direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C5×C42⋊6C4, C42⋊6C20, M4(2)⋊2C20, C20.57C42, C4⋊C4⋊3C20, (C4×C20)⋊26C4, C4.1(C4×C20), C10.38C4≀C2, C20.82(C4⋊C4), (C2×C20).69Q8, (C2×C20).506D4, (C2×C42).7C10, C23.31(C5×D4), (C5×M4(2))⋊14C4, C42⋊C2.2C10, (C22×C10).151D4, (C2×M4(2)).6C10, C20.157(C22⋊C4), (C10×M4(2)).18C2, (C22×C20).570C22, C10.42(C2.C42), C4.2(C5×C4⋊C4), (C5×C4⋊C4)⋊17C4, C2.3(C5×C4≀C2), (C2×C4×C20).30C2, C22.3(C5×C4⋊C4), (C2×C4).12(C5×Q8), (C2×C4).65(C2×C20), (C2×C4).142(C5×D4), C4.25(C5×C22⋊C4), (C2×C10).48(C4⋊C4), (C2×C20).433(C2×C4), C22.28(C5×C22⋊C4), C2.4(C5×C2.C42), (C5×C42⋊C2).16C2, (C22×C4).103(C2×C10), (C2×C10).135(C22⋊C4), SmallGroup(320,144)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×C42⋊6C4
G = < a,b,c,d | a5=b4=c4=d4=1, ab=ba, ac=ca, ad=da, dbd-1=bc=cb, dcd-1=c-1 >
Subgroups: 170 in 110 conjugacy classes, 58 normal (42 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, C23, C10, C10, C10, C42, C42, C22⋊C4, C4⋊C4, C2×C8, M4(2), M4(2), C22×C4, C22×C4, C20, C20, C2×C10, C2×C10, C2×C42, C42⋊C2, C2×M4(2), C40, C2×C20, C2×C20, C22×C10, C42⋊6C4, C4×C20, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C2×C40, C5×M4(2), C5×M4(2), C22×C20, C22×C20, C2×C4×C20, C5×C42⋊C2, C10×M4(2), C5×C42⋊6C4
Quotients: C1, C2, C4, C22, C5, C2×C4, D4, Q8, C10, C42, C22⋊C4, C4⋊C4, C20, C2×C10, C2.C42, C4≀C2, C2×C20, C5×D4, C5×Q8, C42⋊6C4, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C5×C2.C42, C5×C4≀C2, C5×C42⋊6C4
►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)
(6 22 56 63)(7 23 57 64)(8 24 58 65)(9 25 59 61)(10 21 60 62)(16 32 77 73)(17 33 78 74)(18 34 79 75)(19 35 80 71)(20 31 76 72)
(1 54 29 12)(2 55 30 13)(3 51 26 14)(4 52 27 15)(5 53 28 11)(6 63 56 22)(7 64 57 23)(8 65 58 24)(9 61 59 25)(10 62 60 21)(16 73 77 32)(17 74 78 33)(18 75 79 34)(19 71 80 35)(20 72 76 31)(36 41 70 46)(37 42 66 47)(38 43 67 48)(39 44 68 49)(40 45 69 50)
(1 16 66 7)(2 17 67 8)(3 18 68 9)(4 19 69 10)(5 20 70 6)(11 72 41 63)(12 73 42 64)(13 74 43 65)(14 75 44 61)(15 71 45 62)(21 52 35 50)(22 53 31 46)(23 54 32 47)(24 55 33 48)(25 51 34 49)(26 79 39 59)(27 80 40 60)(28 76 36 56)(29 77 37 57)(30 78 38 58)
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), (6,22,56,63)(7,23,57,64)(8,24,58,65)(9,25,59,61)(10,21,60,62)(16,32,77,73)(17,33,78,74)(18,34,79,75)(19,35,80,71)(20,31,76,72), (1,54,29,12)(2,55,30,13)(3,51,26,14)(4,52,27,15)(5,53,28,11)(6,63,56,22)(7,64,57,23)(8,65,58,24)(9,61,59,25)(10,62,60,21)(16,73,77,32)(17,74,78,33)(18,75,79,34)(19,71,80,35)(20,72,76,31)(36,41,70,46)(37,42,66,47)(38,43,67,48)(39,44,68,49)(40,45,69,50), (1,16,66,7)(2,17,67,8)(3,18,68,9)(4,19,69,10)(5,20,70,6)(11,72,41,63)(12,73,42,64)(13,74,43,65)(14,75,44,61)(15,71,45,62)(21,52,35,50)(22,53,31,46)(23,54,32,47)(24,55,33,48)(25,51,34,49)(26,79,39,59)(27,80,40,60)(28,76,36,56)(29,77,37,57)(30,78,38,58)>;
