direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×D4⋊8D10, D20⋊12C23, C20.47C24, C10.12C25, D10.6C24, C10⋊22+ (1+4), Dic5.7C24, Dic10⋊14C23, C4○D4⋊17D10, (C2×D4)⋊50D10, (C2×C20)⋊7C23, (C2×Q8)⋊39D10, (C4×D5)⋊2C23, D4⋊9(C22×D5), C5⋊D4⋊5C23, (C5×Q8)⋊9C23, Q8⋊8(C22×D5), (C5×D4)⋊10C23, (D4×D5)⋊12C22, (C22×C4)⋊33D10, (C2×C10).3C24, C4.62(C23×D5), C2.13(D5×C24), C5⋊2(C2×2+ (1+4)), C4○D20⋊25C22, (C2×D20)⋊64C22, (D4×C10)⋊53C22, (C22×D20)⋊24C2, (Q8×C10)⋊46C22, (C22×D5)⋊5C23, (C22×C20)⋊28C22, Q8⋊2D5⋊13C22, (C23×D5)⋊18C22, C22.56(C23×D5), (C2×Dic10)⋊78C22, C23.214(C22×D5), (C22×C10).248C23, (C2×Dic5).309C23, (C2×D4×D5)⋊28C2, (C2×C4○D4)⋊13D5, (C2×C4×D5)⋊34C22, (C2×C4)⋊6(C22×D5), (C10×C4○D4)⋊14C2, (C2×C4○D20)⋊37C2, (C2×Q8⋊2D5)⋊21C2, (C5×C4○D4)⋊20C22, (C2×C5⋊D4)⋊54C22, SmallGroup(320,1619)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 3182 in 898 conjugacy classes, 447 normal (14 characteristic)
C1, C2, C2 [×2], C2 [×18], C4 [×8], C4 [×4], C22, C22 [×6], C22 [×54], C5, C2×C4, C2×C4 [×15], C2×C4 [×26], D4 [×12], D4 [×60], Q8 [×4], Q8 [×4], C23 [×3], C23 [×42], D5 [×12], C10, C10 [×2], C10 [×6], C22×C4 [×3], C22×C4 [×6], C2×D4 [×3], C2×D4 [×87], C2×Q8, C2×Q8, C4○D4 [×8], C4○D4 [×40], C24 [×6], Dic5 [×4], C20 [×8], D10 [×12], D10 [×36], C2×C10, C2×C10 [×6], C2×C10 [×6], C22×D4 [×9], C2×C4○D4, C2×C4○D4 [×5], 2+ (1+4) [×16], Dic10 [×4], C4×D5 [×24], D20 [×36], C2×Dic5 [×2], C5⋊D4 [×24], C2×C20, C2×C20 [×15], C5×D4 [×12], C5×Q8 [×4], C22×D5 [×30], C22×D5 [×12], C22×C10 [×3], C2×2+ (1+4), C2×Dic10, C2×C4×D5 [×6], C2×D20 [×33], C4○D20 [×24], D4×D5 [×48], Q8⋊2D5 [×16], C2×C5⋊D4 [×6], C22×C20 [×3], D4×C10 [×3], Q8×C10, C5×C4○D4 [×8], C23×D5 [×6], C22×D20 [×3], C2×C4○D20 [×3], C2×D4×D5 [×6], C2×Q8⋊2D5 [×2], D4⋊8D10 [×16], C10×C4○D4, C2×D4⋊8D10
Quotients:
C1, C2 [×31], C22 [×155], C23 [×155], D5, C24 [×31], D10 [×15], 2+ (1+4) [×2], C25, C22×D5 [×35], C2×2+ (1+4), C23×D5 [×15], D4⋊8D10 [×2], D5×C24, C2×D4⋊8D10
Generators and relations
G = < a,b,c,d,e | a2=b4=c2=d10=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe=b-1, bd=db, dcd-1=b2c, ce=ec, ede=d-1 >
►On 80 points
(1 66)(2 67)(3 68)(4 69)(5 70)(6 61)(7 62)(8 63)(9 64)(10 65)(11 46)(12 47)(13 48)(14 49)(15 50)(16 41)(17 42)(18 43)(19 44)(20 45)(21 71)(22 72)(23 73)(24 74)(25 75)(26 76)(27 77)(28 78)(29 79)(30 80)(31 60)(32 51)(33 52)(34 53)(35 54)(36 55)(37 56)(38 57)(39 58)(40 59)
(1 20 60 22)(2 11 51 23)(3 12 52 24)(4 13 53 25)(5 14 54 26)(6 15 55 27)(7 16 56 28)(8 17 57 29)(9 18 58 30)(10 19 59 21)(31 72 66 45)(32 73 67 46)(33 74 68 47)(34 75 69 48)(35 76 70 49)(36 77 61 50)(37 78 62 41)(38 79 63 42)(39 80 64 43)(40 71 65 44)
(1 66)(2 32)(3 68)(4 34)(5 70)(6 36)(7 62)(8 38)(9 64)(10 40)(11 46)(12 74)(13 48)(14 76)(15 50)(16 78)(17 42)(18 80)(19 44)(20 72)(21 71)(22 45)(23 73)(24 47)(25 75)(26 49)(27 77)(28 41)(29 79)(30 43)(31 60)(33 52)(35 54)(37 56)(39 58)(51 67)(53 69)(55 61)(57 63)(59 65)
