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## G = C5×D4.9D4order 320 = 26·5

### Direct product of C5 and D4.9D4

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C4 — C5×D4.9D4
 Chief series C1 — C2 — C22 — C2×C4 — C2×C20 — Q8×C10 — C5×C8.C22 — C5×D4.9D4
 Lower central C1 — C2 — C2×C4 — C5×D4.9D4
 Upper central C1 — C10 — C2×C20 — C5×D4.9D4

Generators and relations for C5×D4.9D4
G = < a,b,c,d,e | a5=b4=c2=d4=1, e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe-1=b-1, bd=db, dcd-1=b-1c, ece-1=bc, ede-1=b2d-1 >

Subgroups: 306 in 152 conjugacy classes, 54 normal (22 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, C10, C10, C42, C22⋊C4, M4(2), SD16, Q16, C2×D4, C2×D4, C2×Q8, C4○D4, C4○D4, C20, C20, C2×C10, C2×C10, C4.D4, C4≀C2, C4.4D4, C8.C22, 2+ 1+4, C40, C2×C20, C2×C20, C5×D4, C5×D4, C5×Q8, C5×Q8, C22×C10, C22×C10, D4.9D4, C4×C20, C5×C22⋊C4, C5×M4(2), C5×SD16, C5×Q16, D4×C10, D4×C10, Q8×C10, C5×C4○D4, C5×C4○D4, C5×C4.D4, C5×C4≀C2, C5×C4.4D4, C5×C8.C22, C5×2+ 1+4, C5×D4.9D4
Quotients: C1, C2, C22, C5, D4, C23, C10, C2×D4, C2×C10, C22≀C2, C5×D4, C22×C10, D4.9D4, D4×C10, C5×C22≀C2, C5×D4.9D4

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

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

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

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

80 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 4A 4B 4C 4D 4E 4F 4G 5A 5B 5C 5D 8A 8B 10A 10B 10C 10D 10E 10F 10G 10H 10I ··· 10X 20A ··· 20H 20I ··· 20X 20Y 20Z 20AA 20AB 40A ··· 40H order 1 2 2 2 2 2 2 4 4 4 4 4 4 4 5 5 5 5 8 8 10 10 10 10 10 10 10 10 10 ··· 10 20 ··· 20 20 ··· 20 20 20 20 20 40 ··· 40 size 1 1 2 4 4 4 4 2 2 4 4 4 4 8 1 1 1 1 8 8 1 1 1 1 2 2 2 2 4 ··· 4 2 ··· 2 4 ··· 4 8 8 8 8 8 ··· 8

80 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 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 C5×D4 C5×D4 C5×D4 D4.9D4 C5×D4.9D4 kernel C5×D4.9D4 C5×C4.D4 C5×C4≀C2 C5×C4.4D4 C5×C8.C22 C5×2+ 1+4 D4.9D4 C4.D4 C4≀C2 C4.4D4 C8.C22 2+ 1+4 C5×D4 C5×Q8 C22×C10 D4 Q8 C23 C5 C1 # reps 1 1 2 1 2 1 4 4 8 4 8 4 2 2 2 8 8 8 2 8

Matrix representation of C5×D4.9D4 in GL4(𝔽41) generated by

 18 0 0 0 0 18 0 0 0 0 18 0 0 0 0 18
,
 0 40 40 1 0 40 0 1 1 40 0 0 0 39 0 1
,
 40 0 0 0 0 1 0 40 0 0 1 40 0 0 0 40
,
 0 32 0 0 0 32 9 0 9 32 0 0 9 32 9 32
,
 0 32 0 9 0 32 32 9 9 32 0 0 9 32 32 9
G:=sub<GL(4,GF(41))| [18,0,0,0,0,18,0,0,0,0,18,0,0,0,0,18],[0,0,1,0,40,40,40,39,40,0,0,0,1,1,0,1],[40,0,0,0,0,1,0,0,0,0,1,0,0,40,40,40],[0,0,9,9,32,32,32,32,0,9,0,9,0,0,0,32],[0,0,9,9,32,32,32,32,0,32,0,32,9,9,0,9] >;

C5×D4.9D4 in GAP, Magma, Sage, TeX

C_5\times D_4._9D_4
% in TeX

G:=Group("C5xD4.9D4");
// GroupNames label

G:=SmallGroup(320,956);
// by ID

G=gap.SmallGroup(320,956);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,1120,589,1766,7004,3511,1768,172,5052]);
// Polycyclic

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

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