direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×D4⋊6D10, C24⋊9D10, D20⋊9C23, C10.7C25, C20.42C24, D10.3C24, Dic10⋊9C23, C10⋊12+ (1+4), Dic5.3C24, (C2×D4)⋊46D10, (C2×C20)⋊5C23, (C4×D5)⋊1C23, D4⋊7(C22×D5), (C5×D4)⋊8C23, C5⋊D4⋊3C23, C2.8(D5×C24), (C22×D4)⋊13D5, (D4×D5)⋊11C22, (C22×C4)⋊32D10, C4.42(C23×D5), C23⋊4(C22×D5), C5⋊1(C2×2+ (1+4)), C4○D20⋊22C22, (D4×C10)⋊51C22, (C2×D20)⋊61C22, (C22×C10)⋊7C23, (C2×Dic5)⋊5C23, (C22×D5)⋊4C23, D4⋊2D5⋊12C22, C22.8(C23×D5), (C2×C10).327C24, (C23×C10)⋊16C22, (C22×C20)⋊26C22, (C23×D5)⋊17C22, (C2×Dic10)⋊72C22, (C22×Dic5)⋊38C22, (C2×D4×D5)⋊27C2, (D4×C2×C10)⋊11C2, (C2×C4×D5)⋊33C22, (C2×C4)⋊5(C22×D5), (C2×C4○D20)⋊34C2, (C2×D4⋊2D5)⋊29C2, (C2×C5⋊D4)⋊52C22, (C22×C5⋊D4)⋊21C2, SmallGroup(320,1614)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 2926 in 898 conjugacy classes, 447 normal (13 characteristic)
C1, C2, C2 [×2], C2 [×18], C4 [×4], C4 [×8], C22, C22 [×10], C22 [×50], C5, C2×C4 [×6], C2×C4 [×36], D4 [×16], D4 [×56], Q8 [×8], C23, C23 [×12], C23 [×32], D5 [×8], C10, C10 [×2], C10 [×10], C22×C4, C22×C4 [×8], C2×D4 [×12], C2×D4 [×78], C2×Q8 [×2], C4○D4 [×48], C24 [×2], C24 [×4], Dic5 [×8], C20 [×4], D10 [×8], D10 [×24], C2×C10, C2×C10 [×10], C2×C10 [×18], C22×D4, C22×D4 [×8], C2×C4○D4 [×6], 2+ (1+4) [×16], Dic10 [×8], C4×D5 [×16], D20 [×8], C2×Dic5 [×20], C5⋊D4 [×48], C2×C20 [×6], C5×D4 [×16], C22×D5 [×20], C22×D5 [×8], C22×C10, C22×C10 [×12], C22×C10 [×4], C2×2+ (1+4), C2×Dic10 [×2], C2×C4×D5 [×4], C2×D20 [×2], C4○D20 [×16], D4×D5 [×32], D4⋊2D5 [×32], C22×Dic5 [×4], C2×C5⋊D4 [×44], C22×C20, D4×C10 [×12], C23×D5 [×4], C23×C10 [×2], C2×C4○D20 [×2], C2×D4×D5 [×4], C2×D4⋊2D5 [×4], D4⋊6D10 [×16], C22×C5⋊D4 [×4], D4×C2×C10, C2×D4⋊6D10
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⋊6D10 [×2], D5×C24, C2×D4⋊6D10
Generators and relations
G = < a,b,c,d,e | a2=b4=c2=d10=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=dbd-1=b-1, be=eb, dcd-1=ece=b2c, ede=d-1 >
►On 80 points
(1 59)(2 60)(3 51)(4 52)(5 53)(6 54)(7 55)(8 56)(9 57)(10 58)(11 68)(12 69)(13 70)(14 61)(15 62)(16 63)(17 64)(18 65)(19 66)(20 67)(21 45)(22 46)(23 47)(24 48)(25 49)(26 50)(27 41)(28 42)(29 43)(30 44)(31 71)(32 72)(33 73)(34 74)(35 75)(36 76)(37 77)(38 78)(39 79)(40 80)
(1 54 37 72)(2 73 38 55)(3 56 39 74)(4 75 40 57)(5 58 31 76)(6 77 32 59)(7 60 33 78)(8 79 34 51)(9 52 35 80)(10 71 36 53)(11 63 47 28)(12 29 48 64)(13 65 49 30)(14 21 50 66)(15 67 41 22)(16 23 42 68)(17 69 43 24)(18 25 44 70)(19 61 45 26)(20 27 46 62)
(1 66)(2 22)(3 68)(4 24)(5 70)(6 26)(7 62)(8 28)(9 64)(10 30)(11 51)(12 80)(13 53)(14 72)(15 55)(16 74)(17 57)(18 76)(19 59)(20 78)(21 37)(23 39)(25 31)(27 33)(29 35)(32 61)(34 63)(36 65)(38 67)(40 69)(41 73)(42 56)(43 75)(44 58)(45 77)(46 60)(47 79)(48 52)(49 71)(50 54)
