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## G = C10×Dic10order 400 = 24·52

### Direct product of C10 and Dic10

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C10×Dic10
 Chief series C1 — C5 — C10 — C5×C10 — C5×Dic5 — C10×Dic5 — C10×Dic10
 Lower central C5 — C10 — C10×Dic10
 Upper central C1 — C2×C10 — C2×C20

Generators and relations for C10×Dic10
G = < a,b,c | a10=b20=1, c2=b10, ab=ba, ac=ca, cbc-1=b-1 >

Subgroups: 196 in 92 conjugacy classes, 54 normal (18 characteristic)
C1, C2, C2, C4, C4, C22, C5, C5, C2×C4, C2×C4, Q8, C10, C10, C10, C2×Q8, Dic5, C20, C20, C2×C10, C2×C10, C52, Dic10, C2×Dic5, C2×C20, C2×C20, C5×Q8, C5×C10, C5×C10, C2×Dic10, Q8×C10, C5×Dic5, C5×C20, C102, C5×Dic10, C10×Dic5, C10×C20, C10×Dic10
Quotients: C1, C2, C22, C5, Q8, C23, D5, C10, C2×Q8, D10, C2×C10, Dic10, C5×Q8, C22×D5, C22×C10, C5×D5, C2×Dic10, Q8×C10, D5×C10, C5×Dic10, D5×C2×C10, C10×Dic10

Smallest permutation representation of C10×Dic10
On 80 points
Generators in S80
(1 50 9 58 17 46 5 54 13 42)(2 51 10 59 18 47 6 55 14 43)(3 52 11 60 19 48 7 56 15 44)(4 53 12 41 20 49 8 57 16 45)(21 69 33 61 25 73 37 65 29 77)(22 70 34 62 26 74 38 66 30 78)(23 71 35 63 27 75 39 67 31 79)(24 72 36 64 28 76 40 68 32 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 36 11 26)(2 35 12 25)(3 34 13 24)(4 33 14 23)(5 32 15 22)(6 31 16 21)(7 30 17 40)(8 29 18 39)(9 28 19 38)(10 27 20 37)(41 73 51 63)(42 72 52 62)(43 71 53 61)(44 70 54 80)(45 69 55 79)(46 68 56 78)(47 67 57 77)(48 66 58 76)(49 65 59 75)(50 64 60 74)

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

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

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

130 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 4F 5A 5B 5C 5D 5E ··· 5N 10A ··· 10L 10M ··· 10AP 20A ··· 20AV 20AW ··· 20BL order 1 2 2 2 4 4 4 4 4 4 5 5 5 5 5 ··· 5 10 ··· 10 10 ··· 10 20 ··· 20 20 ··· 20 size 1 1 1 1 2 2 10 10 10 10 1 1 1 1 2 ··· 2 1 ··· 1 2 ··· 2 2 ··· 2 10 ··· 10

130 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 type + + + + - + + + - image C1 C2 C2 C2 C5 C10 C10 C10 Q8 D5 D10 D10 Dic10 C5×Q8 C5×D5 D5×C10 D5×C10 C5×Dic10 kernel C10×Dic10 C5×Dic10 C10×Dic5 C10×C20 C2×Dic10 Dic10 C2×Dic5 C2×C20 C5×C10 C2×C20 C20 C2×C10 C10 C10 C2×C4 C4 C22 C2 # reps 1 4 2 1 4 16 8 4 2 2 4 2 8 8 8 16 8 32

Matrix representation of C10×Dic10 in GL4(𝔽41) generated by

 25 0 0 0 0 25 0 0 0 0 18 0 0 0 0 18
,
 23 0 0 0 0 25 0 0 0 0 21 0 0 0 0 2
,
 0 1 0 0 1 0 0 0 0 0 0 1 0 0 40 0
G:=sub<GL(4,GF(41))| [25,0,0,0,0,25,0,0,0,0,18,0,0,0,0,18],[23,0,0,0,0,25,0,0,0,0,21,0,0,0,0,2],[0,1,0,0,1,0,0,0,0,0,0,40,0,0,1,0] >;

C10×Dic10 in GAP, Magma, Sage, TeX

C_{10}\times {\rm Dic}_{10}
% in TeX

G:=Group("C10xDic10");
// GroupNames label

G:=SmallGroup(400,181);
// by ID

G=gap.SmallGroup(400,181);
# by ID

G:=PCGroup([6,-2,-2,-2,-5,-2,-5,240,794,194,11525]);
// Polycyclic

G:=Group<a,b,c|a^10=b^20=1,c^2=b^10,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

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