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, (C22×C4).272D10, C10.147(C22×D4), C23.D5⋊39C22, C2.93(D4⋊6D10), (C2×Dic10)⋊68C22, (D4×C10).271C22, (C23×C10).79C22, C22.313(C23×D5), C23.136(C22×D5), C23.21D10⋊33C2, (C22×C20).277C22, (C22×C10).234C23, (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
Generators and relations for C24.41D10
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 >
Subgroups: 1198 in 334 conjugacy classes, 111 normal (21 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C5, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C23, D5, C10, C10, C10, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C24, Dic5, C20, D10, C2×C10, C2×C10, C2×C10, C42⋊C2, C22≀C2, C4⋊D4, C4.4D4, C4⋊1D4, C22×D4, C2×C4○D4, Dic10, C4×D5, D20, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C5×D4, C22×D5, C22×C10, C22×C10, C22×C10, C22.29C24, C4×Dic5, C4⋊Dic5, C23.D5, C2×Dic10, C2×C4×D5, C2×D20, C4○D20, C2×C5⋊D4, C22×C20, D4×C10, D4×C10, C23×C10, C23.21D10, C20.17D4, C20⋊2D4, C20⋊D4, C24⋊2D5, C2×C4○D20, D4×C2×C10, C24.41D10
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C24, D10, C22×D4, 2+ 1+4, C5⋊D4, C22×D5, C22.29C24, C2×C5⋊D4, C23×D5, D4⋊6D10, C22×C5⋊D4, C24.41D10
►On 80 points
(2 12)(4 14)(6 16)(8 18)(10 20)(21 79)(22 70)(23 61)(24 72)(25 63)(26 74)(27 65)(28 76)(29 67)(30 78)(31 69)(32 80)(33 71)(34 62)(35 73)(36 64)(37 75)(38 66)(39 77)(40 68)(42 52)(44 54)(46 56)(48 58)(50 60)
(21 31)(22 32)(23 33)(24 34)(25 35)(26 36)(27 37)(28 38)(29 39)(30 40)(61 71)(62 72)(63 73)(64 74)(65 75)(66 76)(67 77)(68 78)(69 79)(70 80)
(1 47)(2 48)(3 49)(4 50)(5 51)(6 52)(7 53)(8 54)(9 55)(10 56)(11 57)(12 58)(13 59)(14 60)(15 41)(16 42)(17 43)(18 44)(19 45)(20 46)(21 69)(22 70)(23 71)(24 72)(25 73)(26 74)(27 75)(28 76)(29 77)(30 78)(31 79)(32 80)(33 61)(34 62)(35 63)(36 64)(37 65)(38 66)(39 67)(40 68)
(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 37 11 27)(2 26 12 36)(3 35 13 25)(4 24 14 34)(5 33 15 23)(6 22 16 32)(7 31 17 21)(8 40 18 30)(9 29 19 39)(10 38 20 28)(41 71 51 61)(42 80 52 70)(43 69 53 79)(44 78 54 68)(45 67 55 77)(46 76 56 66)(47 65 57 75)(48 74 58 64)(49 63 59 73)(50 72 60 62)
G:=sub<Sym(80)| (2,12)(4,14)(6,16)(8,18)(10,20)(21,79)(22,70)(23,61)(24,72)(25,63)(26,74)(27,65)(28,76)(29,67)(30,78)(31,69)(32,80)(33,71)(34,62)(35,73)(36,64)(37,75)(38,66)(39,77)(40,68)(42,52)(44,54)(46,56)(48,58)(50,60), (21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40)(61,71)(62,72)(63,73)(64,74)(65,75)(66,76)(67,77)(68,78)(69,79)(70,80), (1,47)(2,48)(3,49)(4,50)(5,51)(6,52)(7,53)(8,54)(9,55)(10,56)(11,57)(12,58)(13,59)(14,60)(15,41)(16,42)(17,43)(18,44)(19,45)(20,46)(21,69)(22,70)(23,