metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C24.24D10, C10.22+ 1+4, C23⋊3(C4×D5), (C22×C4)⋊3D10, C22⋊C4⋊51D10, (C2×C10).30C24, C10.30(C23×C4), Dic5⋊4D4⋊39C2, C2.1(D4⋊6D10), (C2×C20).571C23, (C22×C20)⋊34C22, C5⋊2(C22.11C24), (C4×Dic5)⋊46C22, D10.11(C22×C4), C23.D5⋊67C22, D10⋊C4⋊57C22, C22.19(C23×D5), C10.D4⋊58C22, (C23×C10).56C22, Dic5.11(C22×C4), (C22×Dic5)⋊5C22, (C23×D5).30C22, C23.220(C22×D5), C23.11D10⋊24C2, (C22×C10).122C23, (C2×Dic5).187C23, (C22×D5).160C23, (C4×C5⋊D4)⋊33C2, (C2×C5⋊D4)⋊14C4, C5⋊D4⋊13(C2×C4), (C2×C22⋊C4)⋊7D5, (C2×C4×D5)⋊39C22, C22.24(C2×C4×D5), C2.11(D5×C22×C4), (D5×C22⋊C4)⋊23C2, (C22×D5)⋊8(C2×C4), (C10×C22⋊C4)⋊26C2, (C22×C10)⋊17(C2×C4), (C2×Dic5)⋊12(C2×C4), (C2×C23.D5)⋊15C2, (C22×C5⋊D4).9C2, (C5×C22⋊C4)⋊61C22, (C2×C4).257(C22×D5), (C2×C5⋊D4).96C22, (C2×C10).119(C22×C4), SmallGroup(320,1158)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C24.24D10
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e10=f2=c, ab=ba, ac=ca, eae-1=faf-1=ad=da, bc=cb, fbf-1=bd=db, be=eb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e9 >
Subgroups: 1166 in 338 conjugacy classes, 151 normal (21 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C5, C2×C4, C2×C4, D4, C23, C23, C23, D5, C10, C10, C10, C42, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C24, C24, Dic5, Dic5, C20, D10, D10, C2×C10, C2×C10, C2×C10, C2×C22⋊C4, C2×C22⋊C4, C42⋊C2, C4×D4, C22×D4, C4×D5, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C22×D5, C22×D5, C22×C10, C22×C10, C22×C10, C22.11C24, C4×Dic5, C10.D4, D10⋊C4, C23.D5, C5×C22⋊C4, C2×C4×D5, C22×Dic5, C22×Dic5, C2×C5⋊D4, C22×C20, C23×D5, C23×C10, C23.11D10, D5×C22⋊C4, Dic5⋊4D4, C4×C5⋊D4, C2×C23.D5, C10×C22⋊C4, C22×C5⋊D4, C24.24D10
Quotients: C1, C2, C4, C22, C2×C4, C23, D5, C22×C4, C24, D10, C23×C4, 2+ 1+4, C4×D5, C22×D5, C22.11C24, C2×C4×D5, C23×D5, D5×C22×C4, D4⋊6D10, C24.24D10
►On 80 points
(2 43)(4 45)(6 47)(8 49)(10 51)(12 53)(14 55)(16 57)(18 59)(20 41)(21 65)(23 67)(25 69)(27 71)(29 73)(31 75)(33 77)(35 79)(37 61)(39 63)
(1 68)(2 69)(3 70)(4 71)(5 72)(6 73)(7 74)(8 75)(9 76)(10 77)(11 78)(12 79)(13 80)(14 61)(15 62)(16 63)(17 64)(18 65)(19 66)(20 67)(21 59)(22 60)(23 41)(24 42)(25 43)(26 44)(27 45)(28 46)(29 47)(30 48)(31 49)(32 50)(33 51)(34 52)(35 53)(36 54)(37 55)(38 56)(39 57)(40 58)
(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 42)(2 43)(3 44)(4 45)(5 46)(6 47)(7 48)(8 49)(9 50)(10 51)(11 52)(12 53)(13 54)(14 55)(15 56)(16 57)(17 58)(18 59)(19 60)(20 41)(21 65)(22 66)(23 67)(24 68)(25 69)(26 70)(27 71)(28 72)(29 73)(30 74)(31 75)(32 76)(33 77)(34 78)(35 79)(36 80)(37 61)(38 62)(39 63)(40 64)
(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 10 11 20)(2 19 12 9)(3 8 13 18)(4 17 14 7)(5 6 15 16)(21 70 31 80)(22 79 32 69)(23 68 33 78)(24 77 34 67)(25 66 35 76)(26 75 36 65)(27 64 37 74)(28 73 38 63)(29 62 39 72)(30 71 40 61)(41 42 51 52)(43 60 53 50)(44 49 54 59)(45 58 55 48)(46 47 56 57)
