?
G = C20⋊(C4⋊C4) order 320 = 26·5
The semidirect product of C20 and C4⋊C4 acting via C4⋊C4/C2=C2×C4
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C20⋊(C4⋊C4), C4⋊C4⋊6F5, C4⋊2(C4⋊F5), C2.7(Q8×F5), Dic5⋊(C4⋊C4), (C2×F5).2D4, C2.14(D4×F5), (C2×F5).2Q8, (C4×D5).8Q8, C10.6(C4×Q8), C4⋊Dic5⋊11C4, (C4×D5).29D4, C10.13(C4×D4), D5.2(C4⋊Q8), D10.28(C2×D4), C10.D4⋊2C4, D10.11(C2×Q8), D5.2(C4⋊D4), D5.3(C22⋊Q8), D10.47(C4○D4), C5⋊(C23.65C23), D10.3Q8.7C2, D5.3(C42.C2), (C22×F5).5C22, C22.83(C22×F5), (C22×D5).273C23, (C5×C4⋊C4)⋊7C4, (C2×C4×F5).3C2, C2.13(C2×C4⋊F5), C10.10(C2×C4⋊C4), (D5×C4⋊C4).17C2, (C2×C4⋊F5).12C2, (C2×C4).74(C2×F5), (C2×C20).88(C2×C4), (C2×C4×D5).279C22, (C2×C10).50(C22×C4), (C2×Dic5).65(C2×C4), SmallGroup(320,1050)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 666 in 170 conjugacy classes, 64 normal (34 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×12], C22, C22 [×6], C5, C2×C4, C2×C4 [×2], C2×C4 [×25], C23, D5 [×4], C10 [×3], C42 [×2], C4⋊C4, C4⋊C4 [×9], C22×C4 [×7], Dic5 [×2], Dic5 [×2], C20 [×2], C20 [×2], F5 [×6], D10 [×6], C2×C10, C2.C42 [×2], C2×C42, C2×C4⋊C4 [×4], C4×D5 [×4], C4×D5 [×4], C2×Dic5, C2×Dic5 [×2], C2×C20, C2×C20 [×2], C2×F5 [×4], C2×F5 [×10], C22×D5, C23.65C23, C10.D4 [×2], C4⋊Dic5, C5×C4⋊C4, C4×F5 [×2], C4⋊F5 [×6], C2×C4×D5, C2×C4×D5 [×2], C22×F5 [×2], C22×F5 [×2], D10.3Q8 [×2], D5×C4⋊C4, C2×C4×F5, C2×C4⋊F5, C2×C4⋊F5 [×2], C20⋊(C4⋊C4)
Quotients:
C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], Q8 [×4], C23, C4⋊C4 [×4], C22×C4, C2×D4 [×2], C2×Q8 [×2], C4○D4 [×2], F5, C2×C4⋊C4, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C42.C2, C4⋊Q8, C2×F5 [×3], C23.65C23, C4⋊F5 [×2], C22×F5, C2×C4⋊F5, D4×F5, Q8×F5, C20⋊(C4⋊C4)
Generators and relations
G = < a,b,c | a20=b4=c4=1, bab-1=a-1, cac-1=a13, cbc-1=b-1 >
►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 34 68 53)(2 33 69 52)(3 32 70 51)(4 31 71 50)(5 30 72 49)(6 29 73 48)(7 28 74 47)(8 27 75 46)(9 26 76 45)(10 25 77 44)(11 24 78 43)(12 23 79 42)(13 22 80 41)(14 21 61 60)(15 40 62 59)(16 39 63 58)(17 38 64 57)(18 37 65 56)(19 36 66 55)(20 35 67 54)
(1 29)(2 26 10 22)(3 23 19 35)(4 40 8 28)(5 37 17 21)(6 34)(7 31 15 27)(9 25 13 33)(11 39)(12 36 20 32)(14 30 18 38)(16 24)(41 69 45 77)(42 66 54 70)(43 63)(44 80 52 76)(46 74 50 62)(47 71 59 75)(48 68)(49 65 57 61)(51 79 55 67)(53 73)(56 64 60 72)(58 78)
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), (1,34,68,53)(2,33,69,52)(3,32,70,51)(4,31,71,50)(5,30,72,49)(6,29,73,48)(7,28,74,47)(8,27,75,46)(9,26,76,45)(10,25,77,44)(11,24,78,43)(12,23,79,42)(13,22,80,41)(14,21,61,60)(15,40,62,59)(16,39,63,58)(17,38,64,57)(18,37,65,56)(19,36,66,55)(20,35,67,54), (1,29)(2,26,10,22)(3,23,19,35)(4,40,8,28)(5,37,17,21)(6,34)(7,31,15,27)(9,25,13,33)(11,39)(12,36,20,32)(14,30,18,38)(16,24)(41,69,45,77)(42,66,54,70)(43,63)(44,80,52,76)(46,74,50,62)(47,71,59,75)(48,68)(49,65,57,61)(51,79,55,67)(53,73)(56,64,60,72)(58,78)>;