(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 53)(2 52)(3 51)(4 60)(5 59)(6 58)(7 57)(8 56)(9 55)(10 54)(11 23)(12 22)(13 21)(14 30)(15 29)(16 28)(17 27)(18 26)(19 25)(20 24)(31 77)(32 76)(33 75)(34 74)(35 73)(36 72)(37 71)(38 80)(39 79)(40 78)(41 64)(42 63)(43 62)(44 61)(45 70)(46 69)(47 68)(48 67)(49 66)(50 65)
G:=sub<Sym(80)| (1,59)(2,60)(3,51)(4,52)(5,53)(6,54)(7,55)(8,56)(9,57)(10,58)(11,68)(12,69)(13,70)(14,61)(15,62)(16,63)(17,64)(18,65)(19,66)(20,67)(21,45)(22,46)(23,47)(24,48)(25,49)(26,50)(27,41)(28,42)(29,43)(30,44)(31,71)(32,72)(33,73)(34,74)(35,75)(36,76)(37,77)(38,78)(39,79)(40,80), (1,54,37,72)(2,73,38,55)(3,56,39,74)(4,75,40,57)(5,58,31,76)(6,77,32,59)(7,60,33,78)(8,79,34,51)(9,52,35,80)(10,71,36,53)(11,63,47,28)(12,29,48,64)(13,65,49,30)(14,21,50,66)(15,67,41,22)(16,23,42,68)(17,69,43,24)(18,25,44,70)(19,61,45,26)(20,27,46,62), (1,66)(2,22)(3,68)(4,24)(5,70)(6,26)(7,62)(8,28)(9,64)(10,30)(11,51)(12,80)(13,53)(14,72)(15,55)(16,74)(17,57)(18,76)(19,59)(20,78)(21,37)(23,39)(25,31)(27,33)(29,35)(32,61)(34,63)(36,65)(38,67)(40,69)(41,73)(42,56)(43,75)(44,58)(45,77)(46,60)(47,79)(48,52)(49,71)(50,54), (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), (1,53)(2,52)(3,51)(4,60)(5,59)(6,58)(7,57)(8,56)(9,55)(10,54)(11,23)(12,22)(13,21)(14,30)(15,29)(16,28)(17,27)(18,26)(19,25)(20,24)(31,77)(32,76)(33,75)(34,74)(35,73)(36,72)(37,71)(38,80)(39,79)(40,78)(41,64)(42,63)(43,62)(44,61)(45,70)(46,69)(47,68)(48,67)(49,66)(50,65)>;
G:=Group( (1,59)(2,60)(3,51)(4,52)(5,53)(6,54)(7,55)(8,56)(9,57)(10,58)(11,68)(12,69)(13,70)(14,61)(15,62)(16,63)(17,64)(18,65)(19,66)(20,67)(21,45)(22,46)(23,47)(24,48)(25,49)(26,50)(27,41)(28,42)(29,43)(30,44)(31,71)(32,72)(33,73)(34,74)(35,75)(36,76)(37,77)(38,78)(39,79)(40,80), (1,54,37,72)(2,73,38,55)(3,56,39,74)(4,75,40,57)(5,58,31,76)(6,77,32,59)(7,60,33,78)(8,79,34,51)(9,52,35,80)(10,71,36,53)(11,63,47,28)(12,29,48,64)(13,65,49,30)(14,21,50,66)(15,67,41,22)(16,23,42,68)(17,69,43,24)(18,25,44,70)(19,61,45,26)(20,27,46,62), (1,66)(2,22)(3,68)(4,24)(5,70)(6,26)(7,62)(8,28)(9,64)(10,30)(11,51)(12,80)(13,53)(14,72)(15,55)(16,74)(17,57)(18,76)(19,59)(20,78)(21,37)(23,39)(25,31)(27,33)(29,35)(32,61)(34,63)(36,65)(38,67)(40,69)(41,73)(42,56)(43,75)(44,58)(45,77)(46,60)(47,79)(48,52)(49,71)(50,54), (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), (1,53)(2,52)(3,51)(4,60)(5,59)(6,58)(7,57)(8,56)(9,55)(10,54)(11,23)(12,22)(13,21)(14,30)(15,29)(16,28)(17,27)(18,26)(19,25)(20,24)(31,77)(32,76)(33,75)(34,74)(35,73)(36,72)(37,71)(38,80)(39,79)(40,78)(41,64)(42,63)(43,62)(44,61)(45,70)(46,69)(47,68)(48,67)(49,66)(50,65) );
