metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C24.35D10, C10.292+ (1+4), C22≀C2⋊6D5, C20⋊2D4⋊14C2, (D4×Dic5)⋊13C2, (C2×D4).86D10, C24⋊2D5⋊8C2, C22⋊C4.2D10, Dic5⋊4D4⋊4C2, Dic5⋊D4⋊5C2, (C2×C20).31C23, C4⋊Dic5⋊27C22, C20.17D4⋊12C2, (C2×C10).137C24, C5⋊3(C22.32C24), (C4×Dic5)⋊17C22, D10.12D4⋊14C2, C2.31(D4⋊6D10), C23.D5⋊17C22, D10⋊C4⋊14C22, Dic5.5D4⋊14C2, (C2×Dic10)⋊22C22, (D4×C10).111C22, C23.18D10⋊5C2, C10.D4⋊12C22, C23.D10⋊12C2, (C23×C10).70C22, (C2×Dic5).62C23, (C22×D5).56C23, C23.177(C22×D5), C22.158(C23×D5), Dic5.14D4⋊14C2, C22.10(D4⋊2D5), (C22×C10).182C23, (C22×Dic5)⋊16C22, (C2×C4×D5)⋊10C22, (C5×C22≀C2)⋊8C2, C10.78(C2×C4○D4), C2.29(C2×D4⋊2D5), (C2×C5⋊D4)⋊10C22, (C2×C23.D5)⋊21C2, (C2×C4).31(C22×D5), (C2×C10).44(C4○D4), (C5×C22⋊C4).3C22, SmallGroup(320,1265)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 878 in 250 conjugacy classes, 95 normal (91 characteristic)
C1, C2 [×3], C2 [×6], C4 [×10], C22, C22 [×2], C22 [×18], C5, C2×C4 [×3], C2×C4 [×11], D4 [×9], Q8, C23 [×4], C23 [×5], D5, C10 [×3], C10 [×5], C42 [×2], C22⋊C4 [×3], C22⋊C4 [×11], C4⋊C4 [×6], C22×C4 [×4], C2×D4 [×3], C2×D4 [×4], C2×Q8, C24, Dic5 [×7], C20 [×3], D10 [×3], C2×C10, C2×C10 [×2], C2×C10 [×15], C2×C22⋊C4, C4×D4 [×2], C22≀C2, C22≀C2, C4⋊D4 [×3], C22⋊Q8, C22.D4 [×2], C4.4D4 [×2], C42⋊2C2 [×2], Dic10, C4×D5, C2×Dic5 [×7], C2×Dic5 [×3], C5⋊D4 [×5], C2×C20 [×3], C5×D4 [×4], C22×D5, C22×C10 [×4], C22×C10 [×4], C22.32C24, C4×Dic5 [×2], C10.D4 [×4], C4⋊Dic5 [×2], D10⋊C4 [×2], C23.D5 [×9], C5×C22⋊C4 [×3], C2×Dic10, C2×C4×D5, C22×Dic5 [×3], C2×C5⋊D4 [×4], D4×C10 [×3], C23×C10, Dic5.14D4, C23.D10 [×2], Dic5⋊4D4, D10.12D4, Dic5.5D4, D4×Dic5, C23.18D10, C20.17D4, C20⋊2D4, Dic5⋊D4 [×2], C2×C23.D5, C24⋊2D5, C5×C22≀C2, C24.35D10
Quotients:
C1, C2 [×15], C22 [×35], C23 [×15], D5, C4○D4 [×2], C24, D10 [×7], C2×C4○D4, 2+ (1+4) [×2], C22×D5 [×7], C22.32C24, D4⋊2D5 [×2], C23×D5, C2×D4⋊2D5, D4⋊6D10 [×2], C24.35D10
Generators and relations
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e10=f2=d, ab=ba, eae-1=ac=ca, ad=da, faf-1=acd, bc=cb, ebe-1=bd=db, fbf-1=bcd, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e9 >
►On 80 points
(2 59)(4 41)(6 43)(8 45)(10 47)(12 49)(14 51)(16 53)(18 55)(20 57)(21 62)(22 32)(23 64)(24 34)(25 66)(26 36)(27 68)(28 38)(29 70)(30 40)(31 72)(33 74)(35 76)(37 78)(39 80)(61 71)(63 73)(65 75)(67 77)(69 79)
(1 11)(3 13)(5 15)(7 17)(9 19)(21 72)(22 63)(23 74)(24 65)(25 76)(26 67)(27 78)(28 69)(29 80)(30 71)(31 62)(32 73)(33 64)(34 75)(35 66)(36 77)(37 68)(38 79)(39 70)(40 61)(42 52)(44 54)(46 56)(48 58)(50 60)
(1 58)(2 59)(3 60)(4 41)(5 42)(6 43)(7 44)(8 45)(9 46)(10 47)(11 48)(12 49)(13 50)(14 51)(15 52)(16 53)(17 54)(18 55)(19 56)(20 57)(21 72)(22 73)(23 74)(24 75)(25 76)(26 77)(27 78)(28 79)(29 80)(30 61)(31 62)(32 63)(33 64)(34 65)(35 66)(36 67)(37 68)(38 69)(39 70)(40 71)
(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 17)(8 18)(9 19)(10 20)(21 31)(22 32)(23 33)(24 34)(25 35)(26 36)(27 37)(28 38)(29 39)(30 40)(41 51)(42 52)(43 53)(44 54)(45 55)(46 56)(47 57)(48 58)(49 59)(50 60)(61 71)(62 72)(63 73)(64 74)(65 75)(66 76)(67 77)(68 78)(69 79)(70 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 23 11 33)(2 32 12 22)(3 21 13 31)(4 30 14 40)(5 39 15 29)(6 28 16 38)(7 37 17 27)(8 26 18 36)(9 35 19 25)(10 24 20 34)(41 61 51 71)(42 70 52 80)(43 79 53 69)(44 68 54 78)(45 77 55 67)(46 66 56 76)(47 75 57 65)(48 64 58 74)(49 73 59 63)(50 62 60 72)