(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 5)(2 4)(6 10)(7 9)(11 25)(12 24)(13 23)(14 22)(15 21)(16 30)(17 29)(18 28)(19 27)(20 26)(31 35)(32 34)(36 40)(37 39)(41 80)(42 79)(43 78)(44 77)(45 76)(46 75)(47 74)(48 73)(49 72)(50 71)(51 53)(54 60)(55 59)(56 58)(61 65)(62 64)(66 70)(67 69)
G:=sub<Sym(80)| (1,66)(2,67)(3,68)(4,69)(5,70)(6,61)(7,62)(8,63)(9,64)(10,65)(11,46)(12,47)(13,48)(14,49)(15,50)(16,41)(17,42)(18,43)(19,44)(20,45)(21,71)(22,72)(23,73)(24,74)(25,75)(26,76)(27,77)(28,78)(29,79)(30,80)(31,60)(32,51)(33,52)(34,53)(35,54)(36,55)(37,56)(38,57)(39,58)(40,59), (1,20,60,22)(2,11,51,23)(3,12,52,24)(4,13,53,25)(5,14,54,26)(6,15,55,27)(7,16,56,28)(8,17,57,29)(9,18,58,30)(10,19,59,21)(31,72,66,45)(32,73,67,46)(33,74,68,47)(34,75,69,48)(35,76,70,49)(36,77,61,50)(37,78,62,41)(38,79,63,42)(39,80,64,43)(40,71,65,44), (1,66)(2,32)(3,68)(4,34)(5,70)(6,36)(7,62)(8,38)(9,64)(10,40)(11,46)(12,74)(13,48)(14,76)(15,50)(16,78)(17,42)(18,80)(19,44)(20,72)(21,71)(22,45)(23,73)(24,47)(25,75)(26,49)(27,77)(28,41)(29,79)(30,43)(31,60)(33,52)(35,54)(37,56)(39,58)(51,67)(53,69)(55,61)(57,63)(59,65), (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), (1,5)(2,4)(6,10)(7,9)(11,25)(12,24)(13,23)(14,22)(15,21)(16,30)(17,29)(18,28)(19,27)(20,26)(31,35)(32,34)(36,40)(37,39)(41,80)(42,79)(43,78)(44,77)(45,76)(46,75)(47,74)(48,73)(49,72)(50,71)(51,53)(54,60)(55,59)(56,58)(61,65)(62,64)(66,70)(67,69)>;
G:=Group( (1,66)(2,67)(3,68)(4,69)(5,70)(6,61)(7,62)(8,63)(9,64)(10,65)(11,46)(12,47)(13,48)(14,49)(15,50)(16,41)(17,42)(18,43)(19,44)(20,45)(21,71)(22,72)(23,73)(24,74)(25,75)(26,76)(27,77)(28,78)(29,79)(30,80)(31,60)(32,51)(33,52)(34,53)(35,54)(36,55)(37,56)(38,57)(39,58)(40,59), (1,20,60,22)(2,11,51,23)(3,12,52,24)(4,13,53,25)(5,14,54,26)(6,15,55,27)(7,16,56,28)(8,17,57,29)(9,18,58,30)(10,19,59,21)(31,72,66,45)(32,73,67,46)(33,74,68,47)(34,75,69,48)(35,76,70,49)(36,77,61,50)(37,78,62,41)(38,79,63,42)(39,80,64,43)(40,71,65,44), (1,66)(2,32)(3,68)(4,34)(5,70)(6,36)(7,62)(8,38)(9,64)(10,40)(11,46)(12,74)(13,48)(14,76)(15,50)(16,78)(17,42)(18,80)(19,44)(20,72)(21,71)(22,45)(23,73)(24,47)(25,75)(26,49)(27,77)(28,41)(29,79)(30,43)(31,60)(33,52)(35,54)(37,56)(39,58)(51,67)(53,69)(55,61)(57,63)(59,65), (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), (1,5)(2,4)(6,10)(7,9)(11,25)(12,24)(13,23)(14,22)(15,21)(16,30)(17,29)(18,28)(19,27)(20,26)(31,35)(32,34)(36,40)(37,39)(41,80)(42,79)(43,78)(44,77)(45,76)(46,75)(47,74)(48,73)(49,72)(50,71)(51,53)(54,60)(55,59)(56,58)(61,65)(62,64)(66,70)(67,69) );
