direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C10×2+ (1+4), C20.90C24, C10.24C25, C24⋊6(C2×C10), (C2×C20)⋊11C23, D4⋊4(C22×C10), (C5×D4)⋊15C23, C2.4(C24×C10), (C5×Q8)⋊14C23, Q8⋊4(C22×C10), (D4×C10)⋊68C22, (C22×D4)⋊12C10, C4.13(C23×C10), (C22×C10)⋊4C23, C23⋊2(C22×C10), (C23×C10)⋊6C22, (Q8×C10)⋊60C22, (C2×C10).387C24, (C22×C20)⋊53C22, C22.2(C23×C10), (D4×C2×C10)⋊27C2, C4○D4⋊6(C2×C10), (C2×C4○D4)⋊13C10, (C10×C4○D4)⋊29C2, (C2×D4)⋊17(C2×C10), (C2×C4)⋊2(C22×C10), (C2×Q8)⋊20(C2×C10), (C22×C4)⋊13(C2×C10), (C5×C4○D4)⋊26C22, SmallGroup(320,1632)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1186 in 898 conjugacy classes, 754 normal (8 characteristic)
C1, C2, C2 [×2], C2 [×18], C4 [×12], C22, C22 [×18], C22 [×42], C5, C2×C4 [×42], D4 [×72], Q8 [×8], C23 [×33], C23 [×12], C10, C10 [×2], C10 [×18], C22×C4 [×9], C2×D4 [×90], C2×Q8 [×2], C4○D4 [×48], C24 [×6], C20 [×12], C2×C10, C2×C10 [×18], C2×C10 [×42], C22×D4 [×9], C2×C4○D4 [×6], 2+ (1+4) [×16], C2×C20 [×42], C5×D4 [×72], C5×Q8 [×8], C22×C10 [×33], C22×C10 [×12], C2×2+ (1+4), C22×C20 [×9], D4×C10 [×90], Q8×C10 [×2], C5×C4○D4 [×48], C23×C10 [×6], D4×C2×C10 [×9], C10×C4○D4 [×6], C5×2+ (1+4) [×16], C10×2+ (1+4)
Quotients:
C1, C2 [×31], C22 [×155], C5, C23 [×155], C10 [×31], C24 [×31], C2×C10 [×155], 2+ (1+4) [×2], C25, C22×C10 [×155], C2×2+ (1+4), C23×C10 [×31], C5×2+ (1+4) [×2], C24×C10, C10×2+ (1+4)
Generators and relations
G = < a,b,c,d,e | a10=b4=c2=e2=1, d2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=b2d >
►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 73 30 14)(2 74 21 15)(3 75 22 16)(4 76 23 17)(5 77 24 18)(6 78 25 19)(7 79 26 20)(8 80 27 11)(9 71 28 12)(10 72 29 13)(31 56 49 64)(32 57 50 65)(33 58 41 66)(34 59 42 67)(35 60 43 68)(36 51 44 69)(37 52 45 70)(38 53 46 61)(39 54 47 62)(40 55 48 63)
(1 33)(2 34)(3 35)(4 36)(5 37)(6 38)(7 39)(8 40)(9 31)(10 32)(11 55)(12 56)(13 57)(14 58)(15 59)(16 60)(17 51)(18 52)(19 53)(20 54)(21 42)(22 43)(23 44)(24 45)(25 46)(26 47)(27 48)(28 49)(29 50)(30 41)(61 78)(62 79)(63 80)(64 71)(65 72)(66 73)(67 74)(68 75)(69 76)(70 77)
(1 78 30 19)(2 79 21 20)(3 80 22 11)(4 71 23 12)(5 72 24 13)(6 73 25 14)(7 74 26 15)(8 75 27 16)(9 76 28 17)(10 77 29 18)(31 69 49 51)(32 70 50 52)(33 61 41 53)(34 62 42 54)(35 63 43 55)(36 64 44 56)(37 65 45 57)(38 66 46 58)(39 67 47 59)(40 68 48 60)
(1 53)(2 54)(3 55)(4 56)(5 57)(6 58)(7 59)(8 60)(9 51)(10 52)(11 35)(12 36)(13 37)(14 38)(15 39)(16 40)(17 31)(18 32)(19 33)(20 34)(21 62)(22 63)(23 64)(24 65)(25 66)(26 67)(27 68)(28 69)(29 70)(30 61)(41 78)(42 79)(43 80)(44 71)(45 72)(46 73)(47 74)(48 75)(49 76)(50 77)
