metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C24⋊8D10, C10.892+ (1+4), (C2×D4)⋊40D10, (C22×D4)⋊10D5, (C22×C4)⋊28D10, (C22×C10)⋊13D4, C23⋊4(C5⋊D4), C5⋊5(C23⋊3D4), C23⋊D10⋊30C2, (D4×C10)⋊58C22, C24⋊2D5⋊12C2, Dic5⋊D4⋊41C2, (C2×C20).644C23, (C2×C10).299C24, (C22×C20)⋊44C22, (C23×C10)⋊14C22, C10.146(C22×D4), (C23×D5)⋊15C22, C23.D5⋊64C22, C2.92(D4⋊6D10), D10⋊C4⋊36C22, C10.D4⋊38C22, C23.206(C22×D5), C22.312(C23×D5), C23.23D10⋊28C2, C23.18D10⋊29C2, (C22×C10).233C23, (C2×Dic5).154C23, (C22×Dic5)⋊34C22, (C22×D5).130C23, (D4×C2×C10)⋊17C2, (C2×C10).582(C2×D4), (C2×C5⋊D4)⋊48C22, (C22×C5⋊D4)⋊17C2, (C2×C23.D5)⋊30C2, C22.20(C2×C5⋊D4), C2.19(C22×C5⋊D4), (C2×C4).238(C22×D5), SmallGroup(320,1476)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1294 in 346 conjugacy classes, 111 normal (25 characteristic)
C1, C2, C2 [×2], C2 [×10], C4 [×8], C22, C22 [×6], C22 [×30], C5, C2×C4 [×2], C2×C4 [×12], D4 [×20], C23 [×3], C23 [×6], C23 [×12], D5 [×2], C10, C10 [×2], C10 [×8], C22⋊C4 [×12], C4⋊C4 [×4], C22×C4, C22×C4 [×3], C2×D4 [×4], C2×D4 [×16], C24 [×2], C24, Dic5 [×6], C20 [×2], D10 [×10], C2×C10, C2×C10 [×6], C2×C10 [×20], C2×C22⋊C4, C22≀C2 [×4], C4⋊D4 [×4], C22.D4 [×4], C22×D4, C22×D4, C2×Dic5 [×6], C2×Dic5 [×4], C5⋊D4 [×12], C2×C20 [×2], C2×C20 [×2], C5×D4 [×8], C22×D5 [×2], C22×D5 [×4], C22×C10 [×3], C22×C10 [×6], C22×C10 [×6], C23⋊3D4, C10.D4 [×4], D10⋊C4 [×4], C23.D5 [×8], C22×Dic5, C22×Dic5 [×2], C2×C5⋊D4 [×8], C2×C5⋊D4 [×4], C22×C20, D4×C10 [×4], D4×C10 [×4], C23×D5, C23×C10 [×2], C23.23D10 [×2], C23.18D10 [×2], C23⋊D10 [×2], Dic5⋊D4 [×4], C2×C23.D5, C24⋊2D5 [×2], C22×C5⋊D4, D4×C2×C10, C24⋊8D10
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C24, D10 [×7], C22×D4, 2+ (1+4) [×2], C5⋊D4 [×4], C22×D5 [×7], C23⋊3D4, C2×C5⋊D4 [×6], C23×D5, D4⋊6D10 [×2], C22×C5⋊D4, C24⋊8D10
Generators and relations
G = < a,b,c,d,e,f | a2=b2=c2=d2=e10=f2=1, ab=ba, ac=ca, faf=ad=da, ae=ea, bc=cb, ebe-1=bd=db, fbf=bcd, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef=e-1 >
►On 80 points
(1 37)(2 38)(3 39)(4 40)(5 36)(6 26)(7 27)(8 28)(9 29)(10 30)(11 31)(12 32)(13 33)(14 34)(15 35)(16 23)(17 24)(18 25)(19 21)(20 22)(41 76)(42 77)(43 78)(44 79)(45 80)(46 71)(47 72)(48 73)(49 74)(50 75)(51 63)(52 64)(53 65)(54 66)(55 67)(56 68)(57 69)(58 70)(59 61)(60 62)
(1 77)(2 73)(3 79)(4 75)(5 71)(6 80)(7 76)(8 72)(9 78)(10 74)(11 59)(12 55)(13 51)(14 57)(15 53)(16 54)(17 60)(18 56)(19 52)(20 58)(21 64)(22 70)(23 66)(24 62)(25 68)(26 45)(27 41)(28 47)(29 43)(30 49)(31 61)(32 67)(33 63)(34 69)(35 65)(36 46)(37 42)(38 48)(39 44)(40 50)
(1 13)(2 14)(3 15)(4 11)(5 12)(6 16)(7 17)(8 18)(9 19)(10 20)(21 29)(22 30)(23 26)(24 27)(25 28)(31 40)(32 36)(33 37)(34 38)(35 39)(41 62)(42 63)(43 64)(44 65)(45 66)(46 67)(47 68)(48 69)(49 70)(50 61)(51 77)(52 78)(53 79)(54 80)(55 71)(56 72)(57 73)(58 74)(59 75)(60 76)
(1 8)(2 9)(3 10)(4 6)(5 7)(11 16)(12 17)(13 18)(14 19)(15 20)(21 34)(22 35)(23 31)(24 32)(25 33)(26 40)(27 36)(28 37)(29 38)(30 39)(41 46)(42 47)(43 48)(44 49)(45 50)(51 56)(52 57)(53 58)(54 59)(55 60)(61 66)(62 67)(63 68)(64 69)(65 70)(71 76)(72 77)(73 78)(74 79)(75 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 5)(2 4)(6 9)(7 8)(11 14)(12 13)(16 19)(17 18)(21 31)(22 35)(23 34)(24 33)(25 32)(26 38)(27 37)(28 36)(29 40)(30 39)(41 68)(42 67)(43 66)(44 65)(45 64)(46 63)(47 62)(48 61)(49 70)(50 69)(51 76)(52 75)(53 74)(54 73)(55 72)(56 71)(57 80)(58 79)(59 78)(60 77)
G:=sub<Sym(80)| (1,37)(2,38)(3,39)(4,40)(5,36)(6,26)(7,27)(8,28)(9,29)(10,30)(11,31)(12,32)(13,33)(14,34)(15,35)(16,23)(17,24)(18,25)(19,21)(20,22)(41,76)(42,77)(43,78)(44,79)(45,80)(46,71)(47,72)(48,73)(49,74)(50,75)(51,63)(52,64)(53,65)(54,66)(55,67)(56,68)(57,69)(58,70)(59,61)(60,62), (1,77)(2,73)(3,79)(4,75)(5,71)(6,80)(7,76)(8,72)(9,78)(10,74)(11,59)(12,55)(13,51)(14,57)(15,53)(16,54)(17,60)(18,56)(19,52)(20,58)(21,64)(22,70)(23,66)(24,62)(25,68)(26,45)(27,41)(28,47)(29,43)(30,49)(31,61)(32,67)(33,63)(34,69)(35,65)(36,46)(37,42)(38,48)(39,44)(40,50), (1,13)(2,14)(3,15)(4,11)(5,12)(6,16)(7,17)(8,18)(9,19)(10,20)(21,29)(22,30)(23,26)(24,27)(25,28)(31,40)(32,36)(33,37)(34,38)(35,39)(41,62)(42,63)(43,64)(44,65)(45,66)(46,67)(47,68)(48,69)(49,70)(50,61)(51,77)(52,78)(53,79)(54,80)(55,71)(56,72)(57,73)(58,74)(59,75)(60,76), (1,8)(2,9)(3,10)(4,6)(5,7)(11,16)(12,17)(13,18)(14,19)(15,20)(21,34)(22,35)(23,31)(24,32)(25,33)(26,40)(27,36)(28,37)(29,38)(30,39)(41,46)(42,47)(43,48)(44,49)(45,50)(51,56)(52,57)(53,58)(54,59)(55,60)(61,66)(62,67)(63,68)(64,69)(65,70)(71,76)(72,77)(73,78)(74,79)(75,80), (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), (1,5)(2,4)(6,9)(7,8)(11,14)(12,13)(16,19)(17,18)(21,31)(22,35)(23,34)(24,33)(25,32)(26,38)(27,37)(28,36)(29,40)(30,39)(41,68)(42,67)(43,66)(44,65)(45,64)(46,63)(47,62)(48,61)(49,70)(50,69)(51,76)(52,75)(53,74)(54,73)(55,72)(56,71)(57,80)(58,79)(59,78)(60,77)>;