G=PermutationGroup([(1,59),(2,60),(3,51),(4,52),(5,53),(6,54),(7,55),(8,56),(9,57),(10,58),(11,68),(12,69),(13,70),(14,61),(15,62),(16,63),(17,64),(18,65),(19,66),(20,67),(21,45),(22,46),(23,47),(24,48),(25,49),(26,50),(27,41),(28,42),(29,43),(30,44),(31,71),(32,72),(33,73),(34,74),(35,75),(36,76),(37,77),(38,78),(39,79),(40,80)], [(1,54,37,72),(2,73,38,55),(3,56,39,74),(4,75,40,57),(5,58,31,76),(6,77,32,59),(7,60,33,78),(8,79,34,51),(9,52,35,80),(10,71,36,53),(11,63,47,28),(12,29,48,64),(13,65,49,30),(14,21,50,66),(15,67,41,22),(16,23,42,68),(17,69,43,24),(18,25,44,70),(19,61,45,26),(20,27,46,62)], [(1,66),(2,22),(3,68),(4,24),(5,70),(6,26),(7,62),(8,28),(9,64),(10,30),(11,51),(12,80),(13,53),(14,72),(15,55),(16,74),(17,57),(18,76),(19,59),(20,78),(21,37),(23,39),(25,31),(27,33),(29,35),(32,61),(34,63),(36,65),(38,67),(40,69),(41,73),(42,56),(43,75),(44,58),(45,77),(46,60),(47,79),(48,52),(49,71),(50,54)], [(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)], [(1,53),(2,52),(3,51),(4,60),(5,59),(6,58),(7,57),(8,56),(9,55),(10,54),(11,23),(12,22),(13,21),(14,30),(15,29),(16,28),(17,27),(18,26),(19,25),(20,24),(31,77),(32,76),(33,75),(34,74),(35,73),(36,72),(37,71),(38,80),(39,79),(40,78),(41,64),(42,63),(43,62),(44,61),(45,70),(46,69),(47,68),(48,67),(49,66),(50,65)])
Matrix representation ►G ⊆ GL6(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 9 | 0 |
| 0 | 0 | 0 | 1 | 0 | 9 |
| 0 | 0 | 18 | 0 | 40 | 0 |
| 0 | 0 | 0 | 18 | 0 | 40 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 23 | 0 | 1 |
| 0 | 0 | 23 | 0 | 1 | 0 |
| 6 | 35 | 0 | 0 | 0 | 0 |
| 6 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 9 | 0 |
| 0 | 0 | 0 | 1 | 0 | 9 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,18,0,0,0,0,1,0,18,0,0,9,0,40,0,0,0,0,9,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,40,0,23,0,0,40,0,23,0,0,0,0,0,0,1,0,0,0,0,1,0],[6,6,0,0,0,0,35,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,9,0,40,0,0,0,0,9,0,40],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,1] >;
74 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | ··· | 2M | 2N | ··· | 2U | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 5A | 5B | 10A | ··· | 10N | 10O | ··· | 10AD | 20A | ··· | 20H |
| order | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 10 | ··· | 10 | 2 | 2 | 2 | 2 | 10 | ··· | 10 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
74 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | D5 | D10 | D10 | D10 | 2+ (1+4) | D4⋊6D10 |
| kernel | C2×D4⋊6D10 | C2×C4○D20 | C2×D4×D5 | C2×D4⋊2D5 | D4⋊6D10 | C22×C5⋊D4 | D4×C2×C10 | C22×D4 | C22×C4 | C2×D4 | C24 | C10 | C2 |
| # reps | 1 | 2 | 4 | 4 | 16 | 4 | 1 | 2 | 2 | 24 | 4 | 2 | 8 |
In GAP, Magma, Sage, TeX
C_2\times D_4\rtimes_6D_{10} % in TeX
G:=Group("C2xD4:6D10"); // GroupNames label
G:=SmallGroup(320,1614);
// by ID
G=gap.SmallGroup(320,1614);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,297,1684,12550]);
// 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=d*b*d^-1=b^-1,b*e=e*b,d*c*d^-1=e*c*e=b^2*c,e*d*e=d^-1>;
// generators/relations