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), (1,73,30,14)(2,74,21,15)(3,75,22,16)(4,76,23,17)(5,77,24,18)(6,78,25,19)(7,79,26,20)(8,80,27,11)(9,71,28,12)(10,72,29,13)(31,56,49,64)(32,57,50,65)(33,58,41,66)(34,59,42,67)(35,60,43,68)(36,51,44,69)(37,52,45,70)(38,53,46,61)(39,54,47,62)(40,55,48,63), (1,33)(2,34)(3,35)(4,36)(5,37)(6,38)(7,39)(8,40)(9,31)(10,32)(11,55)(12,56)(13,57)(14,58)(15,59)(16,60)(17,51)(18,52)(19,53)(20,54)(21,42)(22,43)(23,44)(24,45)(25,46)(26,47)(27,48)(28,49)(29,50)(30,41)(61,78)(62,79)(63,80)(64,71)(65,72)(66,73)(67,74)(68,75)(69,76)(70,77), (1,78,30,19)(2,79,21,20)(3,80,22,11)(4,71,23,12)(5,72,24,13)(6,73,25,14)(7,74,26,15)(8,75,27,16)(9,76,28,17)(10,77,29,18)(31,69,49,51)(32,70,50,52)(33,61,41,53)(34,62,42,54)(35,63,43,55)(36,64,44,56)(37,65,45,57)(38,66,46,58)(39,67,47,59)(40,68,48,60), (1,53)(2,54)(3,55)(4,56)(5,57)(6,58)(7,59)(8,60)(9,51)(10,52)(11,35)(12,36)(13,37)(14,38)(15,39)(16,40)(17,31)(18,32)(19,33)(20,34)(21,62)(22,63)(23,64)(24,65)(25,66)(26,67)(27,68)(28,69)(29,70)(30,61)(41,78)(42,79)(43,80)(44,71)(45,72)(46,73)(47,74)(48,75)(49,76)(50,77)>;
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), (1,73,30,14)(2,74,21,15)(3,75,22,16)(4,76,23,17)(5,77,24,18)(6,78,25,19)(7,79,26,20)(8,80,27,11)(9,71,28,12)(10,72,29,13)(31,56,49,64)(32,57,50,65)(33,58,41,66)(34,59,42,67)(35,60,43,68)(36,51,44,69)(37,52,45,70)(38,53,46,61)(39,54,47,62)(40,55,48,63), (1,33)(2,34)(3,35)(4,36)(5,37)(6,38)(7,39)(8,40)(9,31)(10,32)(11,55)(12,56)(13,57)(14,58)(15,59)(16,60)(17,51)(18,52)(19,53)(20,54)(21,42)(22,43)(23,44)(24,45)(25,46)(26,47)(27,48)(28,49)(29,50)(30,41)(61,78)(62,79)(63,80)(64,71)(65,72)(66,73)(67,74)(68,75)(69,76)(70,77), (1,78,30,19)(2,79,21,20)(3,80,22,11)(4,71,23,12)(5,72,24,13)(6,73,25,14)(7,74,26,15)(8,75,27,16)(9,76,28,17)(10,77,29,18)(31,69,49,51)(32,70,50,52)(33,61,41,53)(34,62,42,54)(35,63,43,55)(36,64,44,56)(37,65,45,57)(38,66,46,58)(39,67,47,59)(40,68,48,60), (1,53)(2,54)(3,55)(4,56)(5,57)(6,58)(7,59)(8,60)(9,51)(10,52)(11,35)(12,36)(13,37)(14,38)(15,39)(16,40)(17,31)(18,32)(19,33)(20,34)(21,62)(22,63)(23,64)(24,65)(25,66)(26,67)(27,68)(28,69)(29,70)(30,61)(41,78)(42,79)(43,80)(44,71)(45,72)(46,73)(47,74)(48,75)(49,76)(50,77) );
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)], [(1,73,30,14),(2,74,21,15),(3,75,22,16),(4,76,23,17),(5,77,24,18),(6,78,25,19),(7,79,26,20),(8,80,27,11),(9,71,28,12),(10,72,29,13),(31,56,49,64),(32,57,50,65),(33,58,41,66),(34,59,42,67),(35,60,43,68),(36,51,44,69),(37,52,45,70),(38,53,46,61),(39,54,47,62),(40,55,48,63)], [(1,33),(2,34),(3,35),(4,36),(5,37),(6,38),(7,39),(8,40),(9,31),(10,32),(11,55),(12,56),(13,57),(14,58),(15,59),(16,60),(17,51),(18,52),(19,53),(20,54),(21,42),(22,43),(23,44),(24,45),(25,46),(26,47),(27,48),(28,49),(29,50),(30,41),(61,78),(62,79