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G = C5×C42.3C4order 320 = 26·5

Direct product of C5 and C42.3C4

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

Aliases: C5×C42.3C4, C42.3C20, C4⋊Q8.3C10, (C4×C20).23C4, (C2×C20).20D4, (C2×Q8).3C20, C4.10D4.C10, (Q8×C10).16C4, C10.58(C23⋊C4), (Q8×C10).156C22, (C2×C4).4(C5×D4), (C2×C4).4(C2×C20), (C5×C4⋊Q8).18C2, C2.11(C5×C23⋊C4), (C2×Q8).2(C2×C10), (C2×C20).188(C2×C4), (C5×C4.10D4).2C2, C22.15(C5×C22⋊C4), (C2×C10).142(C22⋊C4), SmallGroup(320,161)

Series: Derived Chief Lower central Upper central

C1C2×C4 — C5×C42.3C4
C1C2C22C2×C4C2×Q8Q8×C10C5×C4.10D4 — C5×C42.3C4
C1C2C22C2×C4 — C5×C42.3C4
C1C10C2×C10Q8×C10 — C5×C42.3C4

Generators and relations for C5×C42.3C4
 G = < a,b,c,d | a5=b4=c4=1, d4=c2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1c, dcd-1=b2c >

Subgroups: 114 in 60 conjugacy classes, 26 normal (18 characteristic)
C1, C2, C2, C4 [×6], C22, C5, C8 [×2], C2×C4, C2×C4 [×2], C2×C4 [×2], Q8 [×2], C10, C10, C42, C4⋊C4 [×2], M4(2) [×2], C2×Q8 [×2], C20 [×6], C2×C10, C4.10D4 [×2], C4⋊Q8, C40 [×2], C2×C20, C2×C20 [×2], C2×C20 [×2], C5×Q8 [×2], C42.3C4, C4×C20, C5×C4⋊C4 [×2], C5×M4(2) [×2], Q8×C10 [×2], C5×C4.10D4 [×2], C5×C4⋊Q8, C5×C42.3C4
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], C42.3C4, C5×C22⋊C4, C5×C23⋊C4, C5×C42.3C4

Smallest permutation representation of C5×C42.3C4
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)
(2 32 6 28)(4 30 8 26)(10 80 14 76)(12 78 16 74)(18 46 22 42)(20 44 24 48)(34 51 38 55)(36 49 40 53)(58 68 62 72)(60 66 64 70)
(1 31 5 27)(2 32 6 28)(3 29 7 25)(4 30 8 26)(9 79 13 75)(10 80 14 76)(11 77 15 73)(12 78 16 74)(17 45 21 41)(18 46 22 42)(19 43 23 47)(20 44 24 48)(33 50 37 54)(34 51 38 55)(35 56 39 52)(36 49 40 53)(57 67 61 71)(58 68 62 72)(59 65 63 69)(60 66 64 70)
(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), (2,32,6,28)(4,30,8,26)(10,80,14,76)(12,78,16,74)(18,46,22,42)(20,44,24,48)(34,51,38,55)(36,49,40,53)(58,68,62,72)(60,66,64,70), (1,31,5,27)(2,32,6,28)(3,29,7,25)(4,30,8,26)(9,79,13,75)(10,80,14,76)(11,77,15,73)(12,78,16,74)(17,45,21,41)(18,46,22,42)(19,43,23,47)(20,44,24,48)(33,50,37,54)(34,51,38,55)(35,56,39,52)(36,49,40,53)(57,67,61,71)(58,68,62,72)(59,65,63,69)(60,66,64,70), (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), (2,32,6,28)(4,30,8,26)(10,80,14,76)(12,78,16,74)(18,46,22,42)(20,44,24,48)(34,51,38,55)(36,49,40,53)(58,68,62,72)(60,66,64,70), (1,31,5,27)(2,32,6,28)(3,29,7,25)(4,30,8,26)(9,79,13,75)(10,80,14,76)(11,77,15,73)(12,78,16,74)(17,45,21,41)(18,46,22,42)(19,43,23,47)(20,44,24,48)(33,50,37,54)(34,51,38,55)(35,56,39,52)(36,49,40,53)(57,67,61,71)(58,68,62,72)(59,65,63,69)(60,66,64,70), (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)], [(2,32,6,28),(4,30,8,26),(10,80,14,76),(12,78,16,74),(18,46,22,42),(20,44,24,48),(34,51,38,55),(36,49,40,53),(58,68,62,72),(60,66,64,70)], [(1,31,5,27),(2,32,6,28),(3,29,7,25),(4,30,8,26),(9,79,13,75),(10,80,14,76),(11,77,15,73),(12,78,16,74),(17,45,21,41),(18,46,22,42),(19,43,23,47),(20,44,24,48),(33,50,37,54),(34,51,38,55),(35,56,39,52),(36,49,40,53),(57,67,61,71),(58,68,62,72),(59,65,63,69),(60,66,64,70)], [(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 2A2B4A···4E4F5A5B5C5D8A8B8C8D10A10B10C10D10E10F10G10H20A···20T20U20V20W20X40A···40P
order1224···4455558888101010101010101020···202020202040···40
size1124···4811118888111122224···488888···8

65 irreducible representations

dim1111111111224444
type+++++-
imageC1C2C2C4C4C5C10C10C20C20D4C5×D4C23⋊C4C42.3C4C5×C23⋊C4C5×C42.3C4
kernelC5×C42.3C4C5×C4.10D4C5×C4⋊Q8C4×C20Q8×C10C42.3C4C4.10D4C4⋊Q8C42C2×Q8C2×C20C2×C4C10C5C2C1
# reps1212248488281248

Matrix representation of C5×C42.3C4 in GL4(𝔽41) generated by

16000
01600
00160
00016
,
1000
0100
00032
00320
,
0900
9000
00032
00320
,
0010
0001
0100
40000
G:=sub<GL(4,GF(41))| [16,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16],[1,0,0,0,0,1,0,0,0,0,0,32,0,0,32,0],[0,9,0,0,9,0,0,0,0,0,0,32,0,0,32,0],[0,0,0,40,0,0,1,0,1,0,0,0,0,1,0,0] >;

C5×C42.3C4 in GAP, Magma, Sage, TeX

C_5\times C_4^2._3C_4
% in TeX

G:=Group("C5xC4^2.3C4");
// GroupNames label

G:=SmallGroup(320,161);
// by ID

G=gap.SmallGroup(320,161);
# by ID

G:=PCGroup([7,-2,-2,-5,-2,-2,-2,-2,280,309,1128,2803,2530,248,4911,375,172,10085]);
// 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,d*c*d^-1=b^2*c>;
// generators/relations

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