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G = C426Dