metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C24.31D10, C10.62+ 1+4, C4⋊Dic5⋊7C22, (C2×C10).39C24, C22⋊C4.88D10, C24⋊2D5.1C2, Dic5⋊4D4⋊41C2, D10.12D4⋊3C2, C20.48D4⋊17C2, (C2×C20).575C23, Dic5.5D4⋊3C2, (C2×Dic10)⋊4C22, (C4×Dic5)⋊49C22, (C22×C4).172D10, C23.D10⋊2C2, C22.D20⋊3C2, C2.10(D4⋊6D10), D10⋊C4⋊49C22, C22.78(C23×D5), Dic5.14D4⋊3C2, C5⋊1(C22.45C24), C22.17(C4○D20), C23.21D10⋊3C2, C10.D4⋊50C22, (C23×C10).65C22, (C2×Dic5).12C23, (C22×D5).11C23, C23.222(C22×D5), C23.11D10⋊25C2, C22.23(D4⋊2D5), C23.23D10⋊10C2, (C22×C20).101C22, (C22×C10).129C23, C23.D5.140C22, (C22×Dic5).80C22, (C4×C5⋊D4)⋊2C2, (C2×C4×D5)⋊42C22, (C2×C22⋊C4)⋊18D5, C2.19(C2×C4○D20), C10.17(C2×C4○D4), (C10×C22⋊C4)⋊21C2, C2.12(C2×D4⋊2D5), (C2×C23.D5)⋊18C2, (C2×C5⋊D4).8C22, (C2×C10).40(C4○D4), (C2×C4).262(C22×D5), (C5×C22⋊C4).110C22, SmallGroup(320,1167)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C24.31D10
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, bd=db, be=eb, fbf-1=bcd, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e9 >
Subgroups: 806 in 248 conjugacy classes, 99 normal (91 characteristic)
C1, C2, C2, C4, C22, C22, C22, C5, C2×C4, C2×C4, D4, Q8, C23, C23, D5, C10, C10, C42, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×Q8, C24, Dic5, C20, D10, C2×C10, C2×C10, C2×C10, C2×C22⋊C4, C2×C22⋊C4, C42⋊C2, C4×D4, C22≀C2, C22⋊Q8, C22.D4, C4.4D4, C42⋊2C2, Dic10, C4×D5, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C22×D5, C22×C10, C22×C10, C22.45C24, C4×Dic5, C10.D4, C4⋊Dic5, D10⋊C4, C23.D5, C5×C22⋊C4, C2×Dic10, C2×C4×D5, C22×Dic5, C2×C5⋊D4, C22×C20, C23×C10, C23.11D10, Dic5.14D4, C23.D10, Dic5⋊4D4, D10.12D4, Dic5.5D4, C22.D20, C20.48D4, C23.21D10, C4×C5⋊D4, C23.23D10, C2×C23.D5, C24⋊2D5, C10×C22⋊C4, C24.31D10
Quotients: C1, C2, C22, C23, D5, C4○D4, C24, D10, C2×C4○D4, 2+ 1+4, C22×D5, C22.45C24, C4○D20, D4⋊2D5, C23×D5, C2×C4○D20, C2×D4⋊2D5, D4⋊6D10, C24.31D10
►On 80 points
(2 77)(4 79)(6 61)(8 63)(10 65)(12 67)(14 69)(16 71)(18 73)(20 75)(21 47)(22 32)(23 49)(24 34)(25 51)(26 36)(27 53)(28 38)(29 55)(30 40)(31 57)(33 59)(35 41)(37 43)(39 45)(42 52)(44 54)(46 56)(48 58)(50 60)
(21 47)(22 48)(23 49)(24 50)(25 51)(26 52)(27 53)(28 54)(29 55)(30 56)(31 57)(32 58)(33 59)(34 60)(35 41)(36 42)(37 43)(38 44)(39 45)(40 46)
(1 76)(2 77)(3 78)(4 79)(5 80)(6 61)(7 62)(8 63)(9 64)(10 65)(11 66)(12 67)(13 68)(14 69)(15 70)(16 71)(17 72)(18 73)(19 74)(20 75)(21 57)(22 58)(23 59)(24 60)(25 41)(26 42)(27 43)(28 44)(29 45)(30 46)(31 47)(32 48)(33 49)(34 50)(35 51)(36 52)(37 53)(38 54)(39 55)(40 56)
(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 29 11 39)(2 38 12 28)(3 27 13 37)(4 36 14 26)(5 25 15 35)(6 34 16 24)(7 23 17 33)(8 32 18 22)(9 21 19 31)(10 30 20 40)(41 70 51 80)(42 79 52 69)(43 68 53 78)(44 77 54 67)(45 66 55 76)(46 75 56 65)(47 64 57 74)(48 73 58 63)(49 62 59 72)(50 71 60 61)
G:=sub<Sym(80)| (2,77)(4,79)(6,61)(8,63)(10,65)(12,67)(14,69)(16,71)(18,73)(20,75)(21,47)(22,32)(23,49)(24,34)(25,51)(26,36)(27,53)(28,38)(29,55)(30,40)(31,57)(33,59)(35,41)(37,43)(39,45)(42,52)(44,54)(46,56)(48,58)(50,60), (21,47)(22,48)(23,49)(24,50)(25,51)(26,52)(27,53)(28,54)(29,55)(30,56)(31,57)(32,58)(33,59)(34,60)(35,41)(36,42)(37,43)(38,44)(39,45)(40,46), (1,76)(2,77)(3,78)(4,79)(5,80)(6,61)(7,62)(8,63)(9,64)(10,65)(11,66)(12,67)(13,68)(14,69)(15,70)(16,71)(17,72)(18,73)(19,74)(20,75)(21,57)(22,58)(23,59)(24,60)(25,41)(26,42)(27,43)(28,44)(29,45)(30,46)(31,47)(32,48)(33,49)(34,50)(35,51)(36,52)(37,53)(38,54)(39,55)(40,56), (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), (1,29,11,39)(2,38,12,28)(3,27,13,37)(4,36,14,26)(5,25,15,35)(6,34,16,24)(7,23,17,33)(8,32,18,22)(9,21,19,31)(10,30,20,40)(41,70,51,80)(42,79,52,69)(43,68,53,78)(44,77,54,67)(45,66,55,76)(46,75,56,65)(47,64,57,74)(48,73,58,63)(49,62,59,72)(50,71,60,61)>;
G:=Group( (2,77)(4,79)(6,61)(8,63)(10,65)(12,67)(14,69)(16,71)(18,73)(20,75)(21,47)(22,32)(23,49)(24,34)(25,51)(26,36)(27,53)(28,38)(29,55)(30,40)(31,57)(33,59)(35,41)(37,43)(39,45)(42,52)(44,54)(46,56)(48,58)(50,60), (21,47)(22,48)(23,49)(24,50)(25,51)(26,52)(27,53)(28,54)(29,55)(30,56)(31,57)(32,58)(33,59)(34,60)(35,41)(36,42)(37,43)(38,44)(39,45)(40,46), (1,76)(2,77)(3,78)(4,79)(5,80)(6,61)(7,62)(8,63)(9,64)(10,65)(11,66)(12,67)(13,68)(14,69)(15,70)(16,71)(17,72)(18,73)(19,74)(20,75)(21,57)(22,58)(23,59)(24,60)(25,41)(26,42)(27,43)(28,44)(29,45)(30,46)(31,47)(32,48)(33,49)(34,50)(35,51)(36,52)(37,53)(38,54)(39,55)(40,56), (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), (1,29,11,39)(2,38,12,28)(3,27,13,37)(4,36,14,26)(5,25,15,35)(6,34,16,24)(7,23,17,33)(8,32,18,22)(9,21,19,31)(10,30,20,40)(41,70,51,80)(42,79,52,69)(43,68,53,78)(44,77,54,67)(45,66,55,76)(46,75,56,65)(47,64,57,74)(48,73,58,63)(49,62,59,72)(50,71,60,61) );
