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G = C5×M4(2).C4order 320 = 26·5

Direct product of C5 and M4(2).C4

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Aliases: C5×M4(2).C4, M4(2).1C20, C8.4(C2×C20), C40.86(C2×C4), C4.78(D4×C10), C20.96(C4⋊C4), (C2×C20).44Q8, C8.C43C10, (C2×C20).523D4, C20.483(C2×D4), C23.5(C5×Q8), C22.2(Q8×C10), (C22×C10).5Q8, C4.29(C22×C20), (C5×M4(2)).7C4, (C2×C20).903C23, (C2×C40).270C22, C20.246(C22×C4), (C2×M4(2)).2C10, (C10×M4(2)).34C2, M4(2).10(C2×C10), (C22×C20).415C22, (C5×M4(2)).44C22, C4.16(C5×C4⋊C4), C10.95(C2×C4⋊C4), C2.16(C10×C4⋊C4), (C2×C4).7(C5×Q8), (C2×C8).17(C2×C10), (C2×C4).26(C2×C20), (C2×C4).126(C5×D4), C22.12(C5×C4⋊C4), (C2×C10).15(C2×Q8), (C5×C8.C4)⋊12C2, (C2×C10).56(C4⋊C4), (C2×C20).372(C2×C4), (C2×C4).78(C22×C10), (C22×C4).34(C2×C10), SmallGroup(320,931)

Series: Derived Chief Lower central Upper central

C1C4 — C5×M4(2).C4
C1C2C4C2×C4C2×C20C5×M4(2)C5×C8.C4 — C5×M4(2).C4
C1C2C4 — C5×M4(2).C4
C1C20C22×C20 — C5×M4(2).C4

Generators and relations for C5×M4(2).C4
 G = < a,b,c,d | a5=b8=c2=1, d4=b4, ab=ba, ac=ca, ad=da, cbc=b5, dbd-1=b-1, dcd-1=b4c >

Subgroups: 130 in 102 conjugacy classes, 78 normal (22 characteristic)
C1, C2, C2 [×3], C4 [×2], C4 [×2], C22, C22 [×2], C22, C5, C8 [×4], C8 [×4], C2×C4 [×2], C2×C4 [×4], C23, C10, C10 [×3], C2×C8 [×2], C2×C8 [×2], M4(2) [×8], M4(2) [×2], C22×C4, C20 [×2], C20 [×2], C2×C10, C2×C10 [×2], C2×C10, C8.C4 [×4], C2×M4(2), C2×M4(2) [×2], C40 [×4], C40 [×4], C2×C20 [×2], C2×C20 [×4], C22×C10, M4(2).C4, C2×C40 [×2], C2×C40 [×2], C5×M4(2) [×8], C5×M4(2) [×2], C22×C20, C5×C8.C4 [×4], C10×M4(2), C10×M4(2) [×2], C5×M4(2).C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C5, C2×C4 [×6], D4 [×2], Q8 [×2], C23, C10 [×7], C4⋊C4 [×4], C22×C4, C2×D4, C2×Q8, C20 [×4], C2×C10 [×7], C2×C4⋊C4, C2×C20 [×6], C5×D4 [×2], C5×Q8 [×2], C22×C10, M4(2).C4, C5×C4⋊C4 [×4], C22×C20, D4×C10, Q8×C10, C10×C4⋊C4, C5×M4(2).C4

Smallest permutation representation of C5×M4(2).C4
On 80 points
Generators in S80
(1 45 31 33 14)(2 46 32 34 15)(3 47 25 35 16)(4 48 26 36 9)(5 41 27 37 10)(6 42 28 38 11)(7 43 29 39 12)(8 44 30 40 13)(17 67 76 55 64)(18 68 77 56 57)(19 69 78 49 58)(20 70 79 50 59)(21 71 80 51 60)(22 72 73 52 61)(23 65 74 53 62)(24 66 75 54 63)
(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 5)(3 7)(10 14)(12 16)(17 21)(19 23)(25 29)(27 31)(33 37)(35 39)(41 45)(43 47)(49 53)(51 55)(58 62)(60 64)(65 69)(67 71)(74 78)(76 80)
(1 61 7 63 5 57 3 59)(2 60 8 62 6 64 4 58)(9 49 15 51 13 53 11 55)(10 56 16 50 14 52 12 54)(17 48 19 46 21 44 23 42)(18 47 20 45 22 43 24 41)(25 70 31 72 29 66 27 68)(26 69 32 71 30 65 28 67)(33 73 39 75 37 77 35 79)(34 80 40 74 38 76 36 78)

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

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

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

110 conjugacy classes

class 1 2A2B2C2D4A4B4C4D4E5A5B5C5D8A···8L10A10B10C10D10E···10P20A···20H20I···20T40A···40AV
order122224444455558···81010101010···1020···2020···2040···40
size112221122211114···411112···21···12···24···4

110 irreducible representations

dim1111111122222244
type++++--
imageC1C2C2C4C5C10C10C20D4Q8Q8C5×D4C5×Q8C5×Q8M4(2).C4C5×M4(2).C4
kernelC5×M4(2).C4C5×C8.C4C10×M4(2)C5×M4(2)M4(2).C4C8.C4C2×M4(2)M4(2)C2×C20C2×C20C22×C10C2×C4C2×C4C23C5C1
# reps1438416123221184428

Matrix representation of C5×M4(2).C4 in GL4(𝔽41) generated by

37000
03700
00370
00037
,
092525
101618
003236
00189
,
40000
01021
0011
00040
,
025117
016024
323200
018025
G:=sub<GL(4,GF(41))| [37,0,0,0,0,37,0,0,0,0,37,0,0,0,0,37],[0,1,0,0,9,0,0,0,25,16,32,18,25,18,36,9],[40,0,0,0,0,1,0,0,0,0,1,0,0,21,1,40],[0,0,32,0,25,16,32,18,1,0,0,0,17,24,0,25] >;

C5×M4(2).C4 in GAP, Magma, Sage, TeX

C_5\times M_4(2).C_4
% in TeX

G:=Group("C5xM4(2).C4");
// GroupNames label

G:=SmallGroup(320,931);
// by ID

G=gap.SmallGroup(320,931);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,560,589,288,1731,7004,172,124]);
// Polycyclic

G:=Group<a,b,c,d|a^5=b^8=c^2=1,d^4=b^4,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^5,d*b*d^-1=b^-1,d*c*d^-1=b^4*c>;
// generators/relations

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