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## G = C5×D4○D8order 320 = 26·5

### Direct product of C5 and D4○D8

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C4 — C5×D4○D8
 Chief series C1 — C2 — C4 — C20 — C5×D4 — C5×D8 — C10×D8 — C5×D4○D8
 Lower central C1 — C2 — C4 — C5×D4○D8
 Upper central C1 — C10 — C5×C4○D4 — C5×D4○D8

Generators and relations for C5×D4○D8
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, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C10, C10, C2×C8, M4(2), D8, SD16, Q16, C2×D4, C2×D4, C4○D4, C4○D4, C4○D4, C20, C20, C20, C2×C10, C2×C10, C8○D4, C2×D8, C4○D8, C8⋊C22, 2+ 1+4, C40, C40, C2×C20, C2×C20, C5×D4, C5×D4, C5×Q8, C5×Q8, C22×C10, D4○D8, C2×C40, C5×M4(2), C5×D8, C5×SD16, C5×Q16, D4×C10, D4×C10, C5×C4○D4, C5×C4○D4, C5×C4○D4, C5×C8○D4, C10×D8, C5×C4○D8, C5×C8⋊C22, C5×2+ 1+4, C5×D4○D8
Quotients: C1, C2, C22, C5, D4, C23, C10, C2×D4, C24, C2×C10, C22×D4, C5×D4, C22×C10, D4○D8, D4×C10, C23×C10, D4×C2×C10, C5×D4○D8

Smallest permutation representation of C5×D4○D8
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 2A 2B 2C 2D 2E ··· 2J 4A 4B 4C 4D 4E 4F 5A 5B 5C 5D 8A 8B 8C 8D 8E 10A 10B 10C 10D 10E ··· 10P 10Q ··· 10AN 20A ··· 20P 20Q ··· 20X 40A ··· 40H 40I ··· 40T order 1 2 2 2 2 2 ··· 2 4 4 4 4 4 4 5 5 5 5 8 8 8 8 8 10 10 10 10 10 ··· 10 10 ··· 10 20 ··· 20 20 ··· 20 40 ··· 40 40 ··· 40 size 1 1 2 2 2 4 ··· 4 2 2 2 2 4 4 1 1 1 1 2 2 4 4 4 1 1 1 1 2 ··· 2 4 ··· 4 2 ··· 2 4 ··· 4 2 ··· 2 4 ··· 4

110 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 4 4 type + + + + + + + + + image C1 C2 C2 C2 C2 C2 C5 C10 C10 C10 C10 C10 D4 D4 C5×D4 C5×D4 D4○D8 C5×D4○D8 kernel C5×D4○D8 C5×C8○D4 C10×D8 C5×C4○D8 C5×C8⋊C22 C5×2+ 1+4 D4○D8 C8○D4 C2×D8 C4○D8 C8⋊C22 2+ 1+4 C5×D4 C5×Q8 D4 Q8 C5 C1 # reps 1 1 3 3 6 2 4 4 12 12 24 8 3 1 12 4 2 8

Matrix representation of C5×D4○D8 in GL4(𝔽41) generated by

 16 0 0 0 0 16 0 0 0 0 16 0 0 0 0 16
,
 0 8 40 12 0 8 0 12 1 40 0 0 0 39 0 33
,
 1 0 0 8 0 1 0 8 0 0 40 0 0 0 0 40
,
 29 29 14 27 12 29 14 0 0 0 17 12 0 0 17 0
,
 0 1 8 0 1 0 8 0 0 0 40 0 0 0 39 1
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] >;

C5×D4○D8 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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