G:=sub<Sym(80)| (2,59)(4,41)(6,43)(8,45)(10,47)(12,49)(14,51)(16,53)(18,55)(20,57)(21,62)(22,32)(23,64)(24,34)(25,66)(26,36)(27,68)(28,38)(29,70)(30,40)(31,72)(33,74)(35,76)(37,78)(39,80)(61,71)(63,73)(65,75)(67,77)(69,79), (1,11)(3,13)(5,15)(7,17)(9,19)(21,72)(22,63)(23,74)(24,65)(25,76)(26,67)(27,78)(28,69)(29,80)(30,71)(31,62)(32,73)(33,64)(34,75)(35,66)(36,77)(37,68)(38,79)(39,70)(40,61)(42,52)(44,54)(46,56)(48,58)(50,60), (1,58)(2,59)(3,60)(4,41)(5,42)(6,43)(7,44)(8,45)(9,46)(10,47)(11,48)(12,49)(13,50)(14,51)(15,52)(16,53)(17,54)(18,55)(19,56)(20,57)(21,72)(22,73)(23,74)(24,75)(25,76)(26,77)(27,78)(28,79)(29,80)(30,61)(31,62)(32,63)(33,64)(34,65)(35,66)(36,67)(37,68)(38,69)(39,70)(40,71), (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,19)(10,20)(21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40)(41,51)(42,52)(43,53)(44,54)(45,55)(46,56)(47,57)(48,58)(49,59)(50,60)(61,71)(62,72)(63,73)(64,74)(65,75)(66,76)(67,77)(68,78)(69,79)(70,80), (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), (1,23,11,33)(2,32,12,22)(3,21,13,31)(4,30,14,40)(5,39,15,29)(6,28,16,38)(7,37,17,27)(8,26,18,36)(9,35,19,25)(10,24,20,34)(41,61,51,71)(42,70,52,80)(43,79,53,69)(44,68,54,78)(45,77,55,67)(46,66,56,76)(47,75,57,65)(48,64,58,74)(49,73,59,63)(50,62,60,72)>;
G:=Group( (2,59)(4,41)(6,43)(8,45)(10,47)(12,49)(14,51)(16,53)(18,55)(20,57)(21,62)(22,32)(23,64)(24,34)(25,66)(26,36)(27,68)(28,38)(29,70)(30,40)(31,72)(33,74)(35,76)(37,78)(39,80)(61,71)(63,73)(65,75)(67,77)(69,79), (1,11)(3,13)(5,15)(7,17)(9,19)(21,72)(22,63)(23,74)(24,65)(25,76)(26,67)(27,78)(28,69)(29,80)(30,71)(31,62)(32,73)(33,64)(34,75)(35,66)(36,77)(37,68)(38,79)(39,70)(40,61)(42,52)(44,54)(46,56)(48,58)(50,60), (1,58)(2,59)(3,60)(4,41)(5,42)(6,43)(7,44)(8,45)(9,46)(10,47)(11,48)(12,49)(13,50)(14,51)(15,52)(16,53)(17,54)(18,55)(19,56)(20,57)(21,72)(22,73)(23,74)(24,75)(25,76)(26,77)(27,78)(28,79)(29,80)(30,61)(31,62)(32,63)(33,64)(34,65)(35,66)(36,67)(37,68)(38,69)(39,70)(40,71), (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,19)(10,20)(21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40)(41,51)(42,52)(43,53)(44,54)(45,55)(46,56)(47,57)(48,58)(49,59)(50,60)(61,71)(62,72)(63,73)(64,74)(65,75)(66,76)(67,77)(68,78)(69,79)(70,80), (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), (1,23,11,33)(2,32,12,22)(3,21,13,31)(4,30,14,40)(5,39,15,29)(6,28,16,38)(7,37,17,27)(8,26,18,36)(9,35,19,25)(10,24,20,34)(41,61,51,71)(42,70,52,80)(43,79,53,69)(44,68,54,78)(45,77,55,67)(46,66,56,76)(47,75,57,65)(48,64,58,74)(49,73,59,63)(50,62,60,72) );
G=PermutationGroup([(2,59),(4,41),(6,43),(8,45),(10,47),(12,49),(14,51),(16,53),(18,55),(20,57),(21,62),(22,32),(23,64),(24,34),(25,66),(26,36),(27,68),(28,38),(29,70),(30,40),(31,72),(33,74),(35,76),(37,78),(39,80),(61,71),(63,73),(65,75),(67,77),(69,79)], [(1,11),(3,13),(5,15),(7,17),(9,19),(21,72),(22,63),(23,74),(24,65),(25,76),(26,67),(27,78),(28,69),(29,80),(30,71),(31,62),(32,73),(33,64),(34,75),(35,66),(36,77),(37,68),(38,79),(39,70),(40,61),(42,52),(44,54),(46,56),(48,58),(50,60)], [(1,58),(2,59),