direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C5×C42⋊C2, C42⋊1C10, C4⋊C4⋊6C10, (C4×C20)⋊2C2, (C2×C4)⋊4C20, (C2×C20)⋊14C4, C4.9(C2×C20), C20.67(C2×C4), C22⋊C4.3C10, (C22×C4).4C10, C2.3(C22×C20), C23.6(C2×C10), C22.5(C2×C20), C10.38(C4○D4), C10.44(C22×C4), (C2×C20).79C22, (C2×C10).72C23, (C22×C20).14C2, C22.6(C22×C10), (C22×C10).25C22, (C5×C4⋊C4)⋊15C2, C2.1(C5×C4○D4), (C2×C10).42(C2×C4), (C2×C4).14(C2×C10), (C5×C22⋊C4).6C2, SmallGroup(160,178)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×C42⋊C2
G = < a,b,c,d | a5=b4=c4=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=bc2, cd=dc >
Subgroups: 92 in 76 conjugacy classes, 60 normal (16 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C5, C2×C4, C2×C4, C23, C10, C10, C10, C42, C22⋊C4, C4⋊C4, C22×C4, C20, C20, C2×C10, C2×C10, C2×C10, C42⋊C2, C2×C20, C2×C20, C22×C10, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C22×C20, C5×C42⋊C2
Quotients: C1, C2, C4, C22, C5, C2×C4, C23, C10, C22×C4, C4○D4, C20, C2×C10, C42⋊C2, C2×C20, C22×C10, C22×C20, C5×C4○D4, C5×C42⋊C2
►On 80 points
(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 61 21 56)(2 62 22 57)(3 63 23 58)(4 64 24 59)(5 65 25 60)(6 41 76 36)(7 42 77 37)(8 43 78 38)(9 44 79 39)(10 45 80 40)(11 46 16 51)(12 47 17 52)(13 48 18 53)(14 49 19 54)(15 50 20 55)(26 71 31 66)(27 72 32 67)(28 73 33 68)(29 74 34 69)(30 75 35 70)
(1 46 26 36)(2 47 27 37)(3 48 28 38)(4 49 29 39)(5 50 30 40)(6 61 16 71)(7 62 17 72)(8 63 18 73)(9 64 19 74)(10 65 20 75)(11 66 76 56)(12 67 77 57)(13 68 78 58)(14 69 79 59)(15 70 80 60)(21 51 31 41)(22 52 32 42)(23 53 33 43)(24 54 34 44)(25 55 35 45)
(1 21)(2 22)(3 23)(4 24)(5 25)(6 11)(7 12)(8 13)(9 14)(10 15)(16 76)(17 77)(18 78)(19 79)(20 80)(26 31)(27 32)(28 33)(29 34)(30 35)(36 41)(37 42)(38 43)(39 44)(40 45)(46 51)(47 52)(48 53)(49 54)(50 55)(56 71)(57 72)(58 73)(59 74)(60 75)(61 66)(62 67)(63 68)(64 69)(65 70)
G:=sub<Sym(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), (1,61,21,56)(2,62,22,57)(3,63,23,58)(4,64,24,59)(5,65,25,60)(6,41,76,36)(7,42,77,37)(8,43,78,38)(9,44,79,39)(10,45,80,40)(11,46,16,51)(12,47,17,52)(13,48,18,53)(14,49,19,54)(15,50,20,55)(26,71,31,66)(27,72,32,67)(28,73,33,68)(29,74,34,69)(30,75,35,70), (1,46,26,36)(2,47,27,37)(3,48,28,38)(4,49,29,39)(5,50,30,40)(6,61,16,71)(7,62,17,72)(8,63,18,73)(9,64,19,74)(10,65,20,75)(11,66,76,56)(12,67,77,57)(13,68,78,58)(14,69,79,59)(15,70,80,60)(21,51,31,41)(22,52,32,42)(23,53,33,43)(24,54,34,44)(25,55,35,45), (1,21)(2,22)(3,23)(4,24)(5,25)(6,11)(7,12)(8,13)(9,14)(10,15)(16,76)(17,77)(18,78)(19,79)(20,80)(26,31)(27,32)(28,33)(29,34)(30,35)(36,41)(37,42)(38,43)(39,44)(40,45)(46,51)(47,52)(48,53)(49,54)(50,55)(56,71)(57,72)(58,73)(59,74)(60,75)(61,66)(62,67)(63,68)(64,69)(65,70)>;
G:=Group( (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), (1,61,21,56)(2,62,22,57)(3,63,23,58)(4,64,24,59)(5,65,25,60)(6,41,76,36)(7,42,77,37)(8,43,78,38)(9,44,79,39)(10,45,80,40)(11,46,16,51)(12,47,17,52)(13,48,18,53)(14,49,19,54)(15,50,20,55)(26,71,31,66)(27,72,32,67)(28,73,33,68)(29,74,34,69)(30,75,35,70), (1,46,26,36)(2,47,27,37)(3,48,28,38)(4,49,29,39)(5,50,30,40)(6,61,16,71)(7,62,17,72)(8,63,18,73)(9,64,19,74)(10,65,20,75)(11,66,76,56)(12,67,77,57)(13,68,78,58)(14,69,79,59)(15,70,80,60)(21,51,31,41)(22,52,32,42)(23,53,33,43)(24,54,34,44)(25,55,35,45), (1,21)(2,22)(3,23)(4,24)(5,25)(6,11)(7,12)(8,13)(9,14)(10,15)(16,76)(17,77)(18,78)(19,79)(20,80)(26,31)(27,32)(28,33)(29,34)(30,35)(36,41)(37,42)(38,43)(39,44)(40,45)(46,51)(47,52)(48,53)(49,54)(50,55)(56,71)(57,72)(58,73)(59,74)(60,75)(61,66)(62,67)(63,68)(64,69)(65,70) );
