metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C24.41D10, C10.902+ (1+4), (C2×C20)⋊14D4, C20⋊D4⋊29C2, C20⋊2D4⋊40C2, C20.252(C2×D4), (C22×D4)⋊11D5, (C2×D4).230D10, (C2×D20)⋊57C22, C24⋊2D5⋊13C2, C4⋊Dic5⋊78C22, C20.17D4⋊28C2, (C2×C20).545C23, (C2×C10).300C24, C5⋊6(C22.29C24), (C4×Dic5)⋊42C22, C10.147(C22×D4), (C22×C4).272D10, C2.93(D4⋊6D10), C23.D5⋊39C22, (C2×Dic10)⋊68C22, (D4×C10).271C22, (C23×C10).79C22, C23.136(C22×D5), C22.313(C23×D5), C23.21D10⋊33C2, (C22×C10).234C23, (C22×C20).277C22, (C2×Dic5).155C23, (C22×D5).131C23, (D4×C2×C10)⋊7C2, (C2×C4)⋊6(C5⋊D4), (C2×C4×D5)⋊31C22, C4.97(C2×C5⋊D4), (C2×C4○D20)⋊29C2, (C2×C10).583(C2×D4), (C2×C5⋊D4)⋊28C22, C2.20(C22×C5⋊D4), C22.36(C2×C5⋊D4), (C2×C4).628(C22×D5), SmallGroup(320,1477)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1198 in 334 conjugacy classes, 111 normal (21 characteristic)
C1, C2, C2 [×2], C2 [×8], C4 [×4], C4 [×6], C22, C22 [×2], C22 [×28], C5, C2×C4 [×2], C2×C4 [×4], C2×C4 [×10], D4 [×22], Q8 [×2], C23, C23 [×4], C23 [×10], D5 [×2], C10, C10 [×2], C10 [×6], C42 [×2], C22⋊C4 [×10], C4⋊C4 [×2], C22×C4, C22×C4 [×2], C2×D4 [×4], C2×D4 [×15], C2×Q8, C4○D4 [×4], C24 [×2], Dic5 [×6], C20 [×4], D10 [×6], C2×C10, C2×C10 [×2], C2×C10 [×22], C42⋊C2, C22≀C2 [×4], C4⋊D4 [×4], C4.4D4 [×2], C4⋊1D4 [×2], C22×D4, C2×C4○D4, Dic10 [×2], C4×D5 [×4], D20 [×2], C2×Dic5 [×6], C5⋊D4 [×12], C2×C20 [×2], C2×C20 [×4], C5×D4 [×8], C22×D5 [×2], C22×C10, C22×C10 [×4], C22×C10 [×8], C22.29C24, C4×Dic5 [×2], C4⋊Dic5 [×2], C23.D5 [×10], C2×Dic10, C2×C4×D5 [×2], C2×D20, C4○D20 [×4], C2×C5⋊D4 [×10], C22×C20, D4×C10 [×4], D4×C10 [×4], C23×C10 [×2], C23.21D10, C20.17D4 [×2], C20⋊2D4 [×4], C20⋊D4 [×2], C24⋊2D5 [×4], C2×C4○D20, D4×C2×C10, C24.41D10
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], C22.29C24, C2×C5⋊D4 [×6], C23×D5, D4⋊6D10 [×2], C22×C5⋊D4, C24.41D10
Generators and relations
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e10=f2=d, ab=ba, ac=ca, eae-1=ad=da, faf-1=acd, bc=cb, fbf-1=bd=db, be=eb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e9 >
►On 80 points
(2 12)(4 14)(6 16)(8 18)(10 20)(21 57)(22 48)(23 59)(24 50)(25 41)(26 52)(27 43)(28 54)(29 45)(30 56)(31 47)(32 58)(33 49)(34 60)(35 51)(36 42)(37 53)(38 44)(39 55)(40 46)(62 72)(64 74)(66 76)(68 78)(70 80)
(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)
(1 75)(2 76)(3 77)(4 78)(5 79)(6 80)(7 61)(8 62)(9 63)(10 64)(11 65)(12 66)(13 67)(14 68)(15 69)(16 70)(17 71)(18 72)(19 73)(20 74)(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 24 11 34)(2 33 12 23)(3 22 13 32)(4 31 14 21)(5 40 15 30)(6 29 16 39)(7 38 17 28)(8 27 18 37)(9 36 19 26)(10 25 20 35)(41 74 51 64)(42 63 52 73)(43 72 53 62)(44 61 54 71)(45 70 55 80)(46 79 56 69)(47 68 57 78)(48 77 58 67)(49 66 59 76)(50 75 60 65)
G:=sub<Sym(80)| (2,12)(4,14)(6,16)(8,18)(10,20)(21,57)(22,48)(23,59)(24,50)(25,41)(26,52)(27,43)(28,54)(29,45)(30,56)(31,47)(32,58)(33,49)(34,60)(35,51)(36,42)(37,53)(38,44)(39,55)(40,46)(62,72)(64,74)(66,76)(68,78)(70,80), (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), (1,75)(2,76)(3,77)(4,78)(5,79)(6,80)(7,61)(8,62)(9,63)(10,64)(11,65)(12,66)(13,67)(14,68)(15,69)(16,70)(17,71)(18,72)(19,73)(20,74)(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), (1,24,11,34)(2,33,12,23)(3,22,13,32)(4,31,14,21)(5,40,15,30)(6,29,16,39)(7,38,17,28)(8,27,18,37)(9,36,19,26)(10,25,20,35)(41,74,51,64)(42,63,52,73)(43,72,53,62)(44,61,54,71)(45,70,55,80)(46,79,56,69)(47,68,57,78)(48,77,58,67)(49,66,59,76)(50,75,60,65)>;
G:=Group( (2,12)(4,14)(6,16)(8,18)(10,20)(21,57)(22,48)(23,59)(24,50)(25,41)(26,52)(27,43)(28,54)(29,45)(30,56)(31,47)(32,58)(33,49)(34,60)(35,51)(36,42)(37,53)(38,44)(39,55)(40,46)(62,72)(64,74)(66,76)(68,78)(70,80), (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), (1,75)(2,76)(3,77)(4,78)(5,79)(6,80)(7,61)(8,62)(9,63)(10,64)(11,65)(12,66)(13,67)(14,68)(15,69)(16,70)(17,71)(18,72)(19,73)(20,74)(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), (1,24,11,34)(2,33,12,23)(3,22,13,32)(4,31,14,21)(5,40,15,30)(6,29,16,39)(7,38,17,28)(8,27,18,37)(9,36,19,26)(10,25,20,35)(41,74,51,64)(42,63,52,73)(43,72,53,62)(44,61,54,71)(45,70,55,80)(46,79,56,69)(47,68,57,78)(48,77,58,67)(49,66,59,76)(50,75,60,65) );
G=PermutationGroup([(2,12),(4,14),(6,16),(8,18),(10,20),(21,57),(22,48),(23,59),(24,50),(25,41),(26,52),(27,43),(28,54),(29,45),(30,56),(31,47),(32,58),(33,49),(34,60),(35,51),(36,42),(37,53),(38,44),(39,55),(40,46),(62,72),(64,74),(66,76),(68,78),(70,80)], [(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)], [(1,75),(2,76),(3,77),(4,78),(5,79),(6,80),(7,61),(8,62),(9,63),(10,64),(11,65),(12,66),(13,67),(14,68),(15,69),(16,70),(17,71),(18,72),(19,73),(20,74),(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)], [(1,24,11,34),(2,33,12,23),(3,22,13,32),(4,31,14,21),(5,40,15,30),(6,29,16,39),(7,38,17,28),(8,27,18,37),(9,36,19,26),(10,25,20,35),(41,74,51,64),(42,63,52,73),(43,72,53,62),(44,61,54,71),(45,70,55,80),(46,79,56,69),(47,68,57,78),(48,77,58,67),(49,66,59,76),(50,75,60,65)])
Matrix representation ►G ⊆ GL6(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 |
| 17 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 21 | 40 | 0 | 0 |
| 0 | 0 | 26 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 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 | 26 | 0 | 40 | 0 |
| 0 | 0 | 26 | 0 | 0 | 40 |
| 40 | 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 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 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 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 25 | 0 | 0 |
| 0 | 0 | 2 | 37 | 0 | 0 |
| 0 | 0 | 18 | 38 | 0 | 31 |
| 0 | 0 | 4 | 38 | 10 | 0 |
| 1 | 17 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 13 | 0 | 0 | 40 |
| 0 | 0 | 27 | 0 | 10 | 10 |
| 0 | 0 | 4 | 4 | 0 | 28 |
| 0 | 0 | 6 | 0 | 0 | 28 |
G:=sub<GL(6,GF(41))| [40,17,0,0,0,0,0,1,0,0,0,0,0,0,1,21,26,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,26,26,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,2,18,4,0,0,25,37,38,38,0,0,0,0,0,10,0,0,0,0,31,0],[1,0,0,0,0,0,17,40,0,0,0,0,0,0,13,27,4,6,0,0,0,0,4,0,0,0,0,10,0,0,0,0,40,10,28,28] >;
62 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 4A | 4B | 4C | 4D | 4E | ··· | 4J | 5A | 5B | 10A | ··· | 10N | 10O | ··· | 10AD | 20A | ··· | 20H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 20 | 20 | 2 | 2 | 2 | 2 | 20 | ··· | 20 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
62 irreducible representations
| dim | 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 | D4 | D5 | D10 | D10 | D10 | C5⋊D4 | 2+ (1+4) | D4⋊6D10 |
| kernel | C24.41D10 | C23.21D10 | C20.17D4 | C20⋊2D4 | C20⋊D4 | C24⋊2D5 | C2×C4○D20 | D4×C2×C10 | C2×C20 | C22×D4 | C22×C4 | C2×D4 | C24 | C2×C4 | C10 | C2 |
| # reps | 1 | 1 | 2 | 4 | 2 | 4 | 1 | 1 | 4 | 2 | 2 | 8 | 4 | 16 | 2 | 8 |
In GAP, Magma, Sage, TeX
C_2^4._{41}D_{10} % in TeX
G:=Group("C2^4.41D10"); // GroupNames label
G:=SmallGroup(320,1477);
// by ID
G=gap.SmallGroup(320,1477);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,758,675,570,12550]);
// 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,a*c=c*a,e*a*e^-1=a*d=d*a,f*a*f^-1=a*c*d,b*c=c*b,f*b*f^-1=b*d=d*b,b*e=e*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