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## G = C5×D4○SD16order 320 = 26·5

### Direct product of C5 and D4○SD16

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C4 — C5×D4○SD16
 Chief series C1 — C2 — C4 — C20 — C5×Q8 — C5×SD16 — C10×SD16 — C5×D4○SD16
 Lower central C1 — C2 — C4 — C5×D4○SD16
 Upper central C1 — C10 — C5×C4○D4 — C5×D4○SD16

Generators and relations for C5×D4○SD16
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=d3 >

Subgroups: 410 in 258 conjugacy classes, 158 normal (26 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, D4, D4, D4, Q8, Q8, Q8, C23, C10, C10, C2×C8, M4(2), D8, SD16, SD16, Q16, C2×D4, C2×D4, C2×Q8, C2×Q8, C4○D4, C4○D4, C4○D4, C20, C20, C20, C2×C10, C2×C10, C8○D4, C2×SD16, C4○D8, C8⋊C22, C8.C22, 2+ 1+4, 2- 1+4, C40, C40, C2×C20, C2×C20, C5×D4, C5×D4, C5×D4, C5×Q8, C5×Q8, C5×Q8, C22×C10, D4○SD16, C2×C40, C5×M4(2), C5×D8, C5×SD16, C5×SD16, C5×Q16, D4×C10, D4×C10, Q8×C10, Q8×C10, C5×C4○D4, C5×C4○D4, C5×C4○D4, C5×C8○D4, C10×SD16, C5×C4○D8, C5×C8⋊C22, C5×C8.C22, C5×2+ 1+4, C5×2- 1+4, C5×D4○SD16
Quotients: C1, C2, C22, C5, D4, C23, C10, C2×D4, C24, C2×C10, C22×D4, C5×D4, C22×C10, D4○SD16, D4×C10, C23×C10, D4×C2×C10, C5×D4○SD16

Smallest permutation representation of C5×D4○SD16
On 80 points
Generators in S80
(1 20 67 53 26)(2 21 68 54 27)(3 22 69 55 28)(4 23 70 56 29)(5 24 71 49 30)(6 17 72 50 31)(7 18 65 51 32)(8 19 66 52 25)(9 78 64 37 41)(10 79 57 38 42)(11 80 58 39 43)(12 73 59 40 44)(13 74 60 33 45)(14 75 61 34 46)(15 76 62 35 47)(16 77 63 36 48)
(1 38 5 34)(2 39 6 35)(3 40 7 36)(4 33 8 37)(9 70 13 66)(10 71 14 67)(11 72 15 68)(12 65 16 69)(17 47 21 43)(18 48 22 44)(19 41 23 45)(20 42 24 46)(25 64 29 60)(26 57 30 61)(27 58 31 62)(28 59 32 63)(49 75 53 79)(50 76 54 80)(51 77 55 73)(52 78 56 74)
(1 34)(2 35)(3 36)(4 37)(5 38)(6 39)(7 40)(8 33)(9 70)(10 71)(11 72)(12 65)(13 66)(14 67)(15 68)(16 69)(17 43)(18 44)(19 45)(20 46)(21 47)(22 48)(23 41)(24 42)(25 60)(26 61)(27 62)(28 63)(29 64)(30 57)(31 58)(32 59)(49 79)(50 80)(51 73)(52 74)(53 75)(54 76)(55 77)(56 78)
(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 4)(3 7)(6 8)(9 15)(11 13)(12 16)(17 19)(18 22)(21 23)(25 31)(27 29)(28 32)(33 39)(35 37)(36 40)(41 47)(43 45)(44 48)(50 52)(51 55)(54 56)(58 60)(59 63)(62 64)(65 69)(66 72)(68 70)(73 77)(74 80)(76 78)

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

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

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

110 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 4A 4B 4C 4D 4E 4F 4G 4H 5A 5B 5C 5D 8A 8B 8C 8D 8E 10A 10B 10C 10D 10E ··· 10P 10Q ··· 10AF 20A ··· 20P 20Q ··· 20AF 40A ··· 40H 40I ··· 40T order 1 2 2 2 2 2 2 2 2 4 4 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 4 4 2 2 2 2 4 4 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 1 1 1 1 2 2 2 2 4 4 type + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C5 C10 C10 C10 C10 C10 C10 C10 D4 D4 C5×D4 C5×D4 D4○SD16 C5×D4○SD16 kernel C5×D4○SD16 C5×C8○D4 C10×SD16 C5×C4○D8 C5×C8⋊C22 C5×C8.C22 C5×2+ 1+4 C5×2- 1+4 D4○SD16 C8○D4 C2×SD16 C4○D8 C8⋊C22 C8.C22 2+ 1+4 2- 1+4 C5×D4 C5×Q8 D4 Q8 C5 C1 # reps 1 1 3 3 3 3 1 1 4 4 12 12 12 12 4 4 3 1 12 4 2 8

Matrix representation of C5×D4○SD16 in GL4(𝔽41) generated by

 18 0 0 0 0 18 0 0 0 0 18 0 0 0 0 18
,
 40 0 40 40 0 0 0 1 2 1 1 1 0 40 0 0
,
 1 0 0 0 0 0 0 1 39 40 40 40 0 1 0 0
,
 11 26 0 26 30 0 15 15 0 0 26 15 0 0 26 26
,
 1 1 0 1 0 40 0 0 0 0 1 0 0 0 0 40
G:=sub<GL(4,GF(41))| [18,0,0,0,0,18,0,0,0,0,18,0,0,0,0,18],[40,0,2,0,0,0,1,40,40,0,1,0,40,1,1,0],[1,0,39,0,0,0,40,1,0,0,40,0,0,1,40,0],[11,30,0,0,26,0,0,0,0,15,26,26,26,15,15,26],[1,0,0,0,1,40,0,0,0,0,1,0,1,0,0,40] >;

C5×D4○SD16 in GAP, Magma, Sage, TeX

C_5\times D_4\circ {\rm SD}_{16}
% in TeX

G:=Group("C5xD4oSD16");
// GroupNames label

G:=SmallGroup(320,1579);
// by ID

G=gap.SmallGroup(320,1579);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-5,-2,-2,1120,1149,1193,10085,5052,124]);
// 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=d^3>;
// generators/relations

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