metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C24.32D10, C10.242+ (1+4), (D4×Dic5)⋊11C2, C22≀C2.2D5, C22⋊C4.1D10, (C2×D4).149D10, (C2×C20).26C23, C4⋊Dic5⋊24C22, C20.17D4⋊11C2, (C2×C10).131C24, (C4×Dic5)⋊13C22, C10.D4⋊7C22, C2.26(D4⋊6D10), C23.D5⋊12C22, C5⋊4(C22.45C24), (C2×Dic10)⋊19C22, (D4×C10).110C22, C23.11D10⋊2C2, C23.18D10⋊3C2, C23.D10⋊11C2, (C23×C10).67C22, C22.152(C23×D5), C23.176(C22×D5), Dic5.14D4⋊12C2, C22.17(D4⋊2D5), (C22×C10).180C23, (C2×Dic5).230C23, (C22×Dic5)⋊11C22, C10.76(C2×C4○D4), (C5×C22≀C2).2C2, C2.27(C2×D4⋊2D5), (C2×C23.D5)⋊19C2, (C2×C4).26(C22×D5), (C2×C10).43(C4○D4), (C5×C22⋊C4).2C22, SmallGroup(320,1259)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 758 in 248 conjugacy classes, 99 normal (27 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×11], C22, C22 [×4], C22 [×14], C5, C2×C4, C2×C4 [×2], C2×C4 [×15], D4 [×5], Q8, C23 [×2], C23 [×2], C23 [×5], C10, C10 [×2], C10 [×6], C42 [×3], C22⋊C4, C22⋊C4 [×2], C22⋊C4 [×11], C4⋊C4 [×8], C22×C4 [×5], C2×D4, C2×D4 [×2], C2×Q8, C24, Dic5 [×8], C20 [×3], C2×C10, C2×C10 [×4], C2×C10 [×14], C2×C22⋊C4 [×2], C42⋊C2 [×2], C4×D4 [×2], C22≀C2, C22⋊Q8 [×2], C22.D4 [×3], C4.4D4, C42⋊2C2 [×2], Dic10, C2×Dic5 [×8], C2×Dic5 [×7], C2×C20, C2×C20 [×2], C5×D4 [×5], C22×C10 [×2], C22×C10 [×2], C22×C10 [×5], C22.45C24, C4×Dic5, C4×Dic5 [×2], C10.D4 [×6], C4⋊Dic5 [×2], C23.D5, C23.D5 [×10], C5×C22⋊C4, C5×C22⋊C4 [×2], C2×Dic10, C22×Dic5, C22×Dic5 [×4], D4×C10, D4×C10 [×2], C23×C10, C23.11D10 [×2], Dic5.14D4 [×2], C23.D10 [×2], D4×Dic5 [×2], C23.18D10, C23.18D10 [×2], C20.17D4, C2×C23.D5 [×2], C5×C22≀C2, C24.32D10
Quotients:
C1, C2 [×15], C22 [×35], C23 [×15], D5, C4○D4 [×4], C24, D10 [×7], C2×C4○D4 [×2], 2+ (1+4), C22×D5 [×7], C22.45C24, D4⋊2D5 [×4], C23×D5, C2×D4⋊2D5 [×2], D4⋊6D10, C24.32D10
Generators and relations
G = < a,b,c,d,e,f | a2=b2=c2=d2=e10=1, 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=e-1 >
►On 80 points
(2 17)(4 19)(6 11)(8 13)(10 15)(21 78)(23 80)(25 72)(27 74)(29 76)(32 50)(34 42)(36 44)(38 46)(40 48)(52 69)(54 61)(56 63)(58 65)(60 67)
(1 33)(3 35)(5 37)(7 39)(9 31)(12 47)(14 49)(16 41)(18 43)(20 45)(21 52)(22 79)(23 54)(24 71)(25 56)(26 73)(27 58)(28 75)(29 60)(30 77)(51 68)(53 70)(55 62)(57 64)(59 66)(61 80)(63 72)(65 74)(67 76)(69 78)
(1 16)(2 17)(3 18)(4 19)(5 20)(6 11)(7 12)(8 13)(9 14)(10 15)(21 78)(22 79)(23 80)(24 71)(25 72)(26 73)(27 74)(28 75)(29 76)(30 77)(31 49)(32 50)(33 41)(34 42)(35 43)(36 44)(37 45)(38 46)(39 47)(40 48)(51 68)(52 69)(53 70)(54 61)(55 62)(56 63)(57 64)(58 65)(59 66)(60 67)
(1 33)(2 34)(3 35)(4 36)(5 37)(6 38)(7 39)(8 40)(9 31)(10 32)(11 46)(12 47)(13 48)(14 49)(15 50)(16 41)(17 42)(18 43)(19 44)(20 45)(21 69)(22 70)(23 61)(24 62)(25 63)(26 64)(27 65)(28 66)(29 67)(30 68)(51 77)(52 78)(53 79)(54 80)(55 71)(56 72)(57 73)(58 74)(59 75)(60 76)
(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 76 33 60)(2 75 34 59)(3 74 35 58)(4 73 36 57)(5 72 37 56)(6 71 38 55)(7 80 39 54)(8 79 40 53)(9 78 31 52)(10 77 32 51)(11 24 46 62)(12 23 47 61)(13 22 48 70)(14 21 49 69)(15 30 50 68)(16 29 41 67)(17 28 42 66)(18 27 43 65)(19 26 44 64)(20 25 45 63)
