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## G = C8.Dic10order 320 = 26·5

### 1st non-split extension by C8 of Dic10 acting via Dic10/C10=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C40 — C8.Dic10
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C2×C40 — C20.4C8 — C8.Dic10
 Lower central C5 — C10 — C20 — C40 — C8.Dic10
 Upper central C1 — C2 — C2×C4 — C2×C8 — C4.Q8

Generators and relations for C8.Dic10
G = < a,b,c | a8=b20=1, c2=ab10, bab-1=a3, cac-1=a5, cbc-1=a-1b-1 >

Smallest permutation representation of C8.Dic10
On 80 points
Generators in S80
(1 29 16 24 9 39 11 34)(2 25 12 30 10 35 17 40)(3 21 18 26 6 31 13 36)(4 27 14 22 7 37 19 32)(5 23 20 28 8 33 15 38)(41 56 77 62 51 46 67 72)(42 63 68 57 52 73 78 47)(43 58 79 64 53 48 69 74)(44 65 70 59 54 75 80 49)(45 60 61 66 55 50 71 76)
(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 71 29 66 16 45 24 50 9 61 39 76 11 55 34 60)(2 75 25 70 12 49 30 54 10 65 35 80 17 59 40 44)(3 79 21 74 18 53 26 58 6 69 31 64 13 43 36 48)(4 63 27 78 14 57 22 42 7 73 37 68 19 47 32 52)(5 67 23 62 20 41 28 46 8 77 33 72 15 51 38 56)

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

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

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

44 conjugacy classes

 class 1 2A 2B 4A 4B 4C 4D 5A 5B 8A 8B 8C 8D 8E 10A ··· 10F 16A 16B 16C 16D 20A 20B 20C 20D 20E ··· 20L 40A ··· 40H order 1 2 2 4 4 4 4 5 5 8 8 8 8 8 10 ··· 10 16 16 16 16 20 20 20 20 20 ··· 20 40 ··· 40 size 1 1 2 2 2 8 8 2 2 2 2 4 40 40 2 ··· 2 20 20 20 20 4 4 4 4 8 ··· 8 4 ··· 4

44 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + - + + + - + - image C1 C2 C2 C2 C4 Q8 D4 D5 SD16 SD16 D10 Dic10 C4×D5 C5⋊D4 C8.Q8 Q8⋊D5 D4.D5 C8.Dic10 kernel C8.Dic10 C20.4C8 C40.6C4 C5×C4.Q8 C5⋊2C16 C40 C2×C20 C4.Q8 C20 C2×C10 C2×C8 C8 C8 C2×C4 C5 C4 C22 C1 # reps 1 1 1 1 4 1 1 2 2 2 2 4 4 4 2 2 2 8

Matrix representation of C8.Dic10 in GL4(𝔽241) generated by

 0 218 0 0 220 38 0 0 63 221 222 19 232 22 222 222
,
 91 0 0 0 164 150 0 0 174 207 175 66 174 109 66 66
,
 1 0 90 0 16 36 240 1 83 222 240 0 193 169 41 205
G:=sub<GL(4,GF(241))| [0,220,63,232,218,38,221,22,0,0,222,222,0,0,19,222],[91,164,174,174,0,150,207,109,0,0,175,66,0,0,66,66],[1,16,83,193,0,36,222,169,90,240,240,41,0,1,0,205] >;

C8.Dic10 in GAP, Magma, Sage, TeX

C_8.{\rm Dic}_{10}
% in TeX

G:=Group("C8.Dic10");
// GroupNames label

G:=SmallGroup(320,45);
// by ID

G=gap.SmallGroup(320,45);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,365,36,758,346,80,851,102,12550]);
// Polycyclic

G:=Group<a,b,c|a^8=b^20=1,c^2=a*b^10,b*a*b^-1=a^3,c*a*c^-1=a^5,c*b*c^-1=a^-1*b^-1>;
// generators/relations

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