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), (1,34,68,53)(2,33,69,52)(3,32,70,51)(4,31,71,50)(5,30,72,49)(6,29,73,48)(7,28,74,47)(8,27,75,46)(9,26,76,45)(10,25,77,44)(11,24,78,43)(12,23,79,42)(13,22,80,41)(14,21,61,60)(15,40,62,59)(16,39,63,58)(17,38,64,57)(18,37,65,56)(19,36,66,55)(20,35,67,54), (1,29)(2,26,10,22)(3,23,19,35)(4,40,8,28)(5,37,17,21)(6,34)(7,31,15,27)(9,25,13,33)(11,39)(12,36,20,32)(14,30,18,38)(16,24)(41,69,45,77)(42,66,54,70)(43,63)(44,80,52,76)(46,74,50,62)(47,71,59,75)(48,68)(49,65,57,61)(51,79,55,67)(53,73)(56,64,60,72)(58,78) );
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)], [(1,34,68,53),(2,33,69,52),(3,32,70,51),(4,31,71,50),(5,30,72,49),(6,29,73,48),(7,28,74,47),(8,27,75,46),(9,26,76,45),(10,25,77,44),(11,24,78,43),(12,23,79,42),(13,22,80,41),(14,21,61,60),(15,40,62,59),(16,39,63,58),(17,38,64,57),(18,37,65,56),(19,36,66,55),(20,35,67,54)], [(1,29),(2,26,10,22),(3,23,19,35),(4,40,8,28),(5,37,17,21),(6,34),(7,31,15,27),(9,25,13,33),(11,39),(12,36,20,32),(14,30,18,38),(16,24),(41,69,45,77),(42,66,54,70),(43,63),(44,80,52,76),(46,74,50,62),(47,71,59,75),(48,68),(49,65,57,61),(51,79,55,67),(53,73),(56,64,60,72),(58,78)])
Matrix representation ►G ⊆ GL6(𝔽41)
| 0 | 40 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 0 | 0 | 1 | 0 | 0 | 40 |
| 0 | 0 | 0 | 1 | 0 | 40 |
| 0 | 0 | 0 | 0 | 1 | 40 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 27 | 0 | 14 | 34 |
| 0 | 0 | 27 | 14 | 7 | 0 |
| 0 | 0 | 0 | 7 | 14 | 27 |
| 0 | 0 | 34 | 14 | 0 | 27 |
| 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 14 | 27 | 34 | 0 |
| 0 | 0 | 7 | 27 | 0 | 14 |
| 0 | 0 | 14 | 0 | 27 | 7 |
| 0 | 0 | 0 | 34 | 27 | 14 |
G:=sub<GL(6,GF(41))| [0,1,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,40,40,40,40],[0,40,0,0,0,0,40,0,0,0,0,0,0,0,27,27,0,34,0,0,0,14,7,14,0,0,14,7,14,0,0,0,34,0,27,27],[9,0,0,0,0,0,0,9,0,0,0,0,0,0,14,7,14,0,0,0,27,27,0,34,0,0,34,0,27,27,0,0,0,14,7,14] >;
38 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | ··· | 4N | 4O | ··· | 4T | 5 | 10A | 10B | 10C | 20A | ··· | 20F |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 5 | 10 | 10 | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 5 | 5 | 5 | 5 | 2 | 2 | 4 | 4 | 10 | ··· | 10 | 20 | ··· | 20 | 4 | 4 | 4 | 4 | 8 | ··· | 8 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 8 | 8 |
| type | + | + | + | + | + | + | - | + | - | + | + | + | - | |||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | D4 | Q8 | D4 | Q8 | C4○D4 | F5 | C2×F5 | C4⋊F5 | D4×F5 | Q8×F5 |
| kernel | C20⋊(C4⋊C4) | D10.3Q8 | D5×C4⋊C4 | C2×C4×F5 | C2×C4⋊F5 | C10.D4 | C4⋊Dic5 | C5×C4⋊C4 | C4×D5 | C4×D5 | C2×F5 | C2×F5 | D10 | C4⋊C4 | C2×C4 | C4 | C2 | C2 |
| # reps | 1 | 2 | 1 | 1 | 3 | 4 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 1 | 3 | 4 | 1 | 1 |
In GAP, Magma, Sage, TeX
C_{20}\rtimes (C_4\rtimes C_4) % in TeX
G:=Group("C20:(C4:C4)"); // GroupNames label
G:=SmallGroup(320,1050);
// by ID
G=gap.SmallGroup(320,1050);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,232,422,387,100,6278,1595]);
// Polycyclic
G:=Group<a,b,c|a^20=b^4=c^4=1,b*a*b^-1=a^-1,c*a*c^-1=a^13,c*b*c^-1=b^-1>;
// generators/relations