71)(24,72)(25,73)(26,74)(27,75)(28,76)(29,77)(30,78)(31,79)(32,80)(33,61)(34,62)(35,63)(36,64)(37,65)(38,66)(39,67)(40,68), (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), (1,37,11,27)(2,26,12,36)(3,35,13,25)(4,24,14,34)(5,33,15,23)(6,22,16,32)(7,31,17,21)(8,40,18,30)(9,29,19,39)(10,38,20,28)(41,71,51,61)(42,80,52,70)(43,69,53,79)(44,78,54,68)(45,67,55,77)(46,76,56,66)(47,65,57,75)(48,74,58,64)(49,63,59,73)(50,72,60,62)>;
G:=Group( (2,12)(4,14)(6,16)(8,18)(10,20)(21,79)(22,70)(23,61)(24,72)(25,63)(26,74)(27,65)(28,76)(29,67)(30,78)(31,69)(32,80)(33,71)(34,62)(35,73)(36,64)(37,75)(38,66)(39,77)(40,68)(42,52)(44,54)(46,56)(48,58)(50,60), (21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40)(61,71)(62,72)(63,73)(64,74)(65,75)(66,76)(67,77)(68,78)(69,79)(70,80), (1,47)(2,48)(3,49)(4,50)(5,51)(6,52)(7,53)(8,54)(9,55)(10,56)(11,57)(12,58)(13,59)(14,60)(15,41)(16,42)(17,43)(18,44)(19,45)(20,46)(21,69)(22,70)(23,71)(24,72)(25,73)(26,74)(27,75)(28,76)(29,77)(30,78)(31,79)(32,80)(33,61)(34,62)(35,63)(36,64)(37,65)(38,66)(39,67)(40,68), (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), (1,37,11,27)(2,26,12,36)(3,35,13,25)(4,24,14,34)(5,33,15,23)(6,22,16,32)(7,31,17,21)(8,40,18,30)(9,29,19,39)(10,38,20,28)(41,71,51,61)(42,80,52,70)(43,69,53,79)(44,78,54,68)(45,67,55,77)(46,76,56,66)(47,65,57,75)(48,74,58,64)(49,63,59,73)(50,72,60,62) );
G=PermutationGroup([[(2,12),(4,14),(6,16),(8,18),(10,20),(21,79),(22,70),(23,61),(24,72),(25,63),(26,74),(27,65),(28,76),(29,67),(30,78),(31,69),(32,80),(33,71),(34,62),(35,73),(36,64),(37,75),(38,66),(39,77),(40,68),(42,52),(44,54),(46,56),(48,58),(50,60)], [(21,31),(22,32),(23,33),(24,34),(25,35),(26,36),(27,37),(28,38),(29,39),(30,40),(61,71),(62,72),(63,73),(64,74),(65,75),(66,76),(67,77),(68,78),(69,79),(70,80)], [(1,47),(2,48),(3,49),(4,50),(5,51),(6,52),(7,53),(8,54),(9,55),(10,56),(11,57),(12,58),(13,59),(14,60),(15,41),(16,42),(17,43),(18,44),(19,45),(20,46),(21,69),(22,70),(23,71),(24,72),(25,73),(26,74),(27,75),(28,76),(29,77),(30,78),(31,79),(32,80),(33,61),(34,62),(35,63),(36,64),(37,65),(38,66),(39,67),(40,68)], [(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)], [(1,37,11,27),(2,26,12,36),(3,35,13,25),(4,24,14,34),(5,33,15,23),(6,22,16,32),(7,31,17,21),(8,40,18,30),(9,29,19,39),(10,38,20,28),(41,71,51,61),(42,80,52,70),(43,69,53,79),(44,78,54,68),(45,67,55,77),(46,76,56,66),(47,65,57,75),(48,74,58,64),(49,63,59,73),(50,72,60,62)]])
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 |
Matrix representation of C24.41D10 ►in 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] >;
C24.41D10 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