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## G = C42⋊Dic5order 320 = 26·5

### 2nd semidirect product of C42 and Dic5 acting via Dic5/C5=C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C20 — C42⋊Dic5
 Chief series C1 — C5 — C10 — C2×C10 — C22×C10 — D4×C10 — C23⋊Dic5 — C42⋊Dic5
 Lower central C5 — C10 — C2×C10 — C2×C20 — C42⋊Dic5
 Upper central C1 — C2 — C22 — C2×D4 — C4.4D4

Generators and relations for C42⋊Dic5
G = < a,b,c,d | a4=b4=c10=1, d2=c5, ab=ba, cac-1=a-1b2, dad-1=a-1b-1, cbc-1=b-1, dbd-1=a2b-1, dcd-1=c-1 >

Subgroups: 302 in 70 conjugacy classes, 23 normal (17 characteristic)
C1, C2, C2 [×3], C4 [×5], C22, C22 [×4], C5, C2×C4, C2×C4 [×4], D4, Q8, C23 [×2], C10, C10 [×3], C42, C22⋊C4 [×4], C2×D4, C2×Q8, Dic5 [×2], C20 [×3], C2×C10, C2×C10 [×4], C23⋊C4 [×2], C4.4D4, C2×Dic5 [×2], C2×C20, C2×C20 [×2], C5×D4, C5×Q8, C22×C10 [×2], C423C4, C23.D5 [×2], C4×C20, C5×C22⋊C4 [×2], D4×C10, Q8×C10, C23⋊Dic5 [×2], C5×C4.4D4, C42⋊Dic5
Quotients: C1, C2 [×3], C4 [×2], C22, C2×C4, D4 [×2], D5, C22⋊C4, Dic5 [×2], D10, C23⋊C4, C2×Dic5, C5⋊D4 [×2], C423C4, C23.D5, C23⋊Dic5, C42⋊Dic5

Smallest permutation representation of C42⋊Dic5
On 80 points
Generators in S80
(1 54 19 45)(2 60 20 41)(3 56 16 47)(4 52 17 43)(5 58 18 49)(6 42 11 51)(7 48 12 57)(8 44 13 53)(9 50 14 59)(10 46 15 55)(21 63 36 68)(22 80 37 75)(23 65 38 70)(24 72 39 77)(25 67 40 62)(26 74 31 79)(27 69 32 64)(28 76 33 71)(29 61 34 66)(30 78 35 73)
(1 26 9 21)(2 22 10 27)(3 28 6 23)(4 24 7 29)(5 30 8 25)(11 38 16 33)(12 34 17 39)(13 40 18 35)(14 36 19 31)(15 32 20 37)(41 75 55 64)(42 65 56 76)(43 77 57 66)(44 67 58 78)(45 79 59 68)(46 69 60 80)(47 71 51 70)(48 61 52 72)(49 73 53 62)(50 63 54 74)
(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)
(2 5)(3 4)(6 7)(8 10)(11 17)(12 16)(13 20)(14 19)(15 18)(21 36 26 31)(22 35 27 40)(23 34 28 39)(24 33 29 38)(25 32 30 37)(41 73 46 78)(42 72 47 77)(43 71 48 76)(44 80 49 75)(45 79 50 74)(51 66 56 61)(52 65 57 70)(53 64 58 69)(54 63 59 68)(55 62 60 67)

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

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

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

41 conjugacy classes

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 4E 4F 4G 4H 5A 5B 10A ··· 10F 10G 10H 10I 10J 20A ··· 20L 20M 20N 20O 20P order 1 2 2 2 2 4 4 4 4 4 4 4 4 5 5 10 ··· 10 10 10 10 10 20 ··· 20 20 20 20 20 size 1 1 2 4 4 4 4 4 8 40 40 40 40 2 2 2 ··· 2 8 8 8 8 4 ··· 4 8 8 8 8

41 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 4 4 4 4 type + + + + + - + - + image C1 C2 C2 C4 C4 D4 D5 Dic5 D10 Dic5 C5⋊D4 C23⋊C4 C42⋊3C4 C23⋊Dic5 C42⋊Dic5 kernel C42⋊Dic5 C23⋊Dic5 C5×C4.4D4 C4×C20 Q8×C10 C22×C10 C4.4D4 C42 C2×D4 C2×Q8 C23 C10 C5 C2 C1 # reps 1 2 1 2 2 2 2 2 2 2 8 1 2 4 8

Matrix representation of C42⋊Dic5 in GL4(𝔽41) generated by

 20 21 23 10 9 6 34 13 19 37 3 7 18 36 11 35
,
 18 1 0 28 5 23 13 13 3 3 17 40 38 0 1 24
,
 38 18 0 0 20 17 0 0 26 6 6 23 38 20 18 21
,
 5 6 14 12 6 7 39 25 21 30 1 2 10 21 35 28
G:=sub<GL(4,GF(41))| [20,9,19,18,21,6,37,36,23,34,3,11,10,13,7,35],[18,5,3,38,1,23,3,0,0,13,17,1,28,13,40,24],[38,20,26,38,18,17,6,20,0,0,6,18,0,0,23,21],[5,6,21,10,6,7,30,21,14,39,1,35,12,25,2,28] >;

C42⋊Dic5 in GAP, Magma, Sage, TeX

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

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

G:=SmallGroup(320,99);
// by ID

G=gap.SmallGroup(320,99);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,232,219,1571,570,297,136,1684,12550]);
// Polycyclic

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

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