G:=Group( (1,37)(2,38)(3,39)(4,40)(5,36)(6,26)(7,27)(8,28)(9,29)(10,30)(11,31)(12,32)(13,33)(14,34)(15,35)(16,23)(17,24)(18,25)(19,21)(20,22)(41,76)(42,77)(43,78)(44,79)(45,80)(46,71)(47,72)(48,73)(49,74)(50,75)(51,63)(52,64)(53,65)(54,66)(55,67)(56,68)(57,69)(58,70)(59,61)(60,62), (1,77)(2,73)(3,79)(4,75)(5,71)(6,80)(7,76)(8,72)(9,78)(10,74)(11,59)(12,55)(13,51)(14,57)(15,53)(16,54)(17,60)(18,56)(19,52)(20,58)(21,64)(22,70)(23,66)(24,62)(25,68)(26,45)(27,41)(28,47)(29,43)(30,49)(31,61)(32,67)(33,63)(34,69)(35,65)(36,46)(37,42)(38,48)(39,44)(40,50), (1,13)(2,14)(3,15)(4,11)(5,12)(6,16)(7,17)(8,18)(9,19)(10,20)(21,29)(22,30)(23,26)(24,27)(25,28)(31,40)(32,36)(33,37)(34,38)(35,39)(41,62)(42,63)(43,64)(44,65)(45,66)(46,67)(47,68)(48,69)(49,70)(50,61)(51,77)(52,78)(53,79)(54,80)(55,71)(56,72)(57,73)(58,74)(59,75)(60,76), (1,8)(2,9)(3,10)(4,6)(5,7)(11,16)(12,17)(13,18)(14,19)(15,20)(21,34)(22,35)(23,31)(24,32)(25,33)(26,40)(27,36)(28,37)(29,38)(30,39)(41,46)(42,47)(43,48)(44,49)(45,50)(51,56)(52,57)(53,58)(54,59)(55,60)(61,66)(62,67)(63,68)(64,69)(65,70)(71,76)(72,77)(73,78)(74,79)(75,80), (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), (1,5)(2,4)(6,9)(7,8)(11,14)(12,13)(16,19)(17,18)(21,31)(22,35)(23,34)(24,33)(25,32)(26,38)(27,37)(28,36)(29,40)(30,39)(41,68)(42,67)(43,66)(44,65)(45,64)(46,63)(47,62)(48,61)(49,70)(50,69)(51,76)(52,75)(53,74)(54,73)(55,72)(56,71)(57,80)(58,79)(59,78)(60,77) );
G=PermutationGroup([(1,37),(2,38),(3,39),(4,40),(5,36),(6,26),(7,27),(8,28),(9,29),(10,30),(11,31),(12,32),(13,33),(14,34),(15,35),(16,23),(17,24),(18,25),(19,21),(20,22),(41,76),(42,77),(43,78),(44,79),(45,80),(46,71),(47,72),(48,73),(49,74),(50,75),(51,63),(52,64),(53,65),(54,66),(55,67),(56,68),(57,69),(58,70),(59,61),(60,62)], [(1,77),(2,73),(3,79),(4,75),(5,71),(6,80),(7,76),(8,72),(9,78),(10,74),(11,59),(12,55),(13,51),(14,57),(15,53),(16,54),(17,60),(18,56),(19,52),(20,58),(21,64),(22,70),(23,66),(24,62),(25,68),(26,45),(27,41),(28,47),(29,43),(30,49),(31,61),(32,67),(33,63),(34,69),(35,65),(36,46),(37,42),(38,48),(39,44),(40,50)], [(1,13),(2,14),(3,15),(4,11),(5,12),(6,16),(7,17),(8,18),(9,19),(10,20),(21,29),(22,30),(23,26),(24,27),(25,28),(31,40),(32,36),(33,37),(34,38),(35,39),(41,62),(42,63),(43,64),(44,65),(45,66),(46,67),(47,68),(48,69),(49,70),(50,61),(51,77),(52,78),(53,79),(54,80),(55,71),(56,72),(57,73),(58,74),(59,75),(60,76)], [(1,8),(2,9),(3,10),(4,6),(5,7),(11,16),(12,17),(13,18),(14,19),(15,20),(21,34),(22,35),(23,31),(24,32),(25,33),(26,40),(27,36),(28,37),(29,38),(30,39),(41,46),(42,47),(43,48),(44,49),(45,50),(51,56),(52,57),(53,58),(54,59),(55,60),(61,66),(62,67),(63,68),(64,69),(65,70),(71,76),(72,77),(73,78),(74,79),(75,80)], [(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)], [(1,5),(2,4),(6,9),(7,8),(11,14),(12,13),(16,19),(17,18),(21,31),(22,35),(23,34),(24,33),(25,32),(26,38),(27,37),(28,36),(29,40),(30,39),(41,68),(42,67),(43,66),(44,65),(45,64),(46,63),(47,62),(48,61),(49,70),(50,69),(51,76),(52,75),(53,74),(54,73),(55,72),(56,71),(57,80),(58,79),(59,78),(60,77)])
