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
Generators and relations for C20⋊(C4⋊C4)
G = < a,b,c | a20=b4=c4=1, bab-1=a-1, cac-1=a13, cbc-1=b-1 >
Subgroups: 666 in 170 conjugacy classes, 64 normal (34 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C2×C4, C2×C4, C2×C4, C23, D5, C10, C42, C4⋊C4, C4⋊C4, C22×C4, Dic5, Dic5, C20, C20, F5, D10, C2×C10, C2.C42, C2×C42, C2×C4⋊C4, C4×D5, C4×D5, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C2×F5, C2×F5, C22×D5, C23.65C23, C10.D4, C4⋊Dic5, C5×C4⋊C4, C4×F5, C4⋊F5, C2×C4×D5, C2×C4×D5, C22×F5, C22×F5, D10.3Q8, D5×C4⋊C4, C2×C4×F5, C2×C4⋊F5, C2×C4⋊F5, C20⋊(C4⋊C4)
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C4⋊C4, C22×C4, C2×D4, C2×Q8, C4○D4, F5, C2×C4⋊C4, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C42.C2, C4⋊Q8, C2×F5, C23.65C23, C4⋊F5, C22×F5, C2×C4⋊F5, D4×F5, Q8×F5, C20⋊(C4⋊C4)
►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 21 60 75)(2 40 41 74)(3 39 42 73)(4 38 43 72)(5 37 44 71)(6 36 45 70)(7 35 46 69)(8 34 47 68)(9 33 48 67)(10 32 49 66)(11 31 50 65)(12 30 51 64)(13 29 52 63)(14 28 53 62)(15 27 54 61)(16 26 55 80)(17 25 56 79)(18 24 57 78)(19 23 58 77)(20 22 59 76)
(1 36)(2 33 10 29)(3 30 19 22)(4 27 8 35)(5 24 17 28)(6 21)(7 38 15 34)(9 32 13 40)(11 26)(12 23 20 39)(14 37 18 25)(16 31)(41 67 49 63)(42 64 58 76)(43 61 47 69)(44 78 56 62)(45 75)(46 72 54 68)(48 66 52 74)(50 80)(51 77 59 73)(53 71 57 79)(55 65)(60 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), (1,21,60,75)(2,40,41,74)(3,39,42,73)(4,38,43,72)(5,37,44,71)(6,36,45,70)(7,35,46,69)(8,34,47,68)(9,33,48,67)(10,32,49,66)(11,31,50,65)(12,30,51,64)(13,29,52,63)(14,28,53,62)(15,27,54,61)(16,26,55,80)(17,25,56,79)(18,24,57,78)(19,23,58,77)(20,22,59,76), (1,36)(2,33,10,29)(3,30,19,22)(4,27,8,35)(5,24,17,28)(6,21)(7,38,15,34)(9,32,13,40)(11,26)(12,23,20,39)(14,37,18,25)(16,31)(41,67,49,63)(42,64,58,76)(43,61,47,69)(44,78,56,62)(45,75)(46,72,54,68)(48,66,52,74)(50,80)(51,77,59,73)(53,71,57,79)(55,65)(60,70)>;
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), (1,21,60,75)(2,40,41,74)(3,39,42,73)(4,38,43,72)(5,37,44,71)(6,36,45,70)(7,35,46,69)(8,34,47,68)(9,33,48,67)(10,32,49,66)(11,31,50,65)(12,30,51,64)(13,29,52,63)(14,28,53,62)(15,27,54,61)(16,26,55,80)(17,25,56,79)(18,24,57,78)(19,23,58,77)(20,22,59,76), (1,36)(2,33,10,29)(3,30,19,22)(4,27,8,35)(5,24,17,28)(6,21)(7,38,15,34)(9,32,13,40)(11,26)(12,23,20,39)(14,37,18,25)(16,31)(41,67,49,63)(42,64,58,76)(43,61,47,69)(44,78,56,62)(45,75)(46,72,54,68)(48,66,52,74)(50,80)(51,77,59,73)(53,71,57,79)(55,65)(60,70) );
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)], [(1,21,60,75),(2,40,41,74),(3,39,42,73),(4,38,43,72),(5,37,44,71),(6,36,45,70),(7,35,46,69),(8,34,47,68),(9,33,48,67),(10,32,49,66),(11,31,50,65),(12,30,51,64),(13,29,52,63),(14,28,53,62),(15,27,54,61),(16,26,55,80),(17,25,56,79),(18,24,57,78),(19,23,58,77),(20,22,59,76)], [(1,36),(2,33,10,29),(3,30,19,22),(4,27,8,35),(5,24,17,28),(6,21),(7,38,15,34),(9,32,13,40),(11,26),(12,23,20,39),(14,37,18,25),(16,31),(41,67,49,63),(42,64,58,76),(43,61,47,69),(44,78,56,62),(45,75),(46,72,54,68),(48,66,52,74),(50,80),(51,77,59,73),(53,71,57,79),(55,65),(60,70)]])
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 |
Matrix representation of C20⋊(C4⋊C4) ►in 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] >;
C20⋊(C4⋊C4) 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