metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C24.4Dic5, (C23×C4).7D5, C20.450(C2×D4), (C2×C20).484D4, (C2×C10)⋊14M4(2), (C22×C20).53C4, (C23×C10).15C4, (C23×C20).15C2, C5⋊6(C24.4C4), C20.55D4⋊27C2, (C2×C20).871C23, (C22×C4).403D10, C10.78(C2×M4(2)), C22⋊2(C4.Dic5), C4.21(C23.D5), (C22×C4).13Dic5, C23.28(C2×Dic5), C20.136(C22⋊C4), (C22×C20).543C22, C22.18(C23.D5), C22.48(C22×Dic5), C4.141(C2×C5⋊D4), (C2×C4.Dic5)⋊7C2, (C2×C20).453(C2×C4), (C2×C5⋊2C8)⋊29C22, C2.4(C2×C23.D5), (C2×C4).64(C2×Dic5), C2.11(C2×C4.Dic5), (C2×C4).259(C5⋊D4), C10.108(C2×C22⋊C4), (C2×C4).813(C22×D5), (C2×C10).292(C22×C4), (C22×C10).202(C2×C4), (C2×C10).172(C22⋊C4), SmallGroup(320,834)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C24.4Dic5
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e10=d, f2=e5, ab=ba, faf-1=ac=ca, ad=da, ae=ea, bc=cb, fbf-1=bd=db, be=eb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e9 >
Subgroups: 398 in 190 conjugacy classes, 79 normal (21 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C5, C8, C2×C4, C2×C4, C2×C4, C23, C23, C23, C10, C10, C10, C2×C8, M4(2), C22×C4, C22×C4, C22×C4, C24, C20, C20, C2×C10, C2×C10, C2×C10, C22⋊C8, C2×M4(2), C23×C4, C5⋊2C8, C2×C20, C2×C20, C2×C20, C22×C10, C22×C10, C22×C10, C24.4C4, C2×C5⋊2C8, C4.Dic5, C22×C20, C22×C20, C22×C20, C23×C10, C20.55D4, C2×C4.Dic5, C23×C20, C24.4Dic5
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D5, C22⋊C4, M4(2), C22×C4, C2×D4, Dic5, D10, C2×C22⋊C4, C2×M4(2), C2×Dic5, C5⋊D4, C22×D5, C24.4C4, C4.Dic5, C23.D5, C22×Dic5, C2×C5⋊D4, C2×C4.Dic5, C2×C23.D5, C24.4Dic5
►On 80 points
Generators in S80
(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 64)(42 65)(43 66)(44 67)(45 68)(46 69)(47 70)(48 71)(49 72)(50 73)(51 74)(52 75)(53 76)(54 77)(55 78)(56 79)(57 80)(58 61)(59 62)(60 63)
(1 40)(2 21)(3 22)(4 23)(5 24)(6 25)(7 26)(8 27)(9 28)(10 29)(11 30)(12 31)(13 32)(14 33)(15 34)(16 35)(17 36)(18 37)(19 38)(20 39)(41 74)(42 75)(43 76)(44 77)(45 78)(46 79)(47 80)(48 61)(49 62)(50 63)(51 64)(52 65)(53 66)(54 67)(55 68)(56 69)(57 70)(58 71)(59 72)(60 73)
(1 30)(2 31)(3 32)(4 33)(5 34)(6 35)(7 36)(8 37)(9 38)(10 39)(11 40)(12 21)(13 22)(14 23)(15 24)(16 25)(17 26)(18 27)(19 28)(20 29)(41 74)(42 75)(43 76)(44 77)(45 78)(46 79)(47 80)(48 61)(49 62)(50 63)(51 64)(52 65)(53 66)(54 67)(55 68)(56 69)(57 70)(58 71)(59 72)(60 73)
(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 71 6 76 11 61 16 66)(2 80 7 65 12 70 17 75)(3 69 8 74 13 79 18 64)(4 78 9 63 14 68 19 73)(5 67 10 72 15 77 20 62)(21 57 26 42 31 47 36 52)(22 46 27 51 32 56 37 41)(23 55 28 60 33 45 38 50)(24 44 29 49 34 54 39 59)(25 53 30 58 35 43 40 48)
