metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C24.33D10, C10.272+ (1+4), C5⋊D4⋊5D4, (C2×D4)⋊19D10, C5⋊4(D4⋊5D4), C22≀C2⋊4D5, C22⋊C4⋊6D10, C23⋊D10⋊5C2, C20⋊2D4⋊13C2, (D4×Dic5)⋊12C2, D10.39(C2×D4), (D4×C10)⋊8C22, C22.11(D4×D5), Dic5⋊D4⋊3C2, Dic5⋊4D4⋊3C2, (C2×C20).29C23, C4⋊Dic5⋊26C22, Dic5.43(C2×D4), C10.57(C22×D4), C22⋊4(D4⋊2D5), C23.7(C22×D5), (C2×C10).135C24, (C4×Dic5)⋊15C22, (C22×C10).9C23, D10.12D4⋊13C2, C23.D5⋊50C22, C2.29(D4⋊6D10), D10⋊C4⋊12C22, Dic5.5D4⋊12C2, (C2×Dic10)⋊20C22, C22.D20⋊10C2, C10.D4⋊10C22, (C23×C10).68C22, (C22×D5).54C23, (C23×D5).43C22, C22.156(C23×D5), Dic5.14D4⋊13C2, (C2×Dic5).232C23, (C22×Dic5)⋊14C22, C2.30(C2×D4×D5), (C2×C4×D5)⋊8C22, (D5×C22⋊C4)⋊3C2, (C5×C22≀C2)⋊6C2, (C2×D4⋊2D5)⋊6C2, C10.77(C2×C4○D4), (C2×C10).54(C2×D4), (C2×C5⋊D4)⋊8C22, (C22×C5⋊D4)⋊9C2, (C2×C10)⋊10(C4○D4), C2.28(C2×D4⋊2D5), (C5×C22⋊C4)⋊6C22, (C2×C23.D5)⋊20C2, (C2×C4).29(C22×D5), SmallGroup(320,1263)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1286 in 334 conjugacy classes, 107 normal (91 characteristic)
C1, C2 [×3], C2 [×9], C4 [×10], C22, C22 [×4], C22 [×25], C5, C2×C4 [×3], C2×C4 [×16], D4 [×18], Q8 [×2], C23 [×4], C23 [×12], D5 [×3], C10 [×3], C10 [×6], C42, C22⋊C4 [×3], C22⋊C4 [×9], C4⋊C4 [×4], C22×C4 [×6], C2×D4 [×3], C2×D4 [×10], C2×Q8, C4○D4 [×4], C24, C24, Dic5 [×2], Dic5 [×5], C20 [×3], D10 [×2], D10 [×9], C2×C10, C2×C10 [×4], C2×C10 [×14], C2×C22⋊C4 [×2], C4×D4 [×2], C22≀C2, C22≀C2, C4⋊D4 [×3], C22⋊Q8, C22.D4 [×2], C4.4D4, C22×D4, C2×C4○D4, Dic10 [×2], C4×D5 [×3], C2×Dic5 [×6], C2×Dic5 [×7], C5⋊D4 [×4], C5⋊D4 [×9], C2×C20 [×3], C5×D4 [×5], C22×D5 [×2], C22×D5 [×5], C22×C10 [×4], C22×C10 [×5], D4⋊5D4, C4×Dic5, C10.D4 [×2], C4⋊Dic5 [×2], D10⋊C4 [×4], C23.D5 [×5], C5×C22⋊C4 [×3], C2×Dic10, C2×C4×D5 [×2], D4⋊2D5 [×4], C22×Dic5 [×4], C2×C5⋊D4 [×6], C2×C5⋊D4 [×4], D4×C10 [×3], C23×D5, C23×C10, Dic5.14D4, D5×C22⋊C4, Dic5⋊4D4, D10.12D4, Dic5.5D4, C22.D20, D4×Dic5, C23⋊D10, C20⋊2D4, Dic5⋊D4 [×2], C2×C23.D5, C5×C22≀C2, C2×D4⋊2D5, C22×C5⋊D4, C24.33D10
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C4○D4 [×2], C24, D10 [×7], C22×D4, C2×C4○D4, 2+ (1+4), C22×D5 [×7], D4⋊5D4, D4×D5 [×2], D4⋊2D5 [×2], C23×D5, C2×D4×D5, C2×D4⋊2D5, D4⋊6D10, C24.33D10
Generators and relations
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e10=f2=d, ab=ba, eae-1=faf-1=ac=ca, ad=da, fbf-1=bc=cb, ebe-1=bd=db, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e9 >
►On 80 points
(1 11)(2 64)(3 13)(4 66)(5 15)(6 68)(7 17)(8 70)(9 19)(10 72)(12 74)(14 76)(16 78)(18 80)(20 62)(21 31)(22 47)(23 33)(24 49)(25 35)(26 51)(27 37)(28 53)(29 39)(30 55)(32 57)(34 59)(36 41)(38 43)(40 45)(42 52)(44 54)(46 56)(48 58)(50 60)(61 71)(63 73)(65 75)(67 77)(69 79)
(1 44)(2 55)(3 46)(4 57)(5 48)(6 59)(7 50)(8 41)(9 52)(10 43)(11 54)(12 45)(13 56)(14 47)(15 58)(16 49)(17 60)(18 51)(19 42)(20 53)(21 65)(22 76)(23 67)(24 78)(25 69)(26 80)(27 71)(28 62)(29 73)(30 64)(31 75)(32 66)(33 77)(34 68)(35 79)(36 70)(37 61)(38 72)(39 63)(40 74)
(1 73)(2 74)(3 75)(4 76)(5 77)(6 78)(7 79)(8 80)(9 61)(10 62)(11 63)(12 64)(13 65)(14 66)(15 67)(16 68)(17 69)(18 70)(19 71)(20 72)(21 56)(22 57)(23 58)(24 59)(25 60)(26 41)(27 42)(28 43)(29 44)(30 45)(31 46)(32 47)(33 48)(34 49)(35 50)(36 51)(37 52)(38 53)(39 54)(40 55)