G=PermutationGroup([(1,66),(2,67),(3,68),(4,69),(5,70),(6,61),(7,62),(8,63),(9,64),(10,65),(11,46),(12,47),(13,48),(14,49),(15,50),(16,41),(17,42),(18,43),(19,44),(20,45),(21,71),(22,72),(23,73),(24,74),(25,75),(26,76),(27,77),(28,78),(29,79),(30,80),(31,60),(32,51),(33,52),(34,53),(35,54),(36,55),(37,56),(38,57),(39,58),(40,59)], [(1,20,60,22),(2,11,51,23),(3,12,52,24),(4,13,53,25),(5,14,54,26),(6,15,55,27),(7,16,56,28),(8,17,57,29),(9,18,58,30),(10,19,59,21),(31,72,66,45),(32,73,67,46),(33,74,68,47),(34,75,69,48),(35,76,70,49),(36,77,61,50),(37,78,62,41),(38,79,63,42),(39,80,64,43),(40,71,65,44)], [(1,66),(2,32),(3,68),(4,34),(5,70),(6,36),(7,62),(8,38),(9,64),(10,40),(11,46),(12,74),(13,48),(14,76),(15,50),(16,78),(17,42),(18,80),(19,44),(20,72),(21,71),(22,45),(23,73),(24,47),(25,75),(26,49),(27,77),(28,41),(29,79),(30,43),(31,60),(33,52),(35,54),(37,56),(39,58),(51,67),(53,69),(55,61),(57,63),(59,65)], [(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)], [(1,5),(2,4),(6,10),(7,9),(11,25),(12,24),(13,23),(14,22),(15,21),(16,30),(17,29),(18,28),(19,27),(20,26),(31,35),(32,34),(36,40),(37,39),(41,80),(42,79),(43,78),(44,77),(45,76),(46,75),(47,74),(48,73),(49,72),(50,71),(51,53),(54,60),(55,59),(56,58),(61,65),(62,64),(66,70),(67,69)])
Matrix representation ►G ⊆ GL6(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 7 | 34 | 0 | 0 | 0 | 0 |
| 7 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 16 |
| 0 | 0 | 0 | 0 | 25 | 16 |
| 0 | 0 | 39 | 25 | 0 | 0 |
| 0 | 0 | 16 | 25 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 6 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 35 |
| 0 | 0 | 0 | 0 | 0 | 40 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,40,0,0,0,0,0,0,40,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[7,7,0,0,0,0,34,40,0,0,0,0,0,0,0,0,39,16,0,0,0,0,25,25,0,0,2,25,0,0,0,0,16,16,0,0],[0,40,0,0,0,0,40,0,0,0,0,0,0,0,40,0,0,0,0,0,6,1,0,0,0,0,0,0,1,0,0,0,0,0,35,40] >;
74 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | ··· | 2I | 2J | ··· | 2U | 4A | ··· | 4H | 4I | 4J | 4K | 4L | 5A | 5B | 10A | ··· | 10F | 10G | ··· | 10R | 20A | ··· | 20H | 20I | ··· | 20T |
| order | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 10 | ··· | 10 | 2 | ··· | 2 | 10 | 10 | 10 | 10 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
74 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | D5 | D10 | D10 | D10 | D10 | 2+ (1+4) | D4⋊8D10 |
| kernel | C2×D4⋊8D10 | C22×D20 | C2×C4○D20 | C2×D4×D5 | C2×Q8⋊2D5 | D4⋊8D10 | C10×C4○D4 | C2×C4○D4 | C22×C4 | C2×D4 | C2×Q8 | C4○D4 | C10 | C2 |
| # reps | 1 | 3 | 3 | 6 | 2 | 16 | 1 | 2 | 6 | 6 | 2 | 16 | 2 | 8 |
In GAP, Magma, Sage, TeX
C_2\times D_4\rtimes_8D_{10} % in TeX
G:=Group("C2xD4:8D10"); // GroupNames label
G:=SmallGroup(320,1619);
// by ID
G=gap.SmallGroup(320,1619);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,297,1684,102,12550]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^4=c^2=d^10=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=e*b*e=b^-1,b*d=d*b,d*c*d^-1=b^2*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations