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, (C4×Dic5)⋊49C22, (C2×Dic10)⋊4C22, C23.D10⋊2C2, (C22×C4).172D10, 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), C10.D4⋊50C22, C23.21D10⋊3C2, (C23×C10).65C22, (C2×Dic5).12C23, (C22×D5).11C23, C23.222(C22×D5), C23.11D10⋊25C2, C23.23D10⋊10C2, C22.23(D4⋊2D5), (C22×C10).129C23, (C22×C20).101C22, C23.D5.140C22, (C22×Dic5).80C22, (C4×C5⋊D4)⋊2C2, (C2×C4×D5)⋊42C22, (C2×C22⋊C4)⋊18D5, C10.17(C2×C4○D4), C2.19(C2×C4○D20), (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
Subgroups: 806 in 248 conjugacy classes, 99 normal (91 characteristic)
C1, C2 [×3], C2 [×6], C4 [×11], C22, C22 [×4], C22 [×14], C5, C2×C4 [×4], C2×C4 [×14], D4 [×5], Q8, C23 [×3], C23 [×6], D5, C10 [×3], C10 [×5], C42 [×3], C22⋊C4 [×4], C22⋊C4 [×10], C4⋊C4 [×8], C22×C4 [×2], C22×C4 [×3], C2×D4 [×3], C2×Q8, C24, Dic5 [×7], C20 [×4], D10 [×3], C2×C10, C2×C10 [×4], C2×C10 [×11], C2×C22⋊C4, C2×C22⋊C4, C42⋊C2 [×2], C4×D4 [×2], C22≀C2, C22⋊Q8 [×2], C22.D4 [×3], C4.4D4, C42⋊2C2 [×2], Dic10, C4×D5, C2×Dic5 [×7], C2×Dic5 [×3], C5⋊D4 [×5], C2×C20 [×4], C2×C20 [×3], C22×D5, C22×C10 [×3], C22×C10 [×5], C22.45C24, C4×Dic5 [×3], C10.D4 [×5], C4⋊Dic5 [×3], D10⋊C4 [×3], C23.D5 [×7], C5×C22⋊C4 [×4], C2×Dic10, C2×C4×D5, C22×Dic5 [×2], C2×C5⋊D4 [×3], C22×C20 [×2], C23×C10, C23.11D10, Dic5.14D4, C23.D10 [×2], 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 [×15], C22 [×35], C23 [×15], D5, C4○D4 [×4], C24, D10 [×7], C2×C4○D4 [×2], 2+ (1+4), C22×D5 [×7], C22.45C24, C4○D20 [×2], D4⋊2D5 [×2], C23×D5, C2×C4○D20, C2×D4⋊2D5, D4⋊6D10, C24.31D10
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, bd=db, be=eb, fbf-1=bcd, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e9 >
►On 80 points
(2 35)(4 37)(6 39)(8 21)(10 23)(12 25)(14 27)(16 29)(18 31)(20 33)(41 51)(42 77)(43 53)(44 79)(45 55)(46 61)(47 57)(48 63)(49 59)(50 65)(52 67)(54 69)(56 71)(58 73)(60 75)(62 72)(64 74)(66 76)(68 78)(70 80)
(41 76)(42 77)(43 78)(44 79)(45 80)(46 61)(47 62)(48 63)(49 64)(50 65)(51 66)(52 67)(53 68)(54 69)(55 70)(56 71)(57 72)(58 73)(59 74)(60 75)
(1 34)(2 35)(3 36)(4 37)(5 38)(6 39)(7 40)(8 21)(9 22)(10 23)(11 24)(12 25)(13 26)(14 27)(15 28)(16 29)(17 30)(18 31)(19 32)(20 33)(41 66)(42 67)(43 68)(44 69)(45 70)(46 71)(47 72)(48 73)(49 74)(50 75)(51 76)(52 77)(53 78)(54 79)(55 80)(56 61)(57 62)(58 63)(59 64)(60 65)
(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 61 11 71)(2 70 12 80)(3 79 13 69)(4 68 14 78)(5 77 15 67)(6 66 16 76)(7 75 17 65)(8 64 18 74)(9 73 19 63)(10 62 20 72)(21 59 31 49)(22 48 32 58)(23 57 33 47)(24 46 34 56)(25 55 35 45)(26 44 36 54)(27 53 37 43)(28 42 38 52)(29 51 39 41)(30 60 40 50)
G:=sub<Sym(80)| (2,35)(4,37)(6,39)(8,21)(10,23)(12,25)(14,27)(16,29)(18,31)(20,33)(41,51)(42,77)(43,53)(44,79)(45,55)(46,61)(47,57)(48,63)(49,59)(50,65)(52,67)(54,69)(56,71)(58,73)(60,75)(62,72)(64,74)(66,76)(68,78)(70,80), (41,76)(42,77)(43,78)(44,79)(45,80)(46,61)(47,62)(48,63)(49,64)(50,65)(51,66)(52,67)(53,68)(54,69)(55,70)(56,71)(57,72)(58,73)(59,74)(60,75), (1,34)(2,35)(3,36)(4,37)(5,38)(6,39)(7,40)(8,21)(9,22)(10,23)(11,24)(12,25)(13,26)(14,27)(15,28)(16,29)(17,30)(18,31)(19,32)(20,33)(41,66)(42,67)(43,68)(44,69)(45,70)(46,71)(47,72)(48,73)(49,74)(50,75)(51,76)(52,77)(53,78)(54,79)(55,80)(56,61)(57,62)(58,63)(59,64)(60,65), (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), (1,61,11,71)(2,70,12,80)(3,79,13,69)(4,68,14,78)(5,77,15,67)(6,66,16,76)(7,75,17,65)(8,64,18,74)(9,73,19,63)(10,62,20,72)(21,59,31,49)(22,48,32,58)(23,57,33,47)(24,46,34,56)(25,55,35,45)(26,44,36,54)(27,53,37,43)(28,42,38,52)(29,51,39,41)(30,60,40,50)>;
G:=Group( (2,35)(4,37)(6,39)(8,21)(10,23)(12,25)(14,27)(16,29)(18,31)(20,33)(41,51)(42,77)(43,53)(44,79)(45,55)(46,61)(47,57)(48,63)(49,59)(50,65)(52,67)(54,69)(56,71)(58,73)(60,75)(62,72)(64,74)(66,76)(68,78)(70,80), (41,76)(42,77)(43,78)(44,79)(45,80)(46,61)(47,62)(48,63)(49,64)(50,65)(51,66)(52,67)(53,68)(54,69)(55,70)(56,71)(57,72)(58,73)(59,74)(60,75), (1,34)(2,35)(3,36)(4,37)(5,38)(6,39)(7,40)(8,21)(9,22)(10,23)(11,24)(12,25)(13,26)(14,27)(15,28)(16,29)(17,30)(18,31)(19,32)(20,33)(41,66)(42,67)(43,68)(44,69)(45,70)(46,71)(47,72)(48,73)(49,74)(50,75)(51,76)(52,77)(53,78)(54,79)(55,80)(56,61)(57,62)(58,63)(59,64)(60,65), (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), (1,61,11,71)(2,70,12,80)(3,79,13,69)(4,68,14,78)(5,77,15,67)(6,66,16,76)(7,75,17,65)(8,64,18,74)(9,73,19,63)(10,62,20,72)(21,59,31,49)(22,48,32,58)(23,57,33,47)(24,46,34,56)(25,55,35,45)(26,44,36,54)(27,53,37,43)(28,42,38,52)(29,51,39,41)(30,60,40,50) );
G=PermutationGroup([(2,35),(4,37),(6,39),(8,21),(10,23),(12,25),(14,27),(16,29),(18,31),(20,33),(41,51),(42,77),(43,53),(44,79),(45,55),(46,61),(47,57),(48,63),(49,59),(50,65),(52,67),(54,69),(56,71),(58,73),(60,75),(62,72),(64,74),(66,76),(68,78),(70,80)], [(41,76),(42,77),(43,78),(44,79),(45,80),(46,61),(47,62),(48,63),(49,64),(50,65),(51,66),(52,67),(53,68),(54,69),(55,70),(56,71),(57,72),(58,73),(59,74),(60,75)], [(1,34),(2,35),(3,36),(4,37),(5,38),(6,39),(7,40),(8,21),(9,22),(10,23),(11,24),(12,25),(13,26),(14,27),(15,28),(16,29),(17,30),(18,31),(19,32),(20,33),(41,66),(42,67),(43,68),(44,69),(45,70),(46,71),(47,72),(48,73),(49,74),(50,75),(51,76),(52,77),(53,78),(54,79),(55,80),(56,61),(57,62),(58,63),(59,64),(60,65)], [(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)], [(1,61,11,71),(2,70,12,80),(3,79,13,69),(4,68,14,78),(5,77,15,67),(6,66,16,76),(7,75,17,65),(8,64,18,74),(9,73,19,63),(10,62,20,72),(21,59,31,49),(22,48,32,58),(23,57,33,47),(24,46,34,56),(25,55,35,45),(26,44,36,54),(27,53,37,43),(28,42,38,52),(29,51,39,41),(30,60,40,50)])
Matrix representation ►G ⊆ 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] >;
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 |
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