G=PermutationGroup([[(2,77),(4,79),(6,61),(8,63),(10,65),(12,67),(14,69),(16,71),(18,73),(20,75),(21,47),(22,32),(23,49),(24,34),(25,51),(26,36),(27,53),(28,38),(29,55),(30,40),(31,57),(33,59),(35,41),(37,43),(39,45),(42,52),(44,54),(46,56),(48,58),(50,60)], [(21,47),(22,48),(23,49),(24,50),(25,51),(26,52),(27,53),(28,54),(29,55),(30,56),(31,57),(32,58),(33,59),(34,60),(35,41),(36,42),(37,43),(38,44),(39,45),(40,46)], [(1,76),(2,77),(3,78),(4,79),(5,80),(6,61),(7,62),(8,63),(9,64),(10,65),(11,66),(12,67),(13,68),(14,69),(15,70),(16,71),(17,72),(18,73),(19,74),(20,75),(21,57),(22,58),(23,59),(24,60),(25,41),(26,42),(27,43),(28,44),(29,45),(30,46),(31,47),(32,48),(33,49),(34,50),(35,51),(36,52),(37,53),(38,54),(39,55),(40,56)], [(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)], [(1,29,11,39),(2,38,12,28),(3,27,13,37),(4,36,14,26),(5,25,15,35),(6,34,16,24),(7,23,17,33),(8,32,18,22),(9,21,19,31),(10,30,20,40),(41,70,51,80),(42,79,52,69),(43,68,53,78),(44,77,54,67),(45,66,55,76),(46,75,56,65),(47,64,57,74),(48,73,58,63),(49,62,59,72),(50,71,60,61)]])
65 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | ··· | 4O | 5A | 5B | 10A | ··· | 10N | 10O | ··· | 10V | 20A | ··· | 20P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 20 | 2 | 2 | 2 | 2 | 4 | 4 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
65 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 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D5 | C4○D4 | D10 | D10 | D10 | C4○D20 | 2+ 1+4 | D4⋊2D5 | D4⋊6D10 |
| kernel | C24.31D10 | C23.11D10 | Dic5.14D4 | C23.D10 | Dic5⋊4D4 | D10.12D4 | Dic5.5D4 | C22.D20 | C20.48D4 | C23.21D10 | C4×C5⋊D4 | C23.23D10 | C2×C23.D5 | C24⋊2D5 | C10×C22⋊C4 | C2×C22⋊C4 | C2×C10 | C22⋊C4 | C22×C4 | C24 | C22 | C10 | C22 | C2 |
| # reps | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 8 | 8 | 4 | 2 | 16 | 1 | 4 | 4 |
Matrix representation of C24.31D10 ►in GL4(𝔽41) generated by
| 1 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 24 | 40 |
| 1 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 21 | 0 | 0 | 0 |
| 0 | 39 | 0 | 0 |
| 0 | 0 | 1 | 17 |
| 0 | 0 | 24 | 40 |
| 0 | 39 | 0 | 0 |
| 21 | 0 | 0 | 0 |
| 0 | 0 | 32 | 0 |
| 0 | 0 | 0 | 32 |
G:=sub<GL(4,GF(41))| [1,0,0,0,0,40,0,0,0,0,1,24,0,0,0,40],[1,0,0,0,0,40,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,40,0,0,0,0,40],[40,0,0,0,0,40,0,0,0,0,40,0,0,0,0,40],[21,0,0,0,0,39,0,0,0,0,1,24,0,0,17,40],[0,21,0,0,39,0,0,0,0,0,32,0,0,0,0,32] >;
C24.31D10 in GAP, Magma, Sage, TeX
C_2^4._{31}D_{10} % in TeX
G:=Group("C2^4.31D10"); // GroupNames label
G:=SmallGroup(320,1167);
// by ID
G=gap.SmallGroup(320,1167);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,758,219,1571,12550]);
// 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,b*d=d*b,b*e=e*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