G:=sub<Sym(80)| (2,43)(4,45)(6,47)(8,49)(10,51)(12,53)(14,55)(16,57)(18,59)(20,41)(21,65)(23,67)(25,69)(27,71)(29,73)(31,75)(33,77)(35,79)(37,61)(39,63), (1,68)(2,69)(3,70)(4,71)(5,72)(6,73)(7,74)(8,75)(9,76)(10,77)(11,78)(12,79)(13,80)(14,61)(15,62)(16,63)(17,64)(18,65)(19,66)(20,67)(21,59)(22,60)(23,41)(24,42)(25,43)(26,44)(27,45)(28,46)(29,47)(30,48)(31,49)(32,50)(33,51)(34,52)(35,53)(36,54)(37,55)(38,56)(39,57)(40,58), (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), (1,42)(2,43)(3,44)(4,45)(5,46)(6,47)(7,48)(8,49)(9,50)(10,51)(11,52)(12,53)(13,54)(14,55)(15,56)(16,57)(17,58)(18,59)(19,60)(20,41)(21,65)(22,66)(23,67)(24,68)(25,69)(26,70)(27,71)(28,72)(29,73)(30,74)(31,75)(32,76)(33,77)(34,78)(35,79)(36,80)(37,61)(38,62)(39,63)(40,64), (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), (1,10,11,20)(2,19,12,9)(3,8,13,18)(4,17,14,7)(5,6,15,16)(21,70,31,80)(22,79,32,69)(23,68,33,78)(24,77,34,67)(25,66,35,76)(26,75,36,65)(27,64,37,74)(28,73,38,63)(29,62,39,72)(30,71,40,61)(41,42,51,52)(43,60,53,50)(44,49,54,59)(45,58,55,48)(46,47,56,57)>;
G:=Group( (2,43)(4,45)(6,47)(8,49)(10,51)(12,53)(14,55)(16,57)(18,59)(20,41)(21,65)(23,67)(25,69)(27,71)(29,73)(31,75)(33,77)(35,79)(37,61)(39,63), (1,68)(2,69)(3,70)(4,71)(5,72)(6,73)(7,74)(8,75)(9,76)(10,77)(11,78)(12,79)(13,80)(14,61)(15,62)(16,63)(17,64)(18,65)(19,66)(20,67)(21,59)(22,60)(23,41)(24,42)(25,43)(26,44)(27,45)(28,46)(29,47)(30,48)(31,49)(32,50)(33,51)(34,52)(35,53)(36,54)(37,55)(38,56)(39,57)(40,58), (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), (1,42)(2,43)(3,44)(4,45)(5,46)(6,47)(7,48)(8,49)(9,50)(10,51)(11,52)(12,53)(13,54)(14,55)(15,56)(16,57)(17,58)(18,59)(19,60)(20,41)(21,65)(22,66)(23,67)(24,68)(25,69)(26,70)(27,71)(28,72)(29,73)(30,74)(31,75)(32,76)(33,77)(34,78)(35,79)(36,80)(37,61)(38,62)(39,63)(40,64), (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), (1,10,11,20)(2,19,12,9)(3,8,13,18)(4,17,14,7)(5,6,15,16)(21,70,31,80)(22,79,32,69)(23,68,33,78)(24,77,34,67)(25,66,35,76)(26,75,36,65)(27,64,37,74)(28,73,38,63)(29,62,39,72)(30,71,40,61)(41,42,51,52)(43,60,53,50)(44,49,54,59)(45,58,55,48)(46,47,56,57) );
G=PermutationGroup([[(2,43),(4,45),(6,47),(8,49),(10,51),(12,53),(14,55),(16,57),(18,59),(20,41),(21,65),(23,67),(25,69),(27,71),(29,73),(31,75),(33,77),(35,79),(37,61),(39,63)], [(1,68),(2,69),(3,70),(4,71),(5,72),(6,73),(7,74),(8,75),(9,76),(10,77),(11,78),(12,79),(13,80),(14,61),(15,62),(16,63),(17,64),(18,65),(19,66),(20,67),(21,59),(22,60),(23,41),(24,42),(25,43),(26,44),(27,45),(28,46),(29,47),(30,48),(31,49),(32,50),(33,51),(34,52),(35,53),(36,54),(37,55),(38,56),(39,57),(40,58)], [(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)], [(1,42),(2,43),(3,44),(4,45),(5,46),(6,47),(7,48),(8,49),(9,50),(10