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G = C42⋊Dic5order 320 = 26·5

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

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C422Dic5, (C4×C20)⋊12C4, (Q8×C10)⋊12C4, (C2×Q8)⋊2Dic5, (C2×D4).9D10, C54(C423C4), C4.4D4.2D5, (C22×C10).16D4, C23.7(C5⋊D4), C23⋊Dic5.4C2, C10.44(C23⋊C4), C2.8(C23⋊Dic5), (D4×C10).172C22, C22.14(C23.D5), (C2×C4).1(C2×Dic5), (C2×C20).181(C2×C4), (C5×C4.4D4).9C2, (C2×C10).163(C22⋊C4), SmallGroup(320,99)

Series: Derived Chief Lower central Upper central

C1C2×C20 — C42⋊Dic5
C1C5C10C2×C10C22×C10D4×C10C23⋊Dic5 — C42⋊Dic5
C5C10C2×C10C2×C20 — C42⋊Dic5
C1C2C22C2×D4C4.4D4

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 2A2B2C2D4A4B4C4D4E4F4G4H5A5B10A···10F10G10H10I10J20A···20L20M20N20O20P
order12222444444445510···101010101020···2020202020
size11244444840404040222···288884···48888

41 irreducible representations

dim111112222224444
type+++++-+-+
imageC1C2C2C4C4D4D5Dic5D10Dic5C5⋊D4C23⋊C4C423C4C23⋊Dic5C42⋊Dic5
kernelC42⋊Dic5C23⋊Dic5C5×C4.4D4C4×C20Q8×C10C22×C10C4.4D4C42C2×D4C2×Q8C23C10C5C2C1
# reps121222222281248

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

20212310
963413
193737
18361135
,
181028
5231313
331740
380124
,
381800
201700
266623
38201821
,
561412
673925
213012
10213528
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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