Matrix representation ►G ⊆ GL6(𝔽41)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 24 | 1 | 0 | 0 |
| 0 | 0 | 40 | 17 | 0 | 0 |
| 0 | 0 | 0 | 0 | 23 | 1 |
| 0 | 0 | 0 | 0 | 5 | 18 |
| 16 | 32 | 0 | 0 | 0 | 0 |
| 1 | 25 | 0 | 0 | 0 | 0 |
| 0 | 0 | 24 | 1 | 38 | 23 |
| 0 | 0 | 40 | 17 | 36 | 23 |
| 0 | 0 | 0 | 0 | 18 | 40 |
| 0 | 0 | 0 | 0 | 36 | 23 |
| 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 | 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 | 7 | 7 | 0 | 0 |
| 0 | 0 | 34 | 40 | 0 | 0 |
| 0 | 0 | 0 | 14 | 0 | 34 |
| 0 | 0 | 27 | 39 | 6 | 35 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 40 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 7 | 0 | 0 |
| 0 | 0 | 40 | 34 | 0 | 0 |
| 0 | 0 | 0 | 0 | 6 | 34 |
| 0 | 0 | 0 | 0 | 5 | 35 |
G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,24,40,0,0,0,0,1,17,0,0,0,0,0,0,23,5,0,0,0,0,1,18],[16,1,0,0,0,0,32,25,0,0,0,0,0,0,24,40,0,0,0,0,1,17,0,0,0,0,38,36,18,36,0,0,23,23,40,23],[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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,7,34,0,27,0,0,7,40,14,39,0,0,0,0,0,6,0,0,0,0,34,35],[1,40,0,0,0,0,0,40,0,0,0,0,0,0,7,40,0,0,0,0,7,34,0,0,0,0,0,0,6,5,0,0,0,0,34,35] >;
62 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | ··· | 2I | 2J | 2K | 2L | 2M | 4A | 4B | 4C | ··· | 4H | 5A | 5B | 10A | ··· | 10N | 10O | ··· | 10AD | 20A | ··· | 20H |
| order | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | 4 | 20 | 20 | 4 | 4 | 20 | ··· | 20 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
62 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D5 | D10 | D10 | D10 | C5⋊D4 | 2+ (1+4) | D4⋊6D10 |
| kernel | C24⋊8D10 | C23.23D10 | C23.18D10 | C23⋊D10 | Dic5⋊D4 | C2×C23.D5 | C24⋊2D5 | C22×C5⋊D4 | D4×C2×C10 | C22×C10 | C22×D4 | C22×C4 | C2×D4 | C24 | C23 | C10 | C2 |
| # reps | 1 | 2 | 2 | 2 | 4 | 1 | 2 | 1 | 1 | 4 | 2 | 2 | 8 | 4 | 16 | 2 | 8 |
In GAP, Magma, Sage, TeX
C_2^4\rtimes_8D_{10} % in TeX
G:=Group("C2^4:8D10"); // GroupNames label
G:=SmallGroup(320,1476);
// by ID
G=gap.SmallGroup(320,1476);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,758,675,570,12550]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=e^10=f^2=1,a*b=b*a,a*c=c*a,f*a*f=a*d=d*a,a*e=e*a,b*c=c*b,e*b*e^-1=b*d=d*b,f*b*f=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=e^-1>;
// generators/relations