direct product, metabelian, nilpotent (class 4), monomial, 2-elementary
Aliases: C5×C42⋊3C4, C42⋊3C20, (C4×C20)⋊16C4, (C2×Q8)⋊2C20, (Q8×C10)⋊14C4, C23.4(C5×D4), C23⋊C4.2C10, (C22×C10).4D4, C4.4D4.2C10, C10.56(C23⋊C4), (D4×C10).177C22, (C2×C4).2(C2×C20), C2.9(C5×C23⋊C4), (C2×D4).4(C2×C10), (C5×C23⋊C4).4C2, (C2×C20).186(C2×C4), (C5×C4.4D4).11C2, C22.13(C5×C22⋊C4), (C2×C10).140(C22⋊C4), SmallGroup(320,159)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×C42⋊3C4
G = < a,b,c,d | a5=b4=c4=d4=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1c-1, dcd-1=b2c-1 >
Subgroups: 178 in 70 conjugacy classes, 26 normal (18 characteristic)
C1, C2, C2, C4, C22, C22, C5, C2×C4, C2×C4, D4, Q8, C23, C10, C10, C42, C22⋊C4, C2×D4, C2×Q8, C20, C2×C10, C2×C10, C23⋊C4, C4.4D4, C2×C20, C2×C20, C5×D4, C5×Q8, C22×C10, C42⋊3C4, C4×C20, C5×C22⋊C4, D4×C10, Q8×C10, C5×C23⋊C4, C5×C4.4D4, C5×C42⋊3C4
Quotients: C1, C2, C4, C22, C5, C2×C4, D4, C10, C22⋊C4, C20, C2×C10, C23⋊C4, C2×C20, C5×D4, C42⋊3C4, C5×C22⋊C4, C5×C23⋊C4, C5×C42⋊3C4
►On 80 points
Generators in S80
(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 73 66 56)(2 74 67 57)(3 75 68 58)(4 71 69 59)(5 72 70 60)(6 41 23 27)(7 42 24 28)(8 43 25 29)(9 44 21 30)(10 45 22 26)(11 31 36 19)(12 32 37 20)(13 33 38 16)(14 34 39 17)(15 35 40 18)(46 64 53 76)(47 65 54 77)(48 61 55 78)(49 62 51 79)(50 63 52 80)
(1 54 12 42)(2 55 13 43)(3 51 14 44)(4 52 15 45)(5 53 11 41)(6 60 64 19)(7 56 65 20)(8 57 61 16)(9 58 62 17)(10 59 63 18)(21 75 79 34)(22 71 80 35)(23 72 76 31)(24 73 77 32)(25 74 78 33)(26 69 50 40)(27 70 46 36)(28 66 47 37)(29 67 48 38)(30 68 49 39)
(1 7 42 32)(2 8 43 33)(3 9 44 34)(4 10 45 35)(5 6 41 31)(11 64 53 72)(12 65 54 73)(13 61 55 74)(14 62 51 75)(15 63 52 71)(16 38 25 48)(17 39 21 49)(18 40 22 50)(19 36 23 46)(20 37 24 47)(26 59 69 80)(27 60 70 76)(28 56 66 77)(29 57 67 78)(30 58 68 79)
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), (1,73,66,56)(2,74,67,57)(3,75,68,58)(4,71,69,59)(5,72,70,60)(6,41,23,27)(7,42,24,28)(8,43,25,29)(9,44,21,30)(10,45,22,26)(11,31,36,19)(12,32,37,20)(13,33,38,16)(14,34,39,17)(15,35,40,18)(46,64,53,76)(47,65,54,77)(48,61,55,78)(49,62,51,79)(50,63,52,80), (1,54,12,42)(2,55,13,43)(3,51,14,44)(4,52,15,45)(5,53,11,41)(6,60,64,19)(7,56,65,20)(8,57,61,16)(9,58,62,17)(10,59,63,18)(21,75,79,34)(22,71,80,35)(23,72,76,31)(24,73,77,32)(25,74,78,33)(26,69,50,40)(27,70,46,36)(28,66,47,37)(29,67,48,38)(30,68,49,39), (1,7,42,32)(2,8,43,33)(3,9,44,34)(4,10,45,35)(5,6,41,31)(11,64,53,72)(12,65,54,73)(13,61,55,74)(14,62,51,75)(15,63,52,71)(16,38,25,48)(17,39,21,49)(18,40,22,50)(19,36,23,46)(20,37,24,47)(26,59,69,80)(27,60,70,76)(28,56,66,77)(29,57,67,78)(30,58,68,79)>;
