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## G = C42⋊6Dic5order 320 = 26·5

### 3rd semidirect product of C42 and Dic5 acting via Dic5/C10=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — C42⋊6Dic5
 Chief series C1 — C5 — C10 — C2×C10 — C22×C10 — C22×C20 — C23.21D10 — C42⋊6Dic5
 Lower central C5 — C10 — C20 — C42⋊6Dic5
 Upper central C1 — C2×C4 — C22×C4 — C2×C42

Generators and relations for C426Dic5
G = < a,b,c,d | a4=b4=c10=1, d2=c5, dad-1=ab=ba, ac=ca, bc=cb, dbd-1=b-1, dcd-1=c-1 >

Subgroups: 294 in 110 conjugacy classes, 51 normal (39 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, C23, C10, C10, C10, C42, C42, C22⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, C22×C4, Dic5, C20, C20, C2×C10, C2×C10, C2×C42, C42⋊C2, C2×M4(2), C52C8, C2×Dic5, C2×C20, C2×C20, C22×C10, C426C4, C2×C52C8, C4.Dic5, C4.Dic5, C4×Dic5, C4⋊Dic5, C23.D5, C4×C20, C4×C20, C22×C20, C22×C20, C2×C4.Dic5, C23.21D10, C2×C4×C20, C426Dic5
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, D5, C42, C22⋊C4, C4⋊C4, Dic5, D10, C2.C42, C4≀C2, Dic10, C4×D5, D20, C2×Dic5, C5⋊D4, C426C4, C4×Dic5, C10.D4, C4⋊Dic5, D10⋊C4, C23.D5, D204C4, C10.10C42, C426Dic5

Smallest permutation representation of C426Dic5
On 80 points
Generators in S80
(1 34)(2 35)(3 36)(4 37)(5 38)(6 39)(7 40)(8 31)(9 32)(10 33)(11 42)(12 43)(13 44)(14 45)(15 46)(16 47)(17 48)(18 49)(19 50)(20 41)(21 55 69 71)(22 56 70 72)(23 57 61 73)(24 58 62 74)(25 59 63 75)(26 60 64 76)(27 51 65 77)(28 52 66 78)(29 53 67 79)(30 54 68 80)
(1 45 39 19)(2 46 40 20)(3 47 31 11)(4 48 32 12)(5 49 33 13)(6 50 34 14)(7 41 35 15)(8 42 36 16)(9 43 37 17)(10 44 38 18)(21 60 69 76)(22 51 70 77)(23 52 61 78)(24 53 62 79)(25 54 63 80)(26 55 64 71)(27 56 65 72)(28 57 66 73)(29 58 67 74)(30 59 68 75)
(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 60 6 55)(2 59 7 54)(3 58 8 53)(4 57 9 52)(5 56 10 51)(11 67 16 62)(12 66 17 61)(13 65 18 70)(14 64 19 69)(15 63 20 68)(21 50 26 45)(22 49 27 44)(23 48 28 43)(24 47 29 42)(25 46 30 41)(31 74 36 79)(32 73 37 78)(33 72 38 77)(34 71 39 76)(35 80 40 75)

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

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

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

92 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E ··· 4N 4O 4P 4Q 4R 5A 5B 8A 8B 8C 8D 10A ··· 10N 20A ··· 20AV order 1 2 2 2 2 2 4 4 4 4 4 ··· 4 4 4 4 4 5 5 8 8 8 8 10 ··· 10 20 ··· 20 size 1 1 1 1 2 2 1 1 1 1 2 ··· 2 20 20 20 20 2 2 20 20 20 20 2 ··· 2 2 ··· 2

92 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + - + + - + - + image C1 C2 C2 C2 C4 C4 C4 D4 Q8 D4 D5 Dic5 D10 C4≀C2 Dic10 C4×D5 D20 C5⋊D4 C5⋊D4 D20⋊4C4 kernel C42⋊6Dic5 C2×C4.Dic5 C23.21D10 C2×C4×C20 C4.Dic5 C4⋊Dic5 C4×C20 C2×C20 C2×C20 C22×C10 C2×C42 C42 C22×C4 C10 C2×C4 C2×C4 C2×C4 C2×C4 C23 C2 # reps 1 1 1 1 4 4 4 2 1 1 2 4 2 8 4 8 4 4 4 32

Matrix representation of C426Dic5 in GL4(𝔽41) generated by

 40 22 0 0 0 32 0 0 0 0 1 0 0 0 0 32
,
 9 29 0 0 0 32 0 0 0 0 32 0 0 0 0 9
,
 1 0 0 0 0 1 0 0 0 0 23 0 0 0 0 25
,
 16 15 0 0 24 25 0 0 0 0 0 1 0 0 40 0
G:=sub<GL(4,GF(41))| [40,0,0,0,22,32,0,0,0,0,1,0,0,0,0,32],[9,0,0,0,29,32,0,0,0,0,32,0,0,0,0,9],[1,0,0,0,0,1,0,0,0,0,23,0,0,0,0,25],[16,24,0,0,15,25,0,0,0,0,0,40,0,0,1,0] >;

C426Dic5 in GAP, Magma, Sage, TeX

C_4^2\rtimes_6{\rm Dic}_5
% in TeX

G:=Group("C4^2:6Dic5");
// GroupNames label

G:=SmallGroup(320,81);
// by ID

G=gap.SmallGroup(320,81);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,253,64,1123,1684,102,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^4=c^10=1,d^2=c^5,d*a*d^-1=a*b=b*a,a*c=c*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^-1>;
// generators/relations

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