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## G = Dic5.20C24order 320 = 26·5

### 20th non-split extension by Dic5 of C24 acting via C24/C23=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — Dic5.20C24
 Chief series C1 — C5 — C10 — Dic5 — C5⋊C8 — D5⋊C8 — Q8.F5 — Dic5.20C24
 Lower central C5 — C10 — Dic5.20C24
 Upper central C1 — C2 — C2×Q8

Generators and relations for Dic5.20C24
G = < a,b,c,d,e,f | a10=f2=1, b2=d2=e2=a5, c2=b, bab-1=a-1, cac-1=a3, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, fcf=a5c, ede-1=a5d, df=fd, ef=fe >

Subgroups: 794 in 258 conjugacy classes, 136 normal (13 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, Q8, C23, D5, C10, C10, C2×C8, M4(2), C22×C4, C2×D4, C2×Q8, C4○D4, Dic5, C20, D10, D10, C2×C10, C2×M4(2), C8○D4, C2×C4○D4, C5⋊C8, C4×D5, D20, C2×Dic5, C2×C20, C5×Q8, C22×D5, Q8○M4(2), D5⋊C8, C4.F5, C22.F5, C2×C4×D5, C2×D20, Q82D5, Q8×C10, D5⋊M4(2), Q8.F5, C2×Q82D5, Dic5.20C24
Quotients: C1, C2, C4, C22, C2×C4, C23, C22×C4, C24, F5, C23×C4, C2×F5, Q8○M4(2), C22×F5, C23×F5, Dic5.20C24

Smallest permutation representation of Dic5.20C24
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 16 6 11)(2 15 7 20)(3 14 8 19)(4 13 9 18)(5 12 10 17)(21 34 26 39)(22 33 27 38)(23 32 28 37)(24 31 29 36)(25 40 30 35)(41 54 46 59)(42 53 47 58)(43 52 48 57)(44 51 49 56)(45 60 50 55)(61 74 66 79)(62 73 67 78)(63 72 68 77)(64 71 69 76)(65 80 70 75)
(1 58 16 42 6 53 11 47)(2 55 15 45 7 60 20 50)(3 52 14 48 8 57 19 43)(4 59 13 41 9 54 18 46)(5 56 12 44 10 51 17 49)(21 72 34 68 26 77 39 63)(22 79 33 61 27 74 38 66)(23 76 32 64 28 71 37 69)(24 73 31 67 29 78 36 62)(25 80 40 70 30 75 35 65)
(1 31 6 36)(2 32 7 37)(3 33 8 38)(4 34 9 39)(5 35 10 40)(11 24 16 29)(12 25 17 30)(13 26 18 21)(14 27 19 22)(15 28 20 23)(41 77 46 72)(42 78 47 73)(43 79 48 74)(44 80 49 75)(45 71 50 76)(51 70 56 65)(52 61 57 66)(53 62 58 67)(54 63 59 68)(55 64 60 69)
(1 24 6 29)(2 25 7 30)(3 26 8 21)(4 27 9 22)(5 28 10 23)(11 36 16 31)(12 37 17 32)(13 38 18 33)(14 39 19 34)(15 40 20 35)(41 66 46 61)(42 67 47 62)(43 68 48 63)(44 69 49 64)(45 70 50 65)(51 76 56 71)(52 77 57 72)(53 78 58 73)(54 79 59 74)(55 80 60 75)
(41 46)(42 47)(43 48)(44 49)(45 50)(51 56)(52 57)(53 58)(54 59)(55 60)(61 66)(62 67)(63 68)(64 69)(65 70)(71 76)(72 77)(73 78)(74 79)(75 80)

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

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

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

44 conjugacy classes

 class 1 2A 2B 2C ··· 2H 4A ··· 4F 4G 4H 4I 5 8A ··· 8P 10A 10B 10C 20A ··· 20F order 1 2 2 2 ··· 2 4 ··· 4 4 4 4 5 8 ··· 8 10 10 10 20 ··· 20 size 1 1 2 10 ··· 10 2 ··· 2 5 5 10 4 10 ··· 10 4 4 4 8 ··· 8

44 irreducible representations

 dim 1 1 1 1 1 1 1 4 4 4 4 8 type + + + + + + + + image C1 C2 C2 C2 C4 C4 C4 F5 C2×F5 C2×F5 Q8○M4(2) Dic5.20C24 kernel Dic5.20C24 D5⋊M4(2) Q8.F5 C2×Q8⋊2D5 C2×D20 Q8⋊2D5 Q8×C10 C2×Q8 C2×C4 Q8 C5 C1 # reps 1 6 8 1 6 8 2 1 3 4 2 2

Matrix representation of Dic5.20C24 in GL8(𝔽41)

 6 1 0 0 0 0 0 0 40 0 0 0 0 0 0 0 18 0 0 1 0 0 0 0 18 0 40 34 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40
,
 6 35 0 0 0 0 0 0 40 35 0 0 0 0 0 0 18 3 1 0 0 0 0 0 18 5 34 40 0 0 0 0 0 0 0 0 32 0 0 0 0 0 0 0 0 32 0 0 0 0 0 0 0 0 32 0 0 0 0 0 0 0 0 32
,
 11 5 39 10 0 0 0 0 11 0 10 39 0 0 0 0 35 18 0 36 0 0 0 0 31 18 30 30 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 5 36 1 9 0 0 0 0 32 0 0 0 0 0 0 0 22 19 0 5
,
 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 9 0 0 0 0 0 0 0 4 37 9 40 0 0 0 0 0 0 0 32
,
 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 40 0 0 0 0 0 0 0 5 36 1 9 0 0 0 0 8 0 18 40
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 40 0 0 0 0 0 8 33 0 40

G:=sub<GL(8,GF(41))| [6,40,18,18,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,1,34,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40],[6,40,18,18,0,0,0,0,35,35,3,5,0,0,0,0,0,0,1,34,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,32],[11,11,35,31,0,0,0,0,5,0,18,18,0,0,0,0,39,10,0,30,0,0,0,0,10,39,36,30,0,0,0,0,0,0,0,0,0,5,32,22,0,0,0,0,0,36,0,19,0,0,0,0,1,1,0,0,0,0,0,0,0,9,0,5],[40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,9,4,0,0,0,0,0,9,0,37,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,40,32],[40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,40,5,8,0,0,0,0,1,0,36,0,0,0,0,0,0,0,1,18,0,0,0,0,0,0,9,40],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,8,0,0,0,0,0,1,0,33,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40] >;

Dic5.20C24 in GAP, Magma, Sage, TeX

{\rm Dic}_5._{20}C_2^4
% in TeX

G:=Group("Dic5.20C2^4");
// GroupNames label

G:=SmallGroup(320,1598);
// by ID

G=gap.SmallGroup(320,1598);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,232,387,184,1123,102,6278,818]);
// Polycyclic

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

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