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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⋊C46F5, C42(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⋊Dic511C4, (C4×D5).29D4, C10.13(C4×D4), D5.2(C4⋊Q8), D10.28(C2×D4), C10.D42C4, 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

C1C2×C10 — C20⋊(C4⋊C4)
C1C5D5D10C22×D5C22×F5C2×C4×F5 — C20⋊(C4⋊C4)
C5C2×C10 — C20⋊(C4⋊C4)
C1C22C4⋊C4

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 [×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)

Smallest permutation representation of C20⋊(C4⋊C4)
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 36 59 68)(2 35 60 67)(3 34 41 66)(4 33 42 65)(5 32 43 64)(6 31 44 63)(7 30 45 62)(8 29 46 61)(9 28 47 80)(10 27 48 79)(11 26 49 78)(12 25 50 77)(13 24 51 76)(14 23 52 75)(15 22 53 74)(16 21 54 73)(17 40 55 72)(18 39 56 71)(19 38 57 70)(20 37 58 69)
(1 31)(2 28 10 24)(3 25 19 37)(4 22 8 30)(5 39 17 23)(6 36)(7 33 15 29)(9 27 13 35)(11 21)(12 38 20 34)(14 32 18 40)(16 26)(41 77 57 69)(42 74 46 62)(43 71 55 75)(44 68)(45 65 53 61)(47 79 51 67)(48 76 60 80)(49 73)(50 70 58 66)(52 64 56 72)(54 78)(59 63)

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), (1,36,59,68)(2,35,60,67)(3,34,41,66)(4,33,42,65)(5,32,43,64)(6,31,44,63)(7,30,45,62)(8,29,46,61)(9,28,47,80)(10,27,48,79)(11,26,49,78)(12,25,50,77)(13,24,51,76)(14,23,52,75)(15,22,53,74)(16,21,54,73)(17,40,55,72)(18,39,56,71)(19,38,57,70)(20,37,58,69), (1,31)(2,28,10,24)(3,25,19,37)(4,22,8,30)(5,39,17,23)(6,36)(7,33,15,29)(9,27,13,35)(11,21)(12,38,20,34)(14,32,18,40)(16,26)(41,77,57,69)(42,74,46,62)(43,71,55,75)(44,68)(45,65,53,61)(47,79,51,67)(48,76,60,80)(49,73)(50,70,58,66)(52,64,56,72)(54,78)(59,63)>;

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), (1,36,59,68)(2,35,60,67)(3,34,41,66)(4,33,42,65)(5,32,43,64)(6,31,44,63)(7,30,45,62)(8,29,46,61)(9,28,47,80)(10,27,48,79)(11,26,49,78)(12,25,50,77)(13,24,51,76)(14,23,52,75)(15,22,53,74)(16,21,54,73)(17,40,55,72)(18,39,56,71)(19,38,57,70)(20,37,58,69), (1,31)(2,28,10,24)(3,25,19,37)(4,22,8,30)(5,39,17,23)(6,36)(7,33,15,29)(9,27,13,35)(11,21)(12,38,20,34)(14,32,18,40)(16,26)(41,77,57,69)(42,74,46,62)(43,71,55,75)(44,68)(45,65,53,61)(47,79,51,67)(48,76,60,80)(49,73)(50,70,58,66)(52,64,56,72)(54,78)(59,63) );

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)], [(1,36,59,68),(2,35,60,67),(3,34,41,66),(4,33,42,65),(5,32,43,64),(6,31,44,63),(7,30,45,62),(8,29,46,61),(9,28,47,80),(10,27,48,79),(11,26,49,78),(12,25,50,77),(13,24,51,76),(14,23,52,75),(15,22,53,74),(16,21,54,73),(17,40,55,72),(18,39,56,71),(19,38,57,70),(20,37,58,69)], [(1,31),(2,28,10,24),(3,25,19,37),(4,22,8,30),(5,39,17,23),(6,36),(7,33,15,29),(9,27,13,35),(11,21),(12,38,20,34),(14,32,18,40),(16,26),(41,77,57,69),(42,74,46,62),(43,71,55,75),(44,68),(45,65,53,61),(47,79,51,67),(48,76,60,80),(49,73),(50,70,58,66),(52,64,56,72),(54,78),(59,63)])

38 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E···4N4O···4T 5 10A10B10C20A···20F
order1222222244444···44···4510101020···20
size11115555224410···1020···2044448···8

38 irreducible representations

dim111111112222244488
type++++++-+-+++-
imageC1C2C2C2C2C4C4C4D4Q8D4Q8C4○D4F5C2×F5C4⋊F5D4×F5Q8×F5
kernelC20⋊(C4⋊C4)D10.3Q8D5×C4⋊C4C2×C4×F5C2×C4⋊F5C10.D4C4⋊Dic5C5×C4⋊C4C4×D5C4×D5C2×F5C2×F5D10C4⋊C4C2×C4C4C2C2
# reps121134222222413411

Matrix representation of C20⋊(C4⋊C4) in GL6(𝔽41)

0400000
100000
0000040
0010040
0001040
0000140
,
0400000
4000000
002701434
00271470
00071427
003414027
,
900000
090000
001427340
00727014
00140277
000342714

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

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