G:=sub<Sym(80)| (2,17)(4,19)(6,11)(8,13)(10,15)(21,78)(23,80)(25,72)(27,74)(29,76)(32,50)(34,42)(36,44)(38,46)(40,48)(52,69)(54,61)(56,63)(58,65)(60,67), (1,33)(3,35)(5,37)(7,39)(9,31)(12,47)(14,49)(16,41)(18,43)(20,45)(21,52)(22,79)(23,54)(24,71)(25,56)(26,73)(27,58)(28,75)(29,60)(30,77)(51,68)(53,70)(55,62)(57,64)(59,66)(61,80)(63,72)(65,74)(67,76)(69,78), (1,16)(2,17)(3,18)(4,19)(5,20)(6,11)(7,12)(8,13)(9,14)(10,15)(21,78)(22,79)(23,80)(24,71)(25,72)(26,73)(27,74)(28,75)(29,76)(30,77)(31,49)(32,50)(33,41)(34,42)(35,43)(36,44)(37,45)(38,46)(39,47)(40,48)(51,68)(52,69)(53,70)(54,61)(55,62)(56,63)(57,64)(58,65)(59,66)(60,67), (1,33)(2,34)(3,35)(4,36)(5,37)(6,38)(7,39)(8,40)(9,31)(10,32)(11,46)(12,47)(13,48)(14,49)(15,50)(16,41)(17,42)(18,43)(19,44)(20,45)(21,69)(22,70)(23,61)(24,62)(25,63)(26,64)(27,65)(28,66)(29,67)(30,68)(51,77)(52,78)(53,79)(54,80)(55,71)(56,72)(57,73)(58,74)(59,75)(60,76), (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), (1,76,33,60)(2,75,34,59)(3,74,35,58)(4,73,36,57)(5,72,37,56)(6,71,38,55)(7,80,39,54)(8,79,40,53)(9,78,31,52)(10,77,32,51)(11,24,46,62)(12,23,47,61)(13,22,48,70)(14,21,49,69)(15,30,50,68)(16,29,41,67)(17,28,42,66)(18,27,43,65)(19,26,44,64)(20,25,45,63)>;
G:=Group( (2,17)(4,19)(6,11)(8,13)(10,15)(21,78)(23,80)(25,72)(27,74)(29,76)(32,50)(34,42)(36,44)(38,46)(40,48)(52,69)(54,61)(56,63)(58,65)(60,67), (1,33)(3,35)(5,37)(7,39)(9,31)(12,47)(14,49)(16,41)(18,43)(20,45)(21,52)(22,79)(23,54)(24,71)(25,56)(26,73)(27,58)(28,75)(29,60)(30,77)(51,68)(53,70)(55,62)(57,64)(59,66)(61,80)(63,72)(65,74)(67,76)(69,78), (1,16)(2,17)(3,18)(4,19)(5,20)(6,11)(7,12)(8,13)(9,14)(10,15)(21,78)(22,79)(23,80)(24,71)(25,72)(26,73)(27,74)(28,75)(29,76)(30,77)(31,49)(32,50)(33,41)(34,42)(35,43)(36,44)(37,45)(38,46)(39,47)(40,48)(51,68)(52,69)(53,70)(54,61)(55,62)(56,63)(57,64)(58,65)(59,66)(60,67), (1,33)(2,34)(3,35)(4,36)(5,37)(6,38)(7,39)(8,40)(9,31)(10,32)(11,46)(12,47)(13,48)(14,49)(15,50)(16,41)(17,42)(18,43)(19,44)(20,45)(21,69)(22,70)(23,61)(24,62)(25,63)(26,64)(27,65)(28,66)(29,67)(30,68)(51,77)(52,78)(53,79)(54,80)(55,71)(56,72)(57,73)(58,74)(59,75)(60,76), (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), (1,76,33,60)(2,75,34,59)(3,74,35,58)(4,73,36,57)(5,72,37,56)(6,71,38,55)(7,80,39,54)(8,79,40,53)(9,78,31,52)(10,77,32,51)(11,24,46,62)(12,23,47,61)(13,22,48,70)(14,21,49,69)(15,30,50,68)(16,29,41,67)(17,28,42,66)(18,27,43,65)(19,26,44,64)(20,25,45,63) );
G=PermutationGroup([(2,17),(4,19),(6,11),(8,13),(10,15),(21,78),(23,80),(25,72),(27,74),(29,76),(32,50),(34,42),(36,44),(38,46),(40,48),(52,69),(54,61),(56,63),(58,65),(60,67)], [(1,33),(3,35),(5,37),(7,39),(9,31),(12,47),(14,49),(16,41),(18,43),(20,45),(21,52),(22,79),(23,54),(24,71),(25,56),(26,73),(27,58),(28,75),(29,60),(30,77),(51,68),(53,70),(55,62),(57,64),(59,66),(61,80),(63,72),(65,74),(67,76),(69,78)], [(1,16),(2,17),(3,18),(4,19),(5,20),(6,11),(7,12),(8,13),(9,14),(10,15),(21,78),(22