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## G = Dic5.21C24order 320 = 26·5

### 21st non-split extension by Dic5 of C24 acting via C24/C23=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — Dic5.21C24
 Chief series C1 — C5 — C10 — Dic5 — C5⋊C8 — C2×C5⋊C8 — C2×D5⋊C8 — Dic5.21C24
 Lower central C5 — C10 — Dic5.21C24
 Upper central C1 — C4 — C4○D4

Generators and relations for Dic5.21C24
G = < a,b,c,d,e,f | a10=d2=f2=1, b2=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, cf=fc, de=ed, fdf=a5d, ef=fe >

Subgroups: 778 in 266 conjugacy classes, 138 normal (18 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, D5, D5, C10, C10, C2×C8, M4(2), C22×C4, C2×D4, C2×Q8, C4○D4, C4○D4, Dic5, Dic5, C20, C20, D10, D10, D10, C2×C10, C22×C8, C2×M4(2), C8○D4, C2×C4○D4, C5⋊C8, Dic10, C4×D5, C4×D5, D20, C2×Dic5, C5⋊D4, C2×C20, C5×D4, C5×Q8, C22×D5, C2×C8○D4, D5⋊C8, D5⋊C8, C4.F5, C2×C5⋊C8, C22.F5, C2×C4×D5, C4○D20, D4×D5, D42D5, Q8×D5, Q82D5, C5×C4○D4, C2×D5⋊C8, D5⋊M4(2), D4.F5, Q8.F5, D5×C4○D4, Dic5.21C24
Quotients: C1, C2, C4, C22, C2×C4, C23, C22×C4, C24, F5, C8○D4, C23×C4, C2×F5, C2×C8○D4, C22×F5, C23×F5, Dic5.21C24

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

50 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 5 8A ··· 8H 8I ··· 8T 10A 10B 10C 10D 20A 20B 20C 20D 20E order 1 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 4 4 5 8 ··· 8 8 ··· 8 10 10 10 10 20 20 20 20 20 size 1 1 2 2 2 5 5 10 10 10 1 1 2 2 2 5 5 10 10 10 4 5 ··· 5 10 ··· 10 4 8 8 8 4 4 8 8 8

50 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 4 4 4 4 8 type + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C4 C4 C4 C4 C8○D4 F5 C2×F5 C2×F5 C2×F5 Dic5.21C24 kernel Dic5.21C24 C2×D5⋊C8 D5⋊M4(2) D4.F5 Q8.F5 D5×C4○D4 C4○D20 D4×D5 Q8×D5 C5×C4○D4 D5 C4○D4 C2×C4 D4 Q8 C1 # reps 1 3 3 6 2 1 6 6 2 2 8 1 3 3 1 2

Matrix representation of Dic5.21C24 in GL6(𝔽41)

 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 1 0 0 0 0 40 0 1 0 0 0 40 0 0 1 0 0 40 0 0 0
,
 32 0 0 0 0 0 0 32 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0
,
 27 0 0 0 0 0 0 27 0 0 0 0 0 0 9 32 32 0 0 0 0 32 0 9 0 0 9 0 32 0 0 0 0 32 32 9
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 9 0 0 0 0 0 0 9 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 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 1 0 0 0 0 0 0 1

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,40,40,40,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[32,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1,0,0,0],[27,0,0,0,0,0,0,27,0,0,0,0,0,0,9,0,9,0,0,0,32,32,0,32,0,0,32,0,32,32,0,0,0,9,0,9],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[9,0,0,0,0,0,0,9,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

Dic5.21C24 in GAP, Magma, Sage, TeX

{\rm Dic}_5._{21}C_2^4
% in TeX

G:=Group("Dic5.21C2^4");
// GroupNames label

G:=SmallGroup(320,1601);
// by ID

G=gap.SmallGroup(320,1601);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,184,570,102,6278,818]);
// Polycyclic

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

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