G:=sub<Sym(80)| (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)(41,64)(42,65)(43,66)(44,67)(45,68)(46,69)(47,70)(48,71)(49,72)(50,73)(51,74)(52,75)(53,76)(54,77)(55,78)(56,79)(57,80)(58,61)(59,62)(60,63), (1,40)(2,21)(3,22)(4,23)(5,24)(6,25)(7,26)(8,27)(9,28)(10,29)(11,30)(12,31)(13,32)(14,33)(15,34)(16,35)(17,36)(18,37)(19,38)(20,39)(41,74)(42,75)(43,76)(44,77)(45,78)(46,79)(47,80)(48,61)(49,62)(50,63)(51,64)(52,65)(53,66)(54,67)(55,68)(56,69)(57,70)(58,71)(59,72)(60,73), (1,30)(2,31)(3,32)(4,33)(5,34)(6,35)(7,36)(8,37)(9,38)(10,39)(11,40)(12,21)(13,22)(14,23)(15,24)(16,25)(17,26)(18,27)(19,28)(20,29)(41,74)(42,75)(43,76)(44,77)(45,78)(46,79)(47,80)(48,61)(49,62)(50,63)(51,64)(52,65)(53,66)(54,67)(55,68)(56,69)(57,70)(58,71)(59,72)(60,73), (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), (1,71,6,76,11,61,16,66)(2,80,7,65,12,70,17,75)(3,69,8,74,13,79,18,64)(4,78,9,63,14,68,19,73)(5,67,10,72,15,77,20,62)(21,57,26,42,31,47,36,52)(22,46,27,51,32,56,37,41)(23,55,28,60,33,45,38,50)(24,44,29,49,34,54,39,59)(25,53,30,58,35,43,40,48)>;
G:=Group( (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)(41,64)(42,65)(43,66)(44,67)(45,68)(46,69)(47,70)(48,71)(49,72)(50,73)(51,74)(52,75)(53,76)(54,77)(55,78)(56,79)(57,80)(58,61)(59,62)(60,63), (1,40)(2,21)(3,22)(4,23)(5,24)(6,25)(7,26)(8,27)(9,28)(10,29)(11,30)(12,31)(13,32)(14,33)(15,34)(16,35)(17,36)(18,37)(19,38)(20,39)(41,74)(42,75)(43,76)(44,77)(45,78)(46,79)(47,80)(48,61)(49,62)(50,63)(51,64)(52,65)(53,66)(54,67)(55,68)(56,69)(57,70)(58,71)(59,72)(60,73), (1,30)(2,31)(3,32)(4,33)(5,34)(6,35)(7,36)(8,37)(9,38)(10,39)(11,40)(12,21)(13,22)(14,23)(15,24)(16,25)(17,26)(18,27)(19,28)(20,29)(41,74)(42,75)(43,76)(44,77)(45,78)(46,79)(47,80)(48,61)(49,62)(50,63)(51,64)(52,65)(53,66)(54,67)(55,68)(56,69)(57,70)(58,71)(59,72)(60,73), (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), (1,71,6,76,11,61,16,66)(2,80,7,65,12,70,17,75)(3,69,8,74,13,79,18,64)(4,78,9,63,14,68,19,73)(5,67,10,72,15,77,20,62)(21,57,26,42,31,47,36,52)(22,46,27,51,32,56,37,41)(23,55,28,60,33,45,38,50)(24,44,29,49,34,54,39,59)(25,53,30,58,35,43,40,48) );
G=PermutationGroup([[(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),(41,64),(42,65),(43,66),(44,67),(45,68),(46,69),(47,70),(48,71),(49,72),(50,73),(51,74),(52,75),(53,76),(54,77),(55,78),(56,79),(57,80),(58,61),(59,62),(60,63)], [(1,40),(2,21),(3,22),(4,23),(5,24),(6,25),(7,26),(8,27),(9,28),(10,29),(11,30),(12,31),(13,32),(14,33),(15,34),(16,35),(17,36),(18,37),(19,38),(20,39),(41,74),(42,75),(43,76),(44,77),(45,78),(46,79),(47