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## G = C5×C22⋊SD16order 320 = 26·5

### Direct product of C5 and C22⋊SD16

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C4 — C5×C22⋊SD16
 Chief series C1 — C2 — C22 — C2×C4 — C2×C20 — Q8×C10 — C10×SD16 — C5×C22⋊SD16
 Lower central C1 — C2 — C2×C4 — C5×C22⋊SD16
 Upper central C1 — C2×C10 — C22×C20 — C5×C22⋊SD16

Generators and relations for C5×C22⋊SD16
G = < a,b,c,d,e | a5=b2=c2=d8=e2=1, ab=ba, ac=ca, ad=da, ae=ea, dbd-1=bc=cb, be=eb, cd=dc, ce=ec, ede=d3 >

Subgroups: 402 in 188 conjugacy classes, 62 normal (30 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C5, C8, C2×C4, C2×C4, D4, D4, Q8, C23, C23, C10, C10, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, SD16, C22×C4, C2×D4, C2×D4, C2×Q8, C24, C20, C20, C2×C10, C2×C10, C2×C10, C22⋊C8, D4⋊C4, C22⋊Q8, C2×SD16, C22×D4, C40, C2×C20, C2×C20, C5×D4, C5×D4, C5×Q8, C22×C10, C22×C10, C22⋊SD16, C5×C22⋊C4, C5×C4⋊C4, C5×C4⋊C4, C2×C40, C5×SD16, C22×C20, D4×C10, D4×C10, Q8×C10, C23×C10, C5×C22⋊C8, C5×D4⋊C4, C5×C22⋊Q8, C10×SD16, D4×C2×C10, C5×C22⋊SD16
Quotients: C1, C2, C22, C5, D4, C23, C10, SD16, C2×D4, C2×C10, C22≀C2, C2×SD16, C8⋊C22, C5×D4, C22×C10, C22⋊SD16, C5×SD16, D4×C10, C5×C22≀C2, C10×SD16, C5×C8⋊C22, C5×C22⋊SD16

Smallest permutation representation of C5×C22⋊SD16
On 80 points
Generators in S80
(1 42 75 40 67)(2 43 76 33 68)(3 44 77 34 69)(4 45 78 35 70)(5 46 79 36 71)(6 47 80 37 72)(7 48 73 38 65)(8 41 74 39 66)(9 49 60 30 20)(10 50 61 31 21)(11 51 62 32 22)(12 52 63 25 23)(13 53 64 26 24)(14 54 57 27 17)(15 55 58 28 18)(16 56 59 29 19)
(1 55)(2 6)(3 49)(4 8)(5 51)(7 53)(9 69)(10 14)(11 71)(12 16)(13 65)(15 67)(17 21)(18 40)(19 23)(20 34)(22 36)(24 38)(25 29)(26 73)(27 31)(28 75)(30 77)(32 79)(33 37)(35 39)(41 45)(42 58)(43 47)(44 60)(46 62)(48 64)(50 54)(52 56)(57 61)(59 63)(66 70)(68 72)(74 78)(76 80)
(1 51)(2 52)(3 53)(4 54)(5 55)(6 56)(7 49)(8 50)(9 65)(10 66)(11 67)(12 68)(13 69)(14 70)(15 71)(16 72)(17 35)(18 36)(19 37)(20 38)(21 39)(22 40)(23 33)(24 34)(25 76)(26 77)(27 78)(28 79)(29 80)(30 73)(31 74)(32 75)(41 61)(42 62)(43 63)(44 64)(45 57)(46 58)(47 59)(48 60)
(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 51)(2 54)(3 49)(4 52)(5 55)(6 50)(7 53)(8 56)(9 69)(10 72)(11 67)(12 70)(13 65)(14 68)(15 71)(16 66)(17 33)(18 36)(19 39)(20 34)(21 37)(22 40)(23 35)(24 38)(25 78)(26 73)(27 76)(28 79)(29 74)(30 77)(31 80)(32 75)(41 59)(42 62)(43 57)(44 60)(45 63)(46 58)(47 61)(48 64)

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

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

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

95 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 4A 4B 4C 4D 4E 5A 5B 5C 5D 8A 8B 8C 8D 10A ··· 10L 10M ··· 10T 10U ··· 10AJ 20A ··· 20H 20I 20J 20K 20L 20M ··· 20T 40A ··· 40P order 1 2 2 2 2 2 2 2 2 2 4 4 4 4 4 5 5 5 5 8 8 8 8 10 ··· 10 10 ··· 10 10 ··· 10 20 ··· 20 20 20 20 20 20 ··· 20 40 ··· 40 size 1 1 1 1 2 2 4 4 4 4 2 2 4 8 8 1 1 1 1 4 4 4 4 1 ··· 1 2 ··· 2 4 ··· 4 2 ··· 2 4 4 4 4 8 ··· 8 4 ··· 4

95 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C5 C10 C10 C10 C10 C10 D4 D4 D4 SD16 C5×D4 C5×D4 C5×D4 C5×SD16 C8⋊C22 C5×C8⋊C22 kernel C5×C22⋊SD16 C5×C22⋊C8 C5×D4⋊C4 C5×C22⋊Q8 C10×SD16 D4×C2×C10 C22⋊SD16 C22⋊C8 D4⋊C4 C22⋊Q8 C2×SD16 C22×D4 C2×C20 C5×D4 C22×C10 C2×C10 C2×C4 D4 C23 C22 C10 C2 # reps 1 1 2 1 2 1 4 4 8 4 8 4 1 4 1 4 4 16 4 16 1 4

Matrix representation of C5×C22⋊SD16 in GL4(𝔽41) generated by

 10 0 0 0 0 10 0 0 0 0 1 0 0 0 0 1
,
 40 0 0 0 0 40 0 0 0 0 40 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 40 0 0 0 0 40
,
 15 26 0 0 15 15 0 0 0 0 0 39 0 0 20 0
,
 1 0 0 0 0 40 0 0 0 0 40 0 0 0 0 40
G:=sub<GL(4,GF(41))| [10,0,0,0,0,10,0,0,0,0,1,0,0,0,0,1],[40,0,0,0,0,40,0,0,0,0,40,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,40,0,0,0,0,40],[15,15,0,0,26,15,0,0,0,0,0,20,0,0,39,0],[1,0,0,0,0,40,0,0,0,0,40,0,0,0,0,40] >;

C5×C22⋊SD16 in GAP, Magma, Sage, TeX

C_5\times C_2^2\rtimes {\rm SD}_{16}
% in TeX

G:=Group("C5xC2^2:SD16");
// GroupNames label

G:=SmallGroup(320,951);
// by ID

G=gap.SmallGroup(320,951);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,1120,589,1766,7004,3511,172]);
// Polycyclic

G:=Group<a,b,c,d,e|a^5=b^2=c^2=d^8=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,d*b*d^-1=b*c=c*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^3>;
// generators/relations

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