,51),(11,52),(12,53),(13,54),(14,55),(15,56),(16,57),(17,58),(18,59),(19,60),(20,41),(21,65),(22,66),(23,67),(24,68),(25,69),(26,70),(27,71),(28,72),(29,73),(30,74),(31,75),(32,76),(33,77),(34,78),(35,79),(36,80),(37,61),(38,62),(39,63),(40,64)], [(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)], [(1,10,11,20),(2,19,12,9),(3,8,13,18),(4,17,14,7),(5,6,15,16),(21,70,31,80),(22,79,32,69),(23,68,33,78),(24,77,34,67),(25,66,35,76),(26,75,36,65),(27,64,37,74),(28,73,38,63),(29,62,39,72),(30,71,40,61),(41,42,51,52),(43,60,53,50),(44,49,54,59),(45,58,55,48),(46,47,56,57)]])
74 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | ··· | 2I | 2J | 2K | 2L | 2M | 4A | ··· | 4H | 4I | ··· | 4T | 5A | 5B | 10A | ··· | 10N | 10O | ··· | 10V | 20A | ··· | 20P |
| 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 | 10 | 10 | 10 | 10 | 2 | ··· | 2 | 10 | ··· | 10 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
74 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | D5 | D10 | D10 | D10 | C4×D5 | 2+ 1+4 | D4⋊6D10 |
| kernel | C24.24D10 | C23.11D10 | D5×C22⋊C4 | Dic5⋊4D4 | C4×C5⋊D4 | C2×C23.D5 | C10×C22⋊C4 | C22×C5⋊D4 | C2×C5⋊D4 | C2×C22⋊C4 | C22⋊C4 | C22×C4 | C24 | C23 | C10 | C2 |
| # reps | 1 | 2 | 2 | 4 | 4 | 1 | 1 | 1 | 16 | 2 | 8 | 4 | 2 | 16 | 2 | 8 |
Matrix representation of C24.24D10 ►in GL6(𝔽41)
| 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 | 14 | 2 | 40 | 0 |
| 0 | 0 | 14 | 2 | 0 | 40 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 18 | 40 | 0 | 0 |
| 0 | 0 | 36 | 23 | 0 | 0 |
| 0 | 0 | 9 | 0 | 17 | 40 |
| 0 | 0 | 28 | 32 | 1 | 24 |
| 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 |
| 22 | 22 | 0 | 0 | 0 | 0 |
| 19 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 31 | 22 | 27 | 33 |
| 0 | 0 | 9 | 13 | 35 | 34 |
| 0 | 0 | 8 | 6 | 19 | 19 |
| 0 | 0 | 8 | 13 | 19 | 19 |
| 19 | 19 | 0 | 0 | 0 | 0 |
| 9 | 22 | 0 | 0 | 0 | 0 |
| 0 | 0 | 31 | 22 | 33 | 27 |
| 0 | 0 | 9 | 13 | 34 | 35 |
| 0 | 0 | 8 | 6 | 19 | 19 |
| 0 | 0 | 2 | 13 | 19 | 19 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,14,14,0,0,0,1,2,2,0,0,0,0,40,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,18,36,9,28,0,0,40,23,0,32,0,0,0,0,17,1,0,0,0,0,40,24],[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],[22,19,0,0,0,0,22,32,0,0,0,0,0,0,31,9,8,8,0,0,22,13,6,13,0,0,27,35,19,19,0,0,33,34,19,19],[19,9,0,0,0,0,19,22,0,0,0,0,0,0,31,9,8,2,0,0,22,13,6,13,0,0,33,34,19,19,0,0,27,35,19,19] >;
C24.24D10 in GAP, Magma, Sage, TeX
C_2^4._{24}D_{10} % in TeX
G:=Group("C2^4.24D10"); // GroupNames label
G:=SmallGroup(320,1158);
// by ID
G=gap.SmallGroup(320,1158);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,758,570,80,12550]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^10=f^2=c,a*b=b*a,a*c=c*a,e*a*e^-1=f*a*f^-1=a*d=d*a,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