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## G = C5×D4.8D4order 320 = 26·5

### Direct product of C5 and D4.8D4

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C4 — C5×D4.8D4
 Chief series C1 — C2 — C22 — C2×C4 — C2×C20 — D4×C10 — C5×C8⋊C22 — C5×D4.8D4
 Lower central C1 — C2 — C2×C4 — C5×D4.8D4
 Upper central C1 — C10 — C2×C20 — C5×D4.8D4

Generators and relations for C5×D4.8D4
G = < a,b,c,d,e | a5=b4=c2=e2=1, d4=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=dbd-1=ebe=b-1, dcd-1=bc, ece=b-1c, ede=b2d3 >

Subgroups: 274 in 146 conjugacy classes, 54 normal (22 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C10, C10, C42, C22⋊C4, M4(2), D8, SD16, C2×D4, C2×Q8, C2×Q8, C4○D4, C4○D4, C20, C20, C2×C10, C2×C10, C4.10D4, C4≀C2, C4.4D4, C8⋊C22, 2- 1+4, C40, C2×C20, C2×C20, C2×C20, C5×D4, C5×D4, C5×Q8, C5×Q8, C22×C10, D4.8D4, C4×C20, C5×C22⋊C4, C5×M4(2), C5×D8, C5×SD16, D4×C10, Q8×C10, Q8×C10, C5×C4○D4, C5×C4○D4, C5×C4.10D4, C5×C4≀C2, C5×C4.4D4, C5×C8⋊C22, C5×2- 1+4, C5×D4.8D4
Quotients: C1, C2, C22, C5, D4, C23, C10, C2×D4, C2×C10, C22≀C2, C5×D4, C22×C10, D4.8D4, D4×C10, C5×C22≀C2, C5×D4.8D4

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

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

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

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

80 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C ··· 4H 5A 5B 5C 5D 8A 8B 10A 10B 10C 10D 10E 10F 10G 10H 10I ··· 10P 10Q 10R 10S 10T 20A ··· 20H 20I ··· 20AF 40A ··· 40H order 1 2 2 2 2 2 4 4 4 ··· 4 5 5 5 5 8 8 10 10 10 10 10 10 10 10 10 ··· 10 10 10 10 10 20 ··· 20 20 ··· 20 40 ··· 40 size 1 1 2 4 4 8 2 2 4 ··· 4 1 1 1 1 8 8 1 1 1 1 2 2 2 2 4 ··· 4 8 8 8 8 2 ··· 2 4 ··· 4 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 D4.8D4 C5×D4.8D4 kernel C5×D4.8D4 C5×C4.10D4 C5×C4≀C2 C5×C4.4D4 C5×C8⋊C22 C5×2- 1+4 D4.8D4 C4.10D4 C4≀C2 C4.4D4 C8⋊C22 2- 1+4 C2×C20 C5×D4 C5×Q8 C2×C4 D4 Q8 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×D4.8D4 in GL4(𝔽41) generated by

 18 0 0 0 0 18 0 0 0 0 18 0 0 0 0 18
,
 0 1 0 0 40 0 0 0 9 32 40 39 0 9 1 1
,
 40 1 32 23 0 0 9 0 0 32 0 0 0 9 1 1
,
 0 0 1 0 32 9 1 2 0 1 0 0 40 0 0 32
,
 1 0 0 0 0 40 0 0 9 32 40 39 0 9 0 1
G:=sub<GL(4,GF(41))| [18,0,0,0,0,18,0,0,0,0,18,0,0,0,0,18],[0,40,9,0,1,0,32,9,0,0,40,1,0,0,39,1],[40,0,0,0,1,0,32,9,32,9,0,1,23,0,0,1],[0,32,0,40,0,9,1,0,1,1,0,0,0,2,0,32],[1,0,9,0,0,40,32,9,0,0,40,0,0,0,39,1] >;

C5×D4.8D4 in GAP, Magma, Sage, TeX

C_5\times D_4._8D_4
% in TeX

G:=Group("C5xD4.8D4");
// GroupNames label

G:=SmallGroup(320,955);
// by ID

G=gap.SmallGroup(320,955);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,589,1766,856,7004,3511,1768,172,5052]);
// 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=d*b*d^-1=e*b*e=b^-1,d*c*d^-1=b*c,e*c*e=b^-1*c,e*d*e=b^2*d^3>;
// generators/relations

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