metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42⋊3F5, C20.20C42, D5⋊C8⋊C4, (C4×C20)⋊8C4, (C4×F5)⋊1C4, C4.7(C4⋊F5), C4.27(C4×F5), (C4×Dic5)⋊7C4, (C4×D5).50D4, C20.14(C4⋊C4), (C4×D5).11Q8, D10.1(C4⋊C4), C5⋊1(C4.9C42), Dic5.1(C4⋊C4), (C2×Dic5).91D4, D5⋊M4(2).8C2, (C22×D5).53D4, D10.1(C22⋊C4), C42⋊D5.13C2, Dic5.1(C22⋊C4), C2.4(D10.3Q8), C22.13(C22⋊F5), C10.2(C2.C42), D10.C23.8C2, (C4×D5).42(C2×C4), (C2×C4).116(C2×F5), (C2×C20).137(C2×C4), (C2×C4×D5).280C22, (C2×C10).13(C22⋊C4), SmallGroup(320,201)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42⋊3F5
G = < a,b,c,d | a4=b4=c5=d4=1, ab=ba, ac=ca, dad-1=ab-1, bc=cb, bd=db, dcd-1=c3 >
Subgroups: 402 in 94 conjugacy classes, 34 normal (30 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, C23, D5, C10, C10, C42, C42, C22⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, Dic5, Dic5, C20, C20, F5, D10, D10, C2×C10, C42⋊C2, C2×M4(2), C5⋊C8, C4×D5, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C2×F5, C22×D5, C4.9C42, C4×Dic5, C10.D4, D10⋊C4, C4×C20, D5⋊C8, C4.F5, C4×F5, C4⋊F5, C22.F5, C22⋊F5, C2×C4×D5, C42⋊D5, D5⋊M4(2), D10.C23, C42⋊3F5
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C42, C22⋊C4, C4⋊C4, F5, C2.C42, C2×F5, C4.9C42, C4×F5, C4⋊F5, C22⋊F5, D10.3Q8, C42⋊3F5
►On 80 points
(1 21)(2 22)(3 23)(4 24)(5 25)(6 26)(7 27)(8 28)(9 29)(10 30)(11 31)(12 32)(13 33)(14 34)(15 35)(16 36)(17 37)(18 38)(19 39)(20 40)(41 76 46 71)(42 77 47 72)(43 78 48 73)(44 79 49 74)(45 80 50 75)(51 61 56 66)(52 62 57 67)(53 63 58 68)(54 64 59 69)(55 65 60 70)
(1 16 6 11)(2 17 7 12)(3 18 8 13)(4 19 9 14)(5 20 10 15)(21 36 26 31)(22 37 27 32)(23 38 28 33)(24 39 29 34)(25 40 30 35)(41 56 46 51)(42 57 47 52)(43 58 48 53)(44 59 49 54)(45 60 50 55)(61 76 66 71)(62 77 67 72)(63 78 68 73)(64 79 69 74)(65 80 70 75)
(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 43)(2 45 5 41)(3 42 4 44)(6 48)(7 50 10 46)(8 47 9 49)(11 53)(12 55 15 51)(13 52 14 54)(16 58)(17 60 20 56)(18 57 19 59)(21 68 26 63)(22 70 30 61)(23 67 29 64)(24 69 28 62)(25 66 27 65)(31 78 36 73)(32 80 40 71)(33 77 39 74)(34 79 38 72)(35 76 37 75)
G:=sub<Sym(80)| (1,21)(2,22)(3,23)(4,24)(5,25)(6,26)(7,27)(8,28)(9,29)(10,30)(11,31)(12,32)(13,33)(14,34)(15,35)(16,36)(17,37)(18,38)(19,39)(20,40)(41,76,46,71)(42,77,47,72)(43,78,48,73)(44,79,49,74)(45,80,50,75)(51,61,56,66)(52,62,57,67)(53,63,58,68)(54,64,59,69)(55,65,60,70), (1,16,6,11)(2,17,7,12)(3,18,8,13)(4,19,9,14)(5,20,10,15)(21,36,26,31)(22,37,27,32)(23,38,28,33)(24,39,29,34)(25,40,30,35)(41,56,46,51)(42,57,47,52)(43,58,48,53)(44,59,49,54)(45,60,50,55)(61,76,66,71)(62,77,67,72)(63,78,68,73)(64,79,69,74)(65,80,70,75), (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), (1,43)(2,45,5,41)(3,42,4,44)(6,48)(7,50,10,46)(8,47,9,49)(11,53)(12,55,15,51)(13,52,14,54)(16,58)(17,60,20,56)(18,57,19,59)(21,68,26,63)(22,70,30,61)(23,67,29,64)(24,69,28,62)(25,66,27,65)(31,78,36,73)(32,80,40,71)(33,77,39,74)(34,79,38,72)(35,76,37,75)>;