),(63,80),(64,71),(65,72),(66,73),(67,74),(68,75),(69,76),(70,77)], [(1,78,30,19),(2,79,21,20),(3,80,22,11),(4,71,23,12),(5,72,24,13),(6,73,25,14),(7,74,26,15),(8,75,27,16),(9,76,28,17),(10,77,29,18),(31,69,49,51),(32,70,50,52),(33,61,41,53),(34,62,42,54),(35,63,43,55),(36,64,44,56),(37,65,45,57),(38,66,46,58),(39,67,47,59),(40,68,48,60)], [(1,53),(2,54),(3,55),(4,56),(5,57),(6,58),(7,59),(8,60),(9,51),(10,52),(11,35),(12,36),(13,37),(14,38),(15,39),(16,40),(17,31),(18,32),(19,33),(20,34),(21,62),(22,63),(23,64),(24,65),(25,66),(26,67),(27,68),(28,69),(29,70),(30,61),(41,78),(42,79),(43,80),(44,71),(45,72),(46,73),(47,74),(48,75),(49,76),(50,77)])
Matrix representation ►G ⊆ GL5(𝔽41)
| 40 | 0 | 0 | 0 | 0 |
| 0 | 37 | 0 | 0 | 0 |
| 0 | 0 | 37 | 0 | 0 |
| 0 | 0 | 0 | 37 | 0 |
| 0 | 0 | 0 | 0 | 37 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 39 |
| 0 | 1 | 0 | 40 | 1 |
| 0 | 0 | 1 | 0 | 40 |
| 0 | 1 | 0 | 0 | 1 |
| 40 | 0 | 0 | 0 | 0 |
| 0 | 1 | 2 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 1 | 0 | 40 |
| 0 | 0 | 40 | 40 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 39 |
| 0 | 0 | 0 | 1 | 1 |
| 0 | 40 | 40 | 0 | 40 |
| 0 | 1 | 0 | 0 | 1 |
| 40 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 2 | 0 |
| 0 | 0 | 0 | 40 | 40 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 40 | 40 | 0 |
G:=sub<GL(5,GF(41))| [40,0,0,0,0,0,37,0,0,0,0,0,37,0,0,0,0,0,37,0,0,0,0,0,37],[1,0,0,0,0,0,40,1,0,1,0,0,0,1,0,0,0,40,0,0,0,39,1,40,1],[40,0,0,0,0,0,1,0,0,0,0,2,40,1,40,0,0,0,0,40,0,0,0,40,0],[1,0,0,0,0,0,40,0,40,1,0,0,0,40,0,0,0,1,0,0,0,39,1,40,1],[40,0,0,0,0,0,40,0,0,0,0,0,0,0,40,0,2,40,1,40,0,0,40,0,0] >;
170 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | ··· | 2U | 4A | ··· | 4L | 5A | 5B | 5C | 5D | 10A | ··· | 10L | 10M | ··· | 10CF | 20A | ··· | 20AV |
| order | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 5 | 5 | 5 | 5 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
170 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 4 |
| type | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C5 | C10 | C10 | C10 | 2+ (1+4) | C5×2+ (1+4) |
| kernel | C10×2+ (1+4) | D4×C2×C10 | C10×C4○D4 | C5×2+ (1+4) | C2×2+ (1+4) | C22×D4 | C2×C4○D4 | 2+ (1+4) | C10 | C2 |
| # reps | 1 | 9 | 6 | 16 | 4 | 36 | 24 | 64 | 2 | 8 |
In GAP, Magma, Sage, TeX
C_{10}\times 2_+^{(1+4)} % in TeX
G:=Group("C10xES+(2,2)"); // GroupNames label
G:=SmallGroup(320,1632);
// by ID
G=gap.SmallGroup(320,1632);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-5,-2,2269,1731,4707]);
// Polycyclic
G:=Group<a,b,c,d,e|a^10=b^4=c^2=e^2=1,d^2=b^2,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,c*d=d*c,c*e=e*c,e*d*e=b^2*d>;
// generators/relations