G=PermutationGroup([[(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)], [(1,61,21,56),(2,62,22,57),(3,63,23,58),(4,64,24,59),(5,65,25,60),(6,41,76,36),(7,42,77,37),(8,43,78,38),(9,44,79,39),(10,45,80,40),(11,46,16,51),(12,47,17,52),(13,48,18,53),(14,49,19,54),(15,50,20,55),(26,71,31,66),(27,72,32,67),(28,73,33,68),(29,74,34,69),(30,75,35,70)], [(1,46,26,36),(2,47,27,37),(3,48,28,38),(4,49,29,39),(5,50,30,40),(6,61,16,71),(7,62,17,72),(8,63,18,73),(9,64,19,74),(10,65,20,75),(11,66,76,56),(12,67,77,57),(13,68,78,58),(14,69,79,59),(15,70,80,60),(21,51,31,41),(22,52,32,42),(23,53,33,43),(24,54,34,44),(25,55,35,45)], [(1,21),(2,22),(3,23),(4,24),(5,25),(6,11),(7,12),(8,13),(9,14),(10,15),(16,76),(17,77),(18,78),(19,79),(20,80),(26,31),(27,32),(28,33),(29,34),(30,35),(36,41),(37,42),(38,43),(39,44),(40,45),(46,51),(47,52),(48,53),(49,54),(50,55),(56,71),(57,72),(58,73),(59,74),(60,75),(61,66),(62,67),(63,68),(64,69),(65,70)]])
C5×C42⋊C2 is a maximal subgroup of
C42⋊1Dic5 C20.32C42 C20.60(C4⋊C4) C20.64(C4⋊C4) C4⋊C4.233D10 C20.35C42 C20.76(C4⋊C4) C42.43D10 C42.187D10 C4⋊C4⋊36D10 C4○D20⋊10C4 C4⋊C4.236D10 C4.(C2×D20) C42⋊4D10 (C2×D20)⋊25C4 C42.87D10 C42.88D10 C42.89D10 C42.90D10 C42⋊7D10 C42.188D10 C42.91D10 C42⋊8D10 C42⋊9D10 C42.92D10 C42⋊10D10 C42.93D10 C42.94D10 C42.95D10 C42.96D10 C42.97D10 C42.98D10 C42.99D10 C42.100D10 C4○D4×C20
C5×C42⋊C2 is a maximal quotient of
C22⋊C4×C20 C4⋊C4×C20
100 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | ··· | 4N | 5A | 5B | 5C | 5D | 10A | ··· | 10L | 10M | ··· | 10T | 20A | ··· | 20P | 20Q | ··· | 20BD |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 5 | 5 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 |
100 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | + | + | |||||||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C5 | C10 | C10 | C10 | C10 | C20 | C4○D4 | C5×C4○D4 |
| kernel | C5×C42⋊C2 | C4×C20 | C5×C22⋊C4 | C5×C4⋊C4 | C22×C20 | C2×C20 | C42⋊C2 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×C4 | C10 | C2 |
| # reps | 1 | 2 | 2 | 2 | 1 | 8 | 4 | 8 | 8 | 8 | 4 | 32 | 4 | 16 |
Matrix representation of C5×C42⋊C2 ►in GL3(𝔽41) generated by
| 1 | 0 | 0 |
| 0 | 10 | 0 |
| 0 | 0 | 10 |
| 9 | 0 | 0 |
| 0 | 32 | 39 |
| 0 | 0 | 9 |
| 1 | 0 | 0 |
| 0 | 9 | 0 |
| 0 | 0 | 9 |
| 1 | 0 | 0 |
| 0 | 40 | 0 |
| 0 | 9 | 1 |
G:=sub<GL(3,GF(41))| [1,0,0,0,10,0,0,0,10],[9,0,0,0,32,0,0,39,9],[1,0,0,0,9,0,0,0,9],[1,0,0,0,40,9,0,0,1] >;
C5×C42⋊C2 in GAP, Magma, Sage, TeX
C_5\times C_4^2\rtimes C_2
% in TeX
G:=Group("C5xC4^2:C2"); // GroupNames label
G:=SmallGroup(160,178);
// by ID
G=gap.SmallGroup(160,178);
# by ID
G:=PCGroup([6,-2,-2,-2,-5,-2,-2,480,505,194]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^4=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=b*c^2,c*d=d*c>;
// generators/relations