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## G = C10.Dic10order 400 = 24·52

### 2nd non-split extension by C10 of Dic10 acting via Dic10/Dic5=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5×C10 — C10.Dic10
 Chief series C1 — C5 — C52 — C5×C10 — C102 — C10×Dic5 — C10.Dic10
 Lower central C52 — C5×C10 — C10.Dic10
 Upper central C1 — C22

Generators and relations for C10.Dic10
G = < a,b,c | a10=b20=1, c2=b10, bab-1=a-1, ac=ca, cbc-1=a5b-1 >

Subgroups: 316 in 68 conjugacy classes, 28 normal (12 characteristic)
C1, C2, C4, C22, C5, C5, C2×C4, C10, C10, C4⋊C4, Dic5, C20, C2×C10, C2×C10, C52, C2×Dic5, C2×Dic5, C2×C20, C5×C10, C10.D4, C5×Dic5, C526C4, C102, C10×Dic5, C2×C526C4, C10.Dic10
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, D5, C4⋊C4, D10, Dic10, C4×D5, C5⋊D4, C10.D4, D52, Dic52D5, C522D4, C522Q8, C10.Dic10

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

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

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

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

58 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 4F 5A 5B 5C 5D 5E 5F 5G 5H 10A ··· 10L 10M ··· 10X 20A ··· 20P order 1 2 2 2 4 4 4 4 4 4 5 5 5 5 5 5 5 5 10 ··· 10 10 ··· 10 20 ··· 20 size 1 1 1 1 10 10 10 10 50 50 2 2 2 2 4 4 4 4 2 ··· 2 4 ··· 4 10 ··· 10

58 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 2 4 4 4 4 type + + + + - + + - + + - - image C1 C2 C2 C4 D4 Q8 D5 D10 Dic10 C4×D5 C5⋊D4 D52 Dic5⋊2D5 C52⋊2D4 C52⋊2Q8 kernel C10.Dic10 C10×Dic5 C2×C52⋊6C4 C52⋊6C4 C5×C10 C5×C10 C2×Dic5 C2×C10 C10 C10 C10 C22 C2 C2 C2 # reps 1 2 1 4 1 1 4 4 8 8 8 4 4 4 4

Matrix representation of C10.Dic10 in GL6(𝔽41)

 40 0 0 0 0 0 0 40 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 40 34 0 0 0 0 7 7
,
 35 23 0 0 0 0 18 20 0 0 0 0 0 0 28 2 0 0 0 0 39 16 0 0 0 0 0 0 1 0 0 0 0 0 34 40
,
 16 39 0 0 0 0 25 25 0 0 0 0 0 0 20 18 0 0 0 0 21 21 0 0 0 0 0 0 40 0 0 0 0 0 0 40

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,7,0,0,0,0,34,7],[35,18,0,0,0,0,23,20,0,0,0,0,0,0,28,39,0,0,0,0,2,16,0,0,0,0,0,0,1,34,0,0,0,0,0,40],[16,25,0,0,0,0,39,25,0,0,0,0,0,0,20,21,0,0,0,0,18,21,0,0,0,0,0,0,40,0,0,0,0,0,0,40] >;

C10.Dic10 in GAP, Magma, Sage, TeX

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

G:=Group("C10.Dic10");
// GroupNames label

G:=SmallGroup(400,75);
// by ID

G=gap.SmallGroup(400,75);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-5,-5,24,121,31,970,11525]);
// Polycyclic

G:=Group<a,b,c|a^10=b^20=1,c^2=b^10,b*a*b^-1=a^-1,a*c=c*a,c*b*c^-1=a^5*b^-1>;
// generators/relations

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