,80),(48,61),(49,62),(50,63),(51,64),(52,65),(53,66),(54,67),(55,68),(56,69),(57,70),(58,71),(59,72),(60,73)], [(1,30),(2,31),(3,32),(4,33),(5,34),(6,35),(7,36),(8,37),(9,38),(10,39),(11,40),(12,21),(13,22),(14,23),(15,24),(16,25),(17,26),(18,27),(19,28),(20,29),(41,74),(42,75),(43,76),(44,77),(45,78),(46,79),(47,80),(48,61),(49,62),(50,63),(51,64),(52,65),(53,66),(54,67),(55,68),(56,69),(57,70),(58,71),(59,72),(60,73)], [(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)], [(1,71,6,76,11,61,16,66),(2,80,7,65,12,70,17,75),(3,69,8,74,13,79,18,64),(4,78,9,63,14,68,19,73),(5,67,10,72,15,77,20,62),(21,57,26,42,31,47,36,52),(22,46,27,51,32,56,37,41),(23,55,28,60,33,45,38,50),(24,44,29,49,34,54,39,59),(25,53,30,58,35,43,40,48)]])
92 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | ··· | 2I | 4A | 4B | 4C | 4D | 4E | ··· | 4J | 5A | 5B | 8A | ··· | 8H | 10A | ··· | 10AD | 20A | ··· | 20AF |
| order | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 8 | ··· | 8 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 2 | 2 | 20 | ··· | 20 | 2 | ··· | 2 | 2 | ··· | 2 |
92 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | - | + | - | |||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | D4 | D5 | M4(2) | Dic5 | D10 | Dic5 | C5⋊D4 | C4.Dic5 |
| kernel | C24.4Dic5 | C20.55D4 | C2×C4.Dic5 | C23×C20 | C22×C20 | C23×C10 | C2×C20 | C23×C4 | C2×C10 | C22×C4 | C22×C4 | C24 | C2×C4 | C22 |
| # reps | 1 | 4 | 2 | 1 | 6 | 2 | 4 | 2 | 8 | 6 | 6 | 2 | 16 | 32 |
Matrix representation of C24.4Dic5 ►in GL4(𝔽41) generated by
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 10 | 40 |
| 40 | 12 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 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 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 21 | 26 | 0 | 0 |
| 0 | 39 | 0 | 0 |
| 0 | 0 | 25 | 0 |
| 0 | 0 | 10 | 23 |
| 16 | 15 | 0 | 0 |
| 30 | 25 | 0 | 0 |
| 0 | 0 | 7 | 15 |
| 0 | 0 | 24 | 34 |
G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,1,10,0,0,0,40],[40,0,0,0,12,1,0,0,0,0,40,0,0,0,0,40],[1,0,0,0,0,1,0,0,0,0,40,0,0,0,0,40],[40,0,0,0,0,40,0,0,0,0,1,0,0,0,0,1],[21,0,0,0,26,39,0,0,0,0,25,10,0,0,0,23],[16,30,0,0,15,25,0,0,0,0,7,24,0,0,15,34] >;
C24.4Dic5 in GAP, Magma, Sage, TeX
C_2^4._4{\rm Dic}_5 % in TeX
G:=Group("C2^4.4Dic5"); // GroupNames label
G:=SmallGroup(320,834);
// by ID
G=gap.SmallGroup(320,834);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,253,758,136,12550]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^10=d,f^2=e^5,a*b=b*a,f*a*f^-1=a*c=c*a,a*d=d*a,a*e=e*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