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), (1,73,66,56)(2,74,67,57)(3,75,68,58)(4,71,69,59)(5,72,70,60)(6,41,23,27)(7,42,24,28)(8,43,25,29)(9,44,21,30)(10,45,22,26)(11,31,36,19)(12,32,37,20)(13,33,38,16)(14,34,39,17)(15,35,40,18)(46,64,53,76)(47,65,54,77)(48,61,55,78)(49,62,51,79)(50,63,52,80), (1,54,12,42)(2,55,13,43)(3,51,14,44)(4,52,15,45)(5,53,11,41)(6,60,64,19)(7,56,65,20)(8,57,61,16)(9,58,62,17)(10,59,63,18)(21,75,79,34)(22,71,80,35)(23,72,76,31)(24,73,77,32)(25,74,78,33)(26,69,50,40)(27,70,46,36)(28,66,47,37)(29,67,48,38)(30,68,49,39), (1,7,42,32)(2,8,43,33)(3,9,44,34)(4,10,45,35)(5,6,41,31)(11,64,53,72)(12,65,54,73)(13,61,55,74)(14,62,51,75)(15,63,52,71)(16,38,25,48)(17,39,21,49)(18,40,22,50)(19,36,23,46)(20,37,24,47)(26,59,69,80)(27,60,70,76)(28,56,66,77)(29,57,67,78)(30,58,68,79) );
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)], [(1,73,66,56),(2,74,67,57),(3,75,68,58),(4,71,69,59),(5,72,70,60),(6,41,23,27),(7,42,24,28),(8,43,25,29),(9,44,21,30),(10,45,22,26),(11,31,36,19),(12,32,37,20),(13,33,38,16),(14,34,39,17),(15,35,40,18),(46,64,53,76),(47,65,54,77),(48,61,55,78),(49,62,51,79),(50,63,52,80)], [(1,54,12,42),(2,55,13,43),(3,51,14,44),(4,52,15,45),(5,53,11,41),(6,60,64,19),(7,56,65,20),(8,57,61,16),(9,58,62,17),(10,59,63,18),(21,75,79,34),(22,71,80,35),(23,72,76,31),(24,73,77,32),(25,74,78,33),(26,69,50,40),(27,70,46,36),(28,66,47,37),(29,67,48,38),(30,68,49,39)], [(1,7,42,32),(2,8,43,33),(3,9,44,34),(4,10,45,35),(5,6,41,31),(11,64,53,72),(12,65,54,73),(13,61,55,74),(14,62,51,75),(15,63,52,71),(16,38,25,48),(17,39,21,49),(18,40,22,50),(19,36,23,46),(20,37,24,47),(26,59,69,80),(27,60,70,76),(28,56,66,77),(29,57,67,78),(30,58,68,79)]])
65 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | ··· | 4H | 5A | 5B | 5C | 5D | 10A | 10B | 10C | 10D | 10E | 10F | 10G | 10H | 10I | ··· | 10P | 20A | ··· | 20L | 20M | ··· | 20AF |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 5 | 5 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 2 | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
65 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | |||||||||||
| image | C1 | C2 | C2 | C4 | C4 | C5 | C10 | C10 | C20 | C20 | D4 | C5×D4 | C23⋊C4 | C42⋊3C4 | C5×C23⋊C4 | C5×C42⋊3C4 |
| kernel | C5×C42⋊3C4 | C5×C23⋊C4 | C5×C4.4D4 | C4×C20 | Q8×C10 | C42⋊3C4 | C23⋊C4 | C4.4D4 | C42 | C2×Q8 | C22×C10 | C23 | C10 | C5 | C2 | C1 |
| # reps | 1 | 2 | 1 | 2 | 2 | 4 | 8 | 4 | 8 | 8 | 2 | 8 | 1 | 2 | 4 | 8 |
Matrix representation of C5×C42⋊3C4 ►in GL4(𝔽41) generated by
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 9 | 0 | 0 | 9 |
| 0 | 9 | 0 | 9 |
| 9 | 9 | 9 | 9 |
| 32 | 32 | 0 | 32 |
| 0 | 0 | 40 | 0 |
| 1 | 1 | 1 | 2 |
| 1 | 0 | 0 | 0 |
| 40 | 40 | 0 | 40 |
| 0 | 0 | 32 | 32 |
| 32 | 32 | 32 | 32 |
| 9 | 0 | 0 | 9 |
| 0 | 9 | 9 | 9 |
G:=sub<GL(4,GF(41))| [16,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16],[9,0,9,32,0,9,9,32,0,0,9,0,9,9,9,32],[0,1,1,40,0,1,0,40,40,1,0,0,0,2,0,40],[0,32,9,0,0,32,0,9,32,32,0,9,32,32,9,9] >;
C5×C42⋊3C4 in GAP, Magma, Sage, TeX
C_5\times C_4^2\rtimes_3C_4
% in TeX
G:=Group("C5xC4^2:3C4"); // GroupNames label
G:=SmallGroup(320,159);
// by ID
G=gap.SmallGroup(320,159);
# by ID
G:=PCGroup([7,-2,-2,-5,-2,-2,-2,-2,280,309,1128,2803,2530,248,4911,375,10085]);
// 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,b*c=c*b,d*b*d^-1=b^-1*c^-1,d*c*d^-1=b^2*c^-1>;
// generators/relations