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## G = C10×C4○D4order 160 = 25·5

### Direct product of C10 and C4○D4

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

Series: Derived Chief Lower central Upper central

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

Generators and relations for C10×C4○D4
G = < a,b,c,d | a10=b4=d2=1, c2=b2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b2c >

Subgroups: 188 in 164 conjugacy classes, 140 normal (12 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C5, C2×C4, C2×C4, D4, Q8, C23, C10, C10, C10, C22×C4, C2×D4, C2×Q8, C4○D4, C20, C2×C10, C2×C10, C2×C10, C2×C4○D4, C2×C20, C2×C20, C5×D4, C5×Q8, C22×C10, C22×C20, D4×C10, Q8×C10, C5×C4○D4, C10×C4○D4
Quotients: C1, C2, C22, C5, C23, C10, C4○D4, C24, C2×C10, C2×C4○D4, C22×C10, C5×C4○D4, C23×C10, C10×C4○D4

Smallest permutation representation of C10×C4○D4
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 41 28 33)(2 42 29 34)(3 43 30 35)(4 44 21 36)(5 45 22 37)(6 46 23 38)(7 47 24 39)(8 48 25 40)(9 49 26 31)(10 50 27 32)(11 55 80 63)(12 56 71 64)(13 57 72 65)(14 58 73 66)(15 59 74 67)(16 60 75 68)(17 51 76 69)(18 52 77 70)(19 53 78 61)(20 54 79 62)
(1 58 28 66)(2 59 29 67)(3 60 30 68)(4 51 21 69)(5 52 22 70)(6 53 23 61)(7 54 24 62)(8 55 25 63)(9 56 26 64)(10 57 27 65)(11 48 80 40)(12 49 71 31)(13 50 72 32)(14 41 73 33)(15 42 74 34)(16 43 75 35)(17 44 76 36)(18 45 77 37)(19 46 78 38)(20 47 79 39)
(1 53)(2 54)(3 55)(4 56)(5 57)(6 58)(7 59)(8 60)(9 51)(10 52)(11 35)(12 36)(13 37)(14 38)(15 39)(16 40)(17 31)(18 32)(19 33)(20 34)(21 64)(22 65)(23 66)(24 67)(25 68)(26 69)(27 70)(28 61)(29 62)(30 63)(41 78)(42 79)(43 80)(44 71)(45 72)(46 73)(47 74)(48 75)(49 76)(50 77)

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

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

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

100 conjugacy classes

 class 1 2A 2B 2C 2D ··· 2I 4A 4B 4C 4D 4E ··· 4J 5A 5B 5C 5D 10A ··· 10L 10M ··· 10AJ 20A ··· 20P 20Q ··· 20AN order 1 2 2 2 2 ··· 2 4 4 4 4 4 ··· 4 5 5 5 5 10 ··· 10 10 ··· 10 20 ··· 20 20 ··· 20 size 1 1 1 1 2 ··· 2 1 1 1 1 2 ··· 2 1 1 1 1 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2

100 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 type + + + + + image C1 C2 C2 C2 C2 C5 C10 C10 C10 C10 C4○D4 C5×C4○D4 kernel C10×C4○D4 C22×C20 D4×C10 Q8×C10 C5×C4○D4 C2×C4○D4 C22×C4 C2×D4 C2×Q8 C4○D4 C10 C2 # reps 1 3 3 1 8 4 12 12 4 32 4 16

Matrix representation of C10×C4○D4 in GL3(𝔽41) generated by

 40 0 0 0 10 0 0 0 10
,
 40 0 0 0 9 0 0 0 9
,
 1 0 0 0 0 1 0 40 0
,
 1 0 0 0 0 40 0 40 0
G:=sub<GL(3,GF(41))| [40,0,0,0,10,0,0,0,10],[40,0,0,0,9,0,0,0,9],[1,0,0,0,0,40,0,1,0],[1,0,0,0,0,40,0,40,0] >;

C10×C4○D4 in GAP, Magma, Sage, TeX

C_{10}\times C_4\circ D_4
% in TeX

G:=Group("C10xC4oD4");
// GroupNames label

G:=SmallGroup(160,231);
// by ID

G=gap.SmallGroup(160,231);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-5,-2,985,374]);
// Polycyclic

G:=Group<a,b,c,d|a^10=b^4=d^2=1,c^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=b^2*c>;
// generators/relations

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