,79),(23,80),(24,71),(25,72),(26,73),(27,74),(28,75),(29,76),(30,77),(31,49),(32,50),(33,41),(34,42),(35,43),(36,44),(37,45),(38,46),(39,47),(40,48),(51,68),(52,69),(53,70),(54,61),(55,62),(56,63),(57,64),(58,65),(59,66),(60,67)], [(1,33),(2,34),(3,35),(4,36),(5,37),(6,38),(7,39),(8,40),(9,31),(10,32),(11,46),(12,47),(13,48),(14,49),(15,50),(16,41),(17,42),(18,43),(19,44),(20,45),(21,69),(22,70),(23,61),(24,62),(25,63),(26,64),(27,65),(28,66),(29,67),(30,68),(51,77),(52,78),(53,79),(54,80),(55,71),(56,72),(57,73),(58,74),(59,75),(60,76)], [(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)], [(1,76,33,60),(2,75,34,59),(3,74,35,58),(4,73,36,57),(5,72,37,56),(6,71,38,55),(7,80,39,54),(8,79,40,53),(9,78,31,52),(10,77,32,51),(11,24,46,62),(12,23,47,61),(13,22,48,70),(14,21,49,69),(15,30,50,68),(16,29,41,67),(17,28,42,66),(18,27,43,65),(19,26,44,64),(20,25,45,63)])
Matrix representation ►G ⊆ GL6(𝔽41)
| 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 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 18 | 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 | 1 |
| 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 |
| 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 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 40 | 9 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 7 | 0 | 0 |
| 0 | 0 | 34 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 32 | 0 | 0 | 0 | 0 | 0 |
| 0 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 27 | 11 | 0 | 0 |
| 0 | 0 | 27 | 14 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 9 |
| 0 | 0 | 0 | 0 | 9 | 0 |
G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40],[40,18,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1],[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],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[40,0,0,0,0,0,9,1,0,0,0,0,0,0,7,34,0,0,0,0,7,40,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[32,0,0,0,0,0,0,32,0,0,0,0,0,0,27,27,0,0,0,0,11,14,0,0,0,0,0,0,0,9,0,0,0,0,9,0] >;
53 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | ··· | 4K | 4L | 4M | 4N | 4O | 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 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 10 | ··· | 10 | 20 | 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 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D5 | C4○D4 | D10 | D10 | D10 | 2+ (1+4) | D4⋊2D5 | D4⋊6D10 |
| kernel | C24.32D10 | C23.11D10 | Dic5.14D4 | C23.D10 | D4×Dic5 | C23.18D10 | C20.17D4 | C2×C23.D5 | C5×C22≀C2 | C22≀C2 | C2×C10 | C22⋊C4 | C2×D4 | C24 | C10 | C22 | C2 |
| # reps | 1 | 2 | 2 | 2 | 2 | 3 | 1 | 2 | 1 | 2 | 8 | 6 | 6 | 2 | 1 | 8 | 4 |
In GAP, Magma, Sage, TeX
C_2^4._{32}D_{10} % in TeX
G:=Group("C2^4.32D10"); // GroupNames label
G:=SmallGroup(320,1259);
// by ID
G=gap.SmallGroup(320,1259);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,219,1571,297,12550]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=e^10=1,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^-1>;
// generators/relations