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G = C5xD4oD8order 320 = 26·5

Direct product of C5 and D4oD8

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

Aliases: C5xD4oD8, C20.85C24, C40.52C23, 2+ 1+4:3C10, C4oD8:4C10, C8oD4:3C10, D8:7(C2xC10), (C2xD8):12C10, (C10xD8):26C2, C8:C22:4C10, Q16:7(C2xC10), (C5xD4).45D4, D4.11(C5xD4), C4.45(D4xC10), (C5xQ8).45D4, Q8.11(C5xD4), (C2xC40):31C22, SD16:4(C2xC10), C20.406(C2xD4), (C5xD8):21C22, C4.8(C23xC10), C22.7(D4xC10), (D4xC10):40C22, M4(2):6(C2xC10), C8.10(C22xC10), (C5xQ16):21C22, D4.5(C22xC10), (C5xD4).38C23, Q8.5(C22xC10), (C5xQ8).39C23, (C2xC20).687C23, (C5xSD16):20C22, C10.206(C22xD4), (C5x2+ 1+4):9C2, (C5xM4(2)):32C22, (C2xC8):4(C2xC10), C2.30(D4xC2xC10), C4oD4:1(C2xC10), (C5xC8oD4):12C2, (C5xC4oD8):11C2, (C2xD4):7(C2xC10), (C5xC8:C22):11C2, (C2xC10).184(C2xD4), (C5xC4oD4):14C22, (C2xC4).48(C22xC10), SmallGroup(320,1578)

Series: Derived Chief Lower central Upper central

C1C4 — C5xD4oD8
C1C2C4C20C5xD4C5xD8C10xD8 — C5xD4oD8
C1C2C4 — C5xD4oD8
C1C10C5xC4oD4 — C5xD4oD8

Generators and relations for C5xD4oD8
 G = < a,b,c,d,e | a5=b4=c2=e2=1, d4=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=b2d3 >

Subgroups: 474 in 268 conjugacy classes, 158 normal (18 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C5, C8, C8, C2xC4, C2xC4, D4, D4, Q8, Q8, C23, C10, C10, C2xC8, M4(2), D8, SD16, Q16, C2xD4, C2xD4, C4oD4, C4oD4, C4oD4, C20, C20, C20, C2xC10, C2xC10, C8oD4, C2xD8, C4oD8, C8:C22, 2+ 1+4, C40, C40, C2xC20, C2xC20, C5xD4, C5xD4, C5xQ8, C5xQ8, C22xC10, D4oD8, C2xC40, C5xM4(2), C5xD8, C5xSD16, C5xQ16, D4xC10, D4xC10, C5xC4oD4, C5xC4oD4, C5xC4oD4, C5xC8oD4, C10xD8, C5xC4oD8, C5xC8:C22, C5x2+ 1+4, C5xD4oD8
Quotients: C1, C2, C22, C5, D4, C23, C10, C2xD4, C24, C2xC10, C22xD4, C5xD4, C22xC10, D4oD8, D4xC10, C23xC10, D4xC2xC10, C5xD4oD8

Smallest permutation representation of C5xD4oD8
On 80 points
Generators in S80
(1 23 69 49 27)(2 24 70 50 28)(3 17 71 51 29)(4 18 72 52 30)(5 19 65 53 31)(6 20 66 54 32)(7 21 67 55 25)(8 22 68 56 26)(9 78 58 36 47)(10 79 59 37 48)(11 80 60 38 41)(12 73 61 39 42)(13 74 62 40 43)(14 75 63 33 44)(15 76 64 34 45)(16 77 57 35 46)
(1 35 5 39)(2 36 6 40)(3 37 7 33)(4 38 8 34)(9 66 13 70)(10 67 14 71)(11 68 15 72)(12 69 16 65)(17 48 21 44)(18 41 22 45)(19 42 23 46)(20 43 24 47)(25 63 29 59)(26 64 30 60)(27 57 31 61)(28 58 32 62)(49 77 53 73)(50 78 54 74)(51 79 55 75)(52 80 56 76)
(9 13)(10 14)(11 15)(12 16)(33 37)(34 38)(35 39)(36 40)(41 45)(42 46)(43 47)(44 48)(57 61)(58 62)(59 63)(60 64)(73 77)(74 78)(75 79)(76 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)
(2 8)(3 7)(4 6)(9 15)(10 14)(11 13)(17 21)(18 20)(22 24)(25 29)(26 28)(30 32)(33 37)(34 36)(38 40)(41 43)(44 48)(45 47)(50 56)(51 55)(52 54)(58 64)(59 63)(60 62)(66 72)(67 71)(68 70)(74 80)(75 79)(76 78)

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

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

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

110 conjugacy classes

class 1 2A2B2C2D2E···2J4A4B4C4D4E4F5A5B5C5D8A8B8C8D8E10A10B10C10D10E···10P10Q···10AN20A···20P20Q···20X40A···40H40I···40T
order122222···24444445555888881010101010···1010···1020···2020···2040···4040···40
size112224···422224411112244411112···24···42···24···42···24···4

110 irreducible representations

dim111111111111222244
type+++++++++
imageC1C2C2C2C2C2C5C10C10C10C10C10D4D4C5xD4C5xD4D4oD8C5xD4oD8
kernelC5xD4oD8C5xC8oD4C10xD8C5xC4oD8C5xC8:C22C5x2+ 1+4D4oD8C8oD4C2xD8C4oD8C8:C222+ 1+4C5xD4C5xQ8D4Q8C5C1
# reps1133624412122483112428

Matrix representation of C5xD4oD8 in GL4(F41) generated by

16000
01600
00160
00016
,
084012
08012
14000
039033
,
1008
0108
00400
00040
,
29291427
1229140
001712
00170
,
0180
1080
00400
00391
G:=sub<GL(4,GF(41))| [16,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16],[0,0,1,0,8,8,40,39,40,0,0,0,12,12,0,33],[1,0,0,0,0,1,0,0,0,0,40,0,8,8,0,40],[29,12,0,0,29,29,0,0,14,14,17,17,27,0,12,0],[0,1,0,0,1,0,0,0,8,8,40,39,0,0,0,1] >;

C5xD4oD8 in GAP, Magma, Sage, TeX

C_5\times D_4\circ D_8
% in TeX

G:=Group("C5xD4oD8");
// GroupNames label

G:=SmallGroup(320,1578);
// by ID

G=gap.SmallGroup(320,1578);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-5,-2,-2,1149,1193,10085,5052,124]);
// Polycyclic

G:=Group<a,b,c,d,e|a^5=b^4=c^2=e^2=1,d^4=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=b^2*d^3>;
// generators/relations

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