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), (6,22,56,63)(7,23,57,64)(8,24,58,65)(9,25,59,61)(10,21,60,62)(16,32,77,73)(17,33,78,74)(18,34,79,75)(19,35,80,71)(20,31,76,72), (1,54,29,12)(2,55,30,13)(3,51,26,14)(4,52,27,15)(5,53,28,11)(6,63,56,22)(7,64,57,23)(8,65,58,24)(9,61,59,25)(10,62,60,21)(16,73,77,32)(17,74,78,33)(18,75,79,34)(19,71,80,35)(20,72,76,31)(36,41,70,46)(37,42,66,47)(38,43,67,48)(39,44,68,49)(40,45,69,50), (1,16,66,7)(2,17,67,8)(3,18,68,9)(4,19,69,10)(5,20,70,6)(11,72,41,63)(12,73,42,64)(13,74,43,65)(14,75,44,61)(15,71,45,62)(21,52,35,50)(22,53,31,46)(23,54,32,47)(24,55,33,48)(25,51,34,49)(26,79,39,59)(27,80,40,60)(28,76,36,56)(29,77,37,57)(30,78,38,58) );
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)], [(6,22,56,63),(7,23,57,64),(8,24,58,65),(9,25,59,61),(10,21,60,62),(16,32,77,73),(17,33,78,74),(18,34,79,75),(19,35,80,71),(20,31,76,72)], [(1,54,29,12),(2,55,30,13),(3,51,26,14),(4,52,27,15),(5,53,28,11),(6,63,56,22),(7,64,57,23),(8,65,58,24),(9,61,59,25),(10,62,60,21),(16,73,77,32),(17,74,78,33),(18,75,79,34),(19,71,80,35),(20,72,76,31),(36,41,70,46),(37,42,66,47),(38,43,67,48),(39,44,68,49),(40,45,69,50)], [(1,16,66,7),(2,17,67,8),(3,18,68,9),(4,19,69,10),(5,20,70,6),(11,72,41,63),(12,73,42,64),(13,74,43,65),(14,75,44,61),(15,71,45,62),(21,52,35,50),(22,53,31,46),(23,54,32,47),(24,55,33,48),(25,51,34,49),(26,79,39,59),(27,80,40,60),(28,76,36,56),(29,77,37,57),(30,78,38,58)]])
140 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | ··· | 4N | 4O | 4P | 4Q | 4R | 5A | 5B | 5C | 5D | 8A | 8B | 8C | 8D | 10A | ··· | 10L | 10M | ··· | 10T | 20A | ··· | 20P | 20Q | ··· | 20BD | 20BE | ··· | 20BT | 40A | ··· | 40P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
140 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | + | |||||||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C4 | C5 | C10 | C10 | C10 | C20 | C20 | C20 | D4 | Q8 | D4 | C4≀C2 | C5×D4 | C5×Q8 | C5×D4 | C5×C4≀C2 |
| kernel | C5×C42⋊6C4 | C2×C4×C20 | C5×C42⋊C2 | C10×M4(2) | C4×C20 | C5×C4⋊C4 | C5×M4(2) | C42⋊6C4 | C2×C42 | C42⋊C2 | C2×M4(2) | C42 | C4⋊C4 | M4(2) | C2×C20 | C2×C20 | C22×C10 | C10 | C2×C4 | C2×C4 | C23 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 16 | 16 | 16 | 2 | 1 | 1 | 8 | 8 | 4 | 4 | 32 |
Matrix representation of C5×C42⋊6C4 ►in GL3(𝔽41) generated by
| 1 | 0 | 0 |
| 0 | 18 | 0 |
| 0 | 0 | 18 |
| 40 | 0 | 0 |
| 0 | 1 | 5 |
| 0 | 0 | 9 |
| 1 | 0 | 0 |
| 0 | 9 | 40 |
| 0 | 0 | 32 |
| 32 | 0 | 0 |
| 0 | 32 | 0 |
| 0 | 2 | 9 |
G:=sub<GL(3,GF(41))| [1,0,0,0,18,0,0,0,18],[40,0,0,0,1,0,0,5,9],[1,0,0,0,9,0,0,40,32],[32,0,0,0,32,2,0,0,9] >;
C5×C42⋊6C4 in GAP, Magma, Sage, TeX
C_5\times C_4^2\rtimes_6C_4
% in TeX
G:=Group("C5xC4^2:6C4"); // GroupNames label
G:=SmallGroup(320,144);
// by ID
G=gap.SmallGroup(320,144);
# by ID
G:=PCGroup([7,-2,-2,-5,-2,-2,-2,-2,280,309,568,5043,248,10085]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^4=c^4=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,d*b*d^-1=b*c=c*b,d*c*d^-1=c^-1>;
// generators/relations