(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 20 11 10)(2 9 12 19)(3 18 13 8)(4 7 14 17)(5 16 15 6)(21 41 31 51)(22 50 32 60)(23 59 33 49)(24 48 34 58)(25 57 35 47)(26 46 36 56)(27 55 37 45)(28 44 38 54)(29 53 39 43)(30 42 40 52)(61 64 71 74)(62 73 72 63)(65 80 75 70)(66 69 76 79)(67 78 77 68)
G:=sub<Sym(80)| (1,11)(2,64)(3,13)(4,66)(5,15)(6,68)(7,17)(8,70)(9,19)(10,72)(12,74)(14,76)(16,78)(18,80)(20,62)(21,31)(22,47)(23,33)(24,49)(25,35)(26,51)(27,37)(28,53)(29,39)(30,55)(32,57)(34,59)(36,41)(38,43)(40,45)(42,52)(44,54)(46,56)(48,58)(50,60)(61,71)(63,73)(65,75)(67,77)(69,79), (1,44)(2,55)(3,46)(4,57)(5,48)(6,59)(7,50)(8,41)(9,52)(10,43)(11,54)(12,45)(13,56)(14,47)(15,58)(16,49)(17,60)(18,51)(19,42)(20,53)(21,65)(22,76)(23,67)(24,78)(25,69)(26,80)(27,71)(28,62)(29,73)(30,64)(31,75)(32,66)(33,77)(34,68)(35,79)(36,70)(37,61)(38,72)(39,63)(40,74), (1,73)(2,74)(3,75)(4,76)(5,77)(6,78)(7,79)(8,80)(9,61)(10,62)(11,63)(12,64)(13,65)(14,66)(15,67)(16,68)(17,69)(18,70)(19,71)(20,72)(21,56)(22,57)(23,58)(24,59)(25,60)(26,41)(27,42)(28,43)(29,44)(30,45)(31,46)(32,47)(33,48)(34,49)(35,50)(36,51)(37,52)(38,53)(39,54)(40,55), (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), (1,20,11,10)(2,9,12,19)(3,18,13,8)(4,7,14,17)(5,16,15,6)(21,41,31,51)(22,50,32,60)(23,59,33,49)(24,48,34,58)(25,57,35,47)(26,46,36,56)(27,55,37,45)(28,44,38,54)(29,53,39,43)(30,42,40,52)(61,64,71,74)(62,73,72,63)(65,80,75,70)(66,69,76,79)(67,78,77,68)>;
G:=Group( (1,11)(2,64)(3,13)(4,66)(5,15)(6,68)(7,17)(8,70)(9,19)(10,72)(12,74)(14,76)(16,78)(18,80)(20,62)(21,31)(22,47)(23,33)(24,49)(25,35)(26,51)(27,37)(28,53)(29,39)(30,55)(32,57)(34,59)(36,41)(38,43)(40,45)(42,52)(44,54)(46,56)(48,58)(50,60)(61,71)(63,73)(65,75)(67,77)(69,79), (1,44)(2,55)(3,46)(4,57)(5,48)(6,59)(7,50)(8,41)(9,52)(10,43)(11,54)(12,45)(13,56)(14,47)(15,58)(16,49)(17,60)(18,51)(19,42)(20,53)(21,65)(22,76)(23,67)(24,78)(25,69)(26,80)(27,71)(28,62)(29,73)(30,64)(31,75)(32,66)(33,77)(34,68)(35,79)(36,70)(37,61)(38,72)(39,63)(40,74), (1,73)(2,74)(3,75)(4,76)(5,77)(6,78)(7,79)(8,80)(9,61)(10,62)(11,63)(12,64)(13,65)(14,66)(15,67)(16,68)(17,69)(18,70)(19,71)(20,72)(21,56)(22,57)(23,58)(24,59)(25,60)(26,41)(27,42)(28,43)(29,44)(30,45)(31,46)(32,47)(33,48)(34,49)(35,50)(36,51)(37,52)(38,53)(39,54)(40,55), (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), (1,20,11,10)(2,9,12,19)(3,18,13,8)(4,7,14,17)(5,16,15,6)(21,41,31,51)(22,50,32,60)(23,59,33,49)(24,48,34,58)(25,57,35,47)(26,46,36,56)(27,55,37,45)(28,44,38,54)(29,53,39,43)(30,42,40,52)(61,64,71,74)(62,73,72,63)(65,80,75,70)(66,69,76,79)(67,78,77,68) );
