Copied to
clipboard

G = C5×C42.C4order 320 = 26·5

Direct product of C5 and C42.C4

direct product, metabelian, nilpotent (class 4), monomial, 2-elementary

Aliases: C5×C42.C4, C42.2C20, (C4×C20).22C4, (C2×D4).3C20, (C2×C20).19D4, (D4×C10).19C4, C4.10D45C10, C4.4D4.3C10, C10.57(C23⋊C4), (Q8×C10).155C22, (C2×C4).3(C5×D4), (C2×C4).3(C2×C20), C2.10(C5×C23⋊C4), (C2×Q8).1(C2×C10), (C2×C20).187(C2×C4), (C5×C4.10D4)⋊12C2, (C5×C4.4D4).12C2, C22.14(C5×C22⋊C4), (C2×C10).141(C22⋊C4), SmallGroup(320,160)

Series: Derived Chief Lower central Upper central

C1C2×C4 — C5×C42.C4
C1C2C22C2×C4C2×Q8Q8×C10C5×C4.10D4 — C5×C42.C4
C1C2C22C2×C4 — C5×C42.C4
C1C10C2×C10Q8×C10 — C5×C42.C4

Generators and relations for C5×C42.C4
 G = < a,b,c,d | a5=b4=c4=1, d4=c2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1c-1, dcd-1=b2c-1 >

Subgroups: 146 in 64 conjugacy classes, 26 normal (18 characteristic)
C1, C2, C2 [×2], C4 [×4], C22, C22 [×3], C5, C8 [×2], C2×C4, C2×C4 [×2], C2×C4, D4, Q8, C23, C10, C10 [×2], C42, C22⋊C4 [×2], M4(2) [×2], C2×D4, C2×Q8, C20 [×4], C2×C10, C2×C10 [×3], C4.10D4 [×2], C4.4D4, C40 [×2], C2×C20, C2×C20 [×2], C2×C20, C5×D4, C5×Q8, C22×C10, C42.C4, C4×C20, C5×C22⋊C4 [×2], C5×M4(2) [×2], D4×C10, Q8×C10, C5×C4.10D4 [×2], C5×C4.4D4, C5×C42.C4
Quotients: C1, C2 [×3], C4 [×2], C22, C5, C2×C4, D4 [×2], C10 [×3], C22⋊C4, C20 [×2], C2×C10, C23⋊C4, C2×C20, C5×D4 [×2], C42.C4, C5×C22⋊C4, C5×C23⋊C4, C5×C42.C4

Smallest permutation representation of C5×C42.C4
On 80 points
Generators in S80
(1 19 9 56 61)(2 20 10 49 62)(3 21 11 50 63)(4 22 12 51 64)(5 23 13 52 57)(6 24 14 53 58)(7 17 15 54 59)(8 18 16 55 60)(25 45 73 33 65)(26 46 74 34 66)(27 47 75 35 67)(28 48 76 36 68)(29 41 77 37 69)(30 42 78 38 70)(31 43 79 39 71)(32 44 80 40 72)
(1 3)(2 26 6 30)(4 28 8 32)(5 7)(9 11)(10 74 14 78)(12 76 16 80)(13 15)(17 23)(18 44 22 48)(19 21)(20 46 24 42)(25 27)(29 31)(33 35)(34 53 38 49)(36 55 40 51)(37 39)(41 43)(45 47)(50 56)(52 54)(57 59)(58 70 62 66)(60 72 64 68)(61 63)(65 67)(69 71)(73 75)(77 79)
(1 31 5 27)(2 28 6 32)(3 29 7 25)(4 26 8 30)(9 79 13 75)(10 76 14 80)(11 77 15 73)(12 74 16 78)(17 45 21 41)(18 42 22 46)(19 43 23 47)(20 48 24 44)(33 50 37 54)(34 55 38 51)(35 56 39 52)(36 53 40 49)(57 67 61 71)(58 72 62 68)(59 65 63 69)(60 70 64 66)
(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)

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

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

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

65 conjugacy classes

class 1 2A2B2C4A···4E5A5B5C5D8A8B8C8D10A10B10C10D10E10F10G10H10I10J10K10L20A···20T40A···40P
order12224···45555888810101010101010101010101020···2040···40
size11284···4111188881111222288884···48···8

65 irreducible representations

dim1111111111224444
type+++++
imageC1C2C2C4C4C5C10C10C20C20D4C5×D4C23⋊C4C42.C4C5×C23⋊C4C5×C42.C4
kernelC5×C42.C4C5×C4.10D4C5×C4.4D4C4×C20D4×C10C42.C4C4.10D4C4.4D4C42C2×D4C2×C20C2×C4C10C5C2C1
# reps1212248488281248

Matrix representation of C5×C42.C4 in GL4(𝔽41) generated by

18000
01800
00180
00018
,
04000
40000
00320
00032
,
03200
32000
00032
00320
,
0010
0001
04000
1000
G:=sub<GL(4,GF(41))| [18,0,0,0,0,18,0,0,0,0,18,0,0,0,0,18],[0,40,0,0,40,0,0,0,0,0,32,0,0,0,0,32],[0,32,0,0,32,0,0,0,0,0,0,32,0,0,32,0],[0,0,0,1,0,0,40,0,1,0,0,0,0,1,0,0] >;

C5×C42.C4 in GAP, Magma, Sage, TeX

C_5\times C_4^2.C_4
% in TeX

G:=Group("C5xC4^2.C4");
// GroupNames label

G:=SmallGroup(320,160);
// by ID

G=gap.SmallGroup(320,160);
# by ID

G:=PCGroup([7,-2,-2,-5,-2,-2,-2,-2,280,309,2803,2530,248,4911,375,172,10085]);
// Polycyclic

G:=Group<a,b,c,d|a^5=b^4=c^4=1,d^4=c^2,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

׿
×
𝔽