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## G = C5×D5⋊C8order 400 = 24·52

### Direct product of C5 and D5⋊C8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5 — C5×D5⋊C8
 Chief series C1 — C5 — C10 — Dic5 — C5×Dic5 — C5×C5⋊C8 — C5×D5⋊C8
 Lower central C5 — C5×D5⋊C8
 Upper central C1 — C20

Generators and relations for C5×D5⋊C8
G = < a,b,c,d | a5=b5=c2=d8=1, ab=ba, ac=ca, ad=da, cbc=b-1, dbd-1=b3, dcd-1=b2c >

Smallest permutation representation of C5×D5⋊C8
On 80 points
Generators in S80
(1 21 75 63 33)(2 22 76 64 34)(3 23 77 57 35)(4 24 78 58 36)(5 17 79 59 37)(6 18 80 60 38)(7 19 73 61 39)(8 20 74 62 40)(9 55 25 41 71)(10 56 26 42 72)(11 49 27 43 65)(12 50 28 44 66)(13 51 29 45 67)(14 52 30 46 68)(15 53 31 47 69)(16 54 32 48 70)
(1 21 75 63 33)(2 64 22 34 76)(3 35 57 77 23)(4 78 36 24 58)(5 17 79 59 37)(6 60 18 38 80)(7 39 61 73 19)(8 74 40 20 62)(9 55 25 41 71)(10 42 56 72 26)(11 65 43 27 49)(12 28 66 50 44)(13 51 29 45 67)(14 46 52 68 30)(15 69 47 31 53)(16 32 70 54 48)
(1 47)(2 16)(3 55)(4 72)(5 43)(6 12)(7 51)(8 68)(9 35)(10 24)(11 79)(13 39)(14 20)(15 75)(17 65)(18 50)(19 29)(21 69)(22 54)(23 25)(26 58)(27 37)(28 80)(30 62)(31 33)(32 76)(34 70)(36 42)(38 66)(40 46)(41 77)(44 60)(45 73)(48 64)(49 59)(52 74)(53 63)(56 78)(57 71)(61 67)
(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)

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

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

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

100 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 5A 5B 5C 5D 5E ··· 5I 8A ··· 8H 10A 10B 10C 10D 10E ··· 10I 10J ··· 10Q 20A ··· 20H 20I ··· 20R 20S ··· 20Z 40A ··· 40AF order 1 2 2 2 4 4 4 4 5 5 5 5 5 ··· 5 8 ··· 8 10 10 10 10 10 ··· 10 10 ··· 10 20 ··· 20 20 ··· 20 20 ··· 20 40 ··· 40 size 1 1 5 5 1 1 5 5 1 1 1 1 4 ··· 4 5 ··· 5 1 1 1 1 4 ··· 4 5 ··· 5 1 ··· 1 4 ··· 4 5 ··· 5 5 ··· 5

100 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 4 4 4 4 4 4 type + + + + + image C1 C2 C2 C4 C4 C5 C8 C10 C10 C20 C20 C40 F5 C2×F5 D5⋊C8 C5×F5 C10×F5 C5×D5⋊C8 kernel C5×D5⋊C8 C5×C5⋊C8 D5×C20 C5×C20 D5×C10 D5⋊C8 C5×D5 C5⋊C8 C4×D5 C20 D10 D5 C20 C10 C5 C4 C2 C1 # reps 1 2 1 2 2 4 8 8 4 8 8 32 1 1 2 4 4 8

Matrix representation of C5×D5⋊C8 in GL4(𝔽41) generated by

 16 0 0 0 0 16 0 0 0 0 16 0 0 0 0 16
,
 16 0 0 0 30 18 0 0 11 0 10 0 2 0 0 37
,
 23 7 0 0 30 18 0 0 28 6 0 5 32 13 33 0
,
 3 0 17 0 0 0 32 1 0 1 38 0 0 0 14 0
G:=sub<GL(4,GF(41))| [16,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16],[16,30,11,2,0,18,0,0,0,0,10,0,0,0,0,37],[23,30,28,32,7,18,6,13,0,0,0,33,0,0,5,0],[3,0,0,0,0,0,1,0,17,32,38,14,0,1,0,0] >;

C5×D5⋊C8 in GAP, Magma, Sage, TeX

C_5\times D_5\rtimes C_8
% in TeX

G:=Group("C5xD5:C8");
// GroupNames label

G:=SmallGroup(400,135);
// by ID

G=gap.SmallGroup(400,135);
# by ID

G:=PCGroup([6,-2,-2,-5,-2,-2,-5,120,247,69,5765,599]);
// Polycyclic

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

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