(3,60),(4,41),(5,42),(6,43),(7,44),(8,45),(9,46),(10,47),(11,48),(12,49),(13,50),(14,51),(15,52),(16,53),(17,54),(18,55),(19,56),(20,57),(21,72),(22,73),(23,74),(24,75),(25,76),(26,77),(27,78),(28,79),(29,80),(30,61),(31,62),(32,63),(33,64),(34,65),(35,66),(36,67),(37,68),(38,69),(39,70),(40,71)], [(1,11),(2,12),(3,13),(4,14),(5,15),(6,16),(7,17),(8,18),(9,19),(10,20),(21,31),(22,32),(23,33),(24,34),(25,35),(26,36),(27,37),(28,38),(29,39),(30,40),(41,51),(42,52),(43,53),(44,54),(45,55),(46,56),(47,57),(48,58),(49,59),(50,60),(61,71),(62,72),(63,73),(64,74),(65,75),(66,76),(67,77),(68,78),(69,79),(70,80)], [(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)], [(1,23,11,33),(2,32,12,22),(3,21,13,31),(4,30,14,40),(5,39,15,29),(6,28,16,38),(7,37,17,27),(8,26,18,36),(9,35,19,25),(10,24,20,34),(41,61,51,71),(42,70,52,80),(43,79,53,69),(44,68,54,78),(45,77,55,67),(46,66,56,76),(47,75,57,65),(48,64,58,74),(49,73,59,63),(50,62,60,72)])
Matrix representation ►G ⊆ GL6(𝔽41)
| 40 | 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 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 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 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 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 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 37 | 0 | 0 |
| 0 | 0 | 37 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 10 |
| 0 | 0 | 0 | 0 | 10 | 0 |
| 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 10 |
| 0 | 0 | 0 | 0 | 10 | 0 |
| 0 | 0 | 0 | 37 | 0 | 0 |
| 0 | 0 | 37 | 0 | 0 | 0 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,40],[40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[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],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,40,0,0,0,0,1,0,0,0,0,0,0,0,0,37,0,0,0,0,37,0,0,0,0,0,0,0,0,10,0,0,0,0,10,0],[9,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,0,37,0,0,0,0,37,0,0,0,0,10,0,0,0,0,10,0,0,0] >;
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | ··· | 4L | 5A | 5B | 10A | ··· | 10F | 10G | ··· | 10R | 10S | 10T | 20A | ··· | 20F |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 10 | ··· | 10 | 10 | ··· | 10 | 10 | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 20 | 4 | 4 | 4 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | 8 | 8 | ··· | 8 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D5 | C4○D4 | D10 | D10 | D10 | 2+ (1+4) | D4⋊2D5 | D4⋊6D10 |
| kernel | C24.35D10 | Dic5.14D4 | C23.D10 | Dic5⋊4D4 | D10.12D4 | Dic5.5D4 | D4×Dic5 | C23.18D10 | C20.17D4 | C20⋊2D4 | Dic5⋊D4 | C2×C23.D5 | C24⋊2D5 | C5×C22≀C2 | C22≀C2 | C2×C10 | C22⋊C4 | C2×D4 | C24 | C10 | C22 | C2 |
| # reps | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 2 | 4 | 6 | 6 | 2 | 2 | 4 | 8 |
In GAP, Magma, Sage, TeX
C_2^4._{35}D_{10} % in TeX
G:=Group("C2^4.35D10"); // GroupNames label
G:=SmallGroup(320,1265);
// by ID
G=gap.SmallGroup(320,1265);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,758,219,675,570,12550]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^10=f^2=d,a*b=b*a,e*a*e^-1=a*c=c*a,a*d=d*a,f*a*f^-1=a*c*d,b*c=c*b,e*b*e^-1=b*d=d*b,f*b*f^-1=b*c*d,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=e^9>;
// generators/relations