C10 [×15], C2×C8⋊C22, D4×C10 [×6], C23×C10, C5×C8⋊C22 [×2], D4×C2×C10, C10×C8⋊C22
Generators and relations
G = < a,b,c,d | a10=b8=c2=d2=1, ab=ba, ac=ca, ad=da, cbc=b3, dbd=b5, cd=dc >
►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 28 66 79 60 31 50 11)(2 29 67 80 51 32 41 12)(3 30 68 71 52 33 42 13)(4 21 69 72 53 34 43 14)(5 22 70 73 54 35 44 15)(6 23 61 74 55 36 45 16)(7 24 62 75 56 37 46 17)(8 25 63 76 57 38 47 18)(9 26 64 77 58 39 48 19)(10 27 65 78 59 40 49 20)
(11 31)(12 32)(13 33)(14 34)(15 35)(16 36)(17 37)(18 38)(19 39)(20 40)(21 72)(22 73)(23 74)(24 75)(25 76)(26 77)(27 78)(28 79)(29 80)(30 71)(41 67)(42 68)(43 69)(44 70)(45 61)(46 62)(47 63)(48 64)(49 65)(50 66)
(1 60)(2 51)(3 52)(4 53)(5 54)(6 55)(7 56)(8 57)(9 58)(10 59)(41 67)(42 68)(43 69)(44 70)(45 61)(46 62)(47 63)(48 64)(49 65)(50 66)
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), (1,28,66,79,60,31,50,11)(2,29,67,80,51,32,41,12)(3,30,68,71,52,33,42,13)(4,21,69,72,53,34,43,14)(5,22,70,73,54,35,44,15)(6,23,61,74,55,36,45,16)(7,24,62,75,56,37,46,17)(8,25,63,76,57,38,47,18)(9,26,64,77,58,39,48,19)(10,27,65,78,59,40,49,20), (11,31)(12,32)(13,33)(14,34)(15,35)(16,36)(17,37)(18,38)(19,39)(20,40)(21,72)(22,73)(23,74)(24,75)(25,76)(26,77)(27,78)(28,79)(29,80)(30,71)(41,67)(42,68)(43,69)(44,70)(45,61)(46,62)(47,63)(48,64)(49,65)(50,66), (1,60)(2,51)(3,52)(4,53)(5,54)(6,55)(7,56)(8,57)(9,58)(10,59)(41,67)(42,68)(43,69)(44,70)(45,61)(46,62)(47,63)(48,64)(49,65)(50,66)>;
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), (1,28,66,79,60,31,50,11)(2,29,67,80,51,32,41,12)(3,30,68,71,52,33,42,13)(4,21,69,72,53,34,43,14)(5,22,70,73,54,35,44,15)(6,23,61,74,55,36,45,16)(7,24,62,75,56,37,46,17)(8,25,63,76,57,38,47,18)(9,26,64,77,58,39,48,19)(10,27,65,78,59,40,49,20), (11,31)(12,32)(13,33)(14,34)(15,35)(16,36)(17,37)(18,38)(19,39)(20,40)(21,72)(22,73)(23,74)(24,75)(25,76)(26,77)(27,78)(28,79)(29,80)(30,71)(41,67)(42,68)(43,69)(44,70)(45,61)(46,62)(47,63)(48,64)(49,65)(50,66), (1,60)(2,51)(3,52)(4,53)(5,54)(6,55)(7,56)(8,57)(9,58)(10,59)(41,67)(42,68)(43,69)(44,70)(45,61)(46,62)(47,63)(48,64)(49,65)(50,66) );
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)], [(1,28,66,79,60,31,50,11),(2,29,67,80,51,32,41,12),(3,30,68,71,52,33,42,13),(4,21,69,72,53,34,43,14),(5,22,70,73,54,35,44,15),(6,23,61,74,55,36,45,16),(7,24,62,75,56,37,46,17),(8,25,63,76,57,38,47,18),(9,26,64,77,58,39,48,19),(10,27,65,78,59,40,49,20)], [(11,31),(12,32),(13,33),(14,34),(15,35),(16,36),(17,37),(18,38),(19,39),(20,40),(21,72),(22,73),(23,74),(24,75),(25,76),(26,77),(27,78),(28,79),(29,80),(30,71),(41,67),(42,68),(43,69),(44,70),(45,61),(46,62),(47,63),(48,64),(49,65),(50,66)], [(1,60),(2,51),(3,52),(4,53),(5,54),(6,55),(7,56),(8,57),(9,58),(10,59),(41,67),(42,68),(43,69),(44,70),(45,61),(46,62),(47,63),(48,64),(49,65),(50,66)])
Matrix representation ►G ⊆ GL6(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 40 | 2 | 0 | 0 | 0 | 0 |
| 40 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 6 | 0 | 0 | 40 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 6 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 1 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 6 | 0 | 0 | 40 |
| 0 | 0 | 6 | 0 | 40 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 29 | 0 | 0 | 1 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[40,40,0,0,0,0,2,1,0,0,0,0,0,0,0,6,0,1,0,0,0,0,1,0,0,0,1,0,0,6,0,0,0,40,0,0],[1,1,0,0,0,0,0,40,0,0,0,0,0,0,1,0,6,6,0,0,0,40,0,0,0,0,0,0,0,40,0,0,0,0,40,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,29,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;
110 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | ··· | 2K | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 5C | 5D | 8A | 8B | 8C | 8D | 10A | ··· | 10L | 10M | ··· | 10T | 10U | ··· | 10AR | 20A | ··· | 20P | 20Q | ··· | 20X | 40A | ··· | 40P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | ··· | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
110 irreducible representations
| dim | 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 | C5 | C10 | C10 | C10 | C10 | C10 | C10 | D4 | D4 | C5×D4 | C5×D4 | C8⋊C22 | C5×C8⋊C22 |
| kernel | C10×C8⋊C22 | C10×M4(2) | C10×D8 | C10×SD16 | C5×C8⋊C22 | D4×C2×C10 | C10×C4○D4 | C2×C8⋊C22 | C2×M4(2) | C2×D8 | C2×SD16 | C8⋊C22 | C22×D4 | C2×C4○D4 | C2×C20 | C22×C10 | C2×C4 | C23 | C10 | C2 |
| # reps | 1 | 1 | 2 | 2 | 8 | 1 | 1 | 4 | 4 | 8 | 8 | 32 | 4 | 4 | 3 | 1 | 12 | 4 | 2 | 8 |
In GAP, Magma, Sage, TeX
C_{10}\times C_8\rtimes C_2^2 % in TeX
G:=Group("C10xC8:C2^2"); // GroupNames label
G:=SmallGroup(320,1575);
// by ID
G=gap.SmallGroup(320,1575);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-5,-2,-2,1149,3446,10085,5052,124]);
// Polycyclic
G:=Group<a,b,c,d|a^10=b^8=c^2=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^3,d*b*d=b^5,c*d=d*c>;
// generators/relations