G=PermutationGroup([(1,11),(2,64),(3,13),(4,66),(5,15),(6,68),(7,17),(8,70),(9,19),(10,72),(12,74),(14,76),(16,78),(18,80),(20,62),(21,31),(22,47),(23,33),(24,49),(25,35),(26,51),(27,37),(28,53),(29,39),(30,55),(32,57),(34,59),(36,41),(38,43),(40,45),(42,52),(44,54),(46,56),(48,58),(50,60),(61,71),(63,73),(65,75),(67,77),(69,79)], [(1,44),(2,55),(3,46),(4,57),(5,48),(6,59),(7,50),(8,41),(9,52),(10,43),(11,54),(12,45),(13,56),(14,47),(15,58),(16,49),(17,60),(18,51),(19,42),(20,53),(21,65),(22,76),(23,67),(24,78),(25,69),(26,80),(27,71),(28,62),(29,73),(30,64),(31,75),(32,66),(33,77),(34,68),(35,79),(36,70),(37,61),(38,72),(39,63),(40,74)], [(1,73),(2,74),(3,75),(4,76),(5,77),(6,78),(7,79),(8,80),(9,61),(10,62),(11,63),(12,64),(13,65),(14,66),(15,67),(16,68),(17,69),(18,70),(19,71),(20,72),(21,56),(22,57),(23,58),(24,59),(25,60),(26,41),(27,42),(28,43),(29,44),(30,45),(31,46),(32,47),(33,48),(34,49),(35,50),(36,51),(37,52),(38,53),(39,54),(40,55)], [(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)], [(1,20,11,10),(2,9,12,19),(3,18,13,8),(4,7,14,17),(5,16,15,6),(21,41,31,51),(22,50,32,60),(23,59,33,49),(24,48,34,58),(25,57,35,47),(26,46,36,56),(27,55,37,45),(28,44,38,54),(29,53,39,43),(30,42,40,52),(61,64,71,74),(62,73,72,63),(65,80,75,70),(66,69,76,79),(67,78,77,68)])
Matrix representation ►G ⊆ GL6(𝔽41)
| 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 | 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 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 2 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 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 | 0 | 0 | 0 | 1 |
| 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 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 34 | 34 | 0 | 0 | 0 | 0 |
| 7 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 32 | 0 |
| 0 | 0 | 0 | 0 | 32 | 9 |
| 34 | 34 | 0 | 0 | 0 | 0 |
| 1 | 7 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 0 | 0 | 9 |
G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,2,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[34,7,0,0,0,0,34,1,0,0,0,0,0,0,0,40,0,0,0,0,1,0,0,0,0,0,0,0,32,32,0,0,0,0,0,9],[34,1,0,0,0,0,34,7,0,0,0,0,0,0,0,1,0,0,0,0,40,0,0,0,0,0,0,0,9,0,0,0,0,0,0,9] >;
53 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 2L | 4A | 4B | 4C | 4D | ··· | 4I | 4J | 4K | 4L | 5A | 5B | 10A | ··· | 10F | 10G | ··· | 10R | 10S | 10T | 20A | ··· | 20F |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 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 | 2 | 2 | 4 | 4 | 10 | 10 | 20 | 4 | 4 | 4 | 10 | ··· | 10 | 20 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | 8 | 8 | ··· | 8 |
53 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D5 | C4○D4 | D10 | D10 | D10 | 2+ (1+4) | D4×D5 | D4⋊2D5 | D4⋊6D10 |
| kernel | C24.33D10 | Dic5.14D4 | D5×C22⋊C4 | Dic5⋊4D4 | D10.12D4 | Dic5.5D4 | C22.D20 | D4×Dic5 | C23⋊D10 | C20⋊2D4 | Dic5⋊D4 | C2×C23.D5 | C5×C22≀C2 | C2×D4⋊2D5 | C22×C5⋊D4 | C5⋊D4 | C22≀C2 | C2×C10 | C22⋊C4 | C2×D4 | C24 | C10 | C22 | C22 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 4 | 2 | 4 | 6 | 6 | 2 | 1 | 4 | 4 | 4 |
In GAP, Magma, Sage, TeX
C_2^4._{33}D_{10} % in TeX
G:=Group("C2^4.33D10"); // GroupNames label
G:=SmallGroup(320,1263);
// by ID
G=gap.SmallGroup(320,1263);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,219,184,1571,297,12550]);
// 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=f*a*f^-1=a*c=c*a,a*d=d*a,f*b*f^-1=b*c=c*b,e*b*e^-1=b*d=d*b,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