G:=Group( (1,21)(2,22)(3,23)(4,24)(5,25)(6,26)(7,27)(8,28)(9,29)(10,30)(11,31)(12,32)(13,33)(14,34)(15,35)(16,36)(17,37)(18,38)(19,39)(20,40)(41,76,46,71)(42,77,47,72)(43,78,48,73)(44,79,49,74)(45,80,50,75)(51,61,56,66)(52,62,57,67)(53,63,58,68)(54,64,59,69)(55,65,60,70), (1,16,6,11)(2,17,7,12)(3,18,8,13)(4,19,9,14)(5,20,10,15)(21,36,26,31)(22,37,27,32)(23,38,28,33)(24,39,29,34)(25,40,30,35)(41,56,46,51)(42,57,47,52)(43,58,48,53)(44,59,49,54)(45,60,50,55)(61,76,66,71)(62,77,67,72)(63,78,68,73)(64,79,69,74)(65,80,70,75), (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), (1,43)(2,45,5,41)(3,42,4,44)(6,48)(7,50,10,46)(8,47,9,49)(11,53)(12,55,15,51)(13,52,14,54)(16,58)(17,60,20,56)(18,57,19,59)(21,68,26,63)(22,70,30,61)(23,67,29,64)(24,69,28,62)(25,66,27,65)(31,78,36,73)(32,80,40,71)(33,77,39,74)(34,79,38,72)(35,76,37,75) );
G=PermutationGroup([[(1,21),(2,22),(3,23),(4,24),(5,25),(6,26),(7,27),(8,28),(9,29),(10,30),(11,31),(12,32),(13,33),(14,34),(15,35),(16,36),(17,37),(18,38),(19,39),(20,40),(41,76,46,71),(42,77,47,72),(43,78,48,73),(44,79,49,74),(45,80,50,75),(51,61,56,66),(52,62,57,67),(53,63,58,68),(54,64,59,69),(55,65,60,70)], [(1,16,6,11),(2,17,7,12),(3,18,8,13),(4,19,9,14),(5,20,10,15),(21,36,26,31),(22,37,27,32),(23,38,28,33),(24,39,29,34),(25,40,30,35),(41,56,46,51),(42,57,47,52),(43,58,48,53),(44,59,49,54),(45,60,50,55),(61,76,66,71),(62,77,67,72),(63,78,68,73),(64,79,69,74),(65,80,70,75)], [(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)], [(1,43),(2,45,5,41),(3,42,4,44),(6,48),(7,50,10,46),(8,47,9,49),(11,53),(12,55,15,51),(13,52,14,54),(16,58),(17,60,20,56),(18,57,19,59),(21,68,26,63),(22,70,30,61),(23,67,29,64),(24,69,28,62),(25,66,27,65),(31,78,36,73),(32,80,40,71),(33,77,39,74),(34,79,38,72),(35,76,37,75)]])
38 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | ··· | 4M | 5 | 8A | 8B | 8C | 8D | 10A | 10B | 10C | 20A | ··· | 20L |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 2 | 10 | 10 | 1 | 1 | 2 | 4 | 4 | 10 | 10 | 20 | ··· | 20 | 4 | 20 | 20 | 20 | 20 | 4 | 4 | 4 | 4 | ··· | 4 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | - | + | + | + | + | + | ||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | D4 | Q8 | D4 | D4 | F5 | C2×F5 | C4.9C42 | C4×F5 | C4⋊F5 | C22⋊F5 | C42⋊3F5 |
| kernel | C42⋊3F5 | C42⋊D5 | D5⋊M4(2) | D10.C23 | C4×Dic5 | C4×C20 | D5⋊C8 | C4×F5 | C4×D5 | C4×D5 | C2×Dic5 | C22×D5 | C42 | C2×C4 | C5 | C4 | C4 | C22 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 8 |
Matrix representation of C42⋊3F5 ►in GL4(𝔽41) generated by
| 24 | 40 | 0 | 0 |
| 1 | 17 | 0 | 0 |
| 0 | 0 | 30 | 9 |
| 0 | 0 | 32 | 11 |
| 32 | 0 | 0 | 0 |
| 0 | 32 | 0 | 0 |
| 0 | 0 | 32 | 0 |
| 0 | 0 | 0 | 32 |
| 0 | 40 | 0 | 0 |
| 1 | 34 | 0 | 0 |
| 0 | 0 | 40 | 7 |
| 0 | 0 | 34 | 7 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 34 | 0 | 0 |
| 0 | 40 | 0 | 0 |
G:=sub<GL(4,GF(41))| [24,1,0,0,40,17,0,0,0,0,30,32,0,0,9,11],[32,0,0,0,0,32,0,0,0,0,32,0,0,0,0,32],[0,1,0,0,40,34,0,0,0,0,40,34,0,0,7,7],[0,0,1,0,0,0,34,40,1,0,0,0,0,1,0,0] >;
C42⋊3F5 in GAP, Magma, Sage, TeX
C_4^2\rtimes_3F_5
% in TeX
G:=Group("C4^2:3F5"); // GroupNames label
G:=SmallGroup(320,201);
// by ID
G=gap.SmallGroup(320,201);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,253,64,387,1123,102,6278,3156]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^5=d^4=1,a*b=b*a,a*c=c*a,d*a*d^-1=a*b^-1,b*c=c*b,b*d=d*b,d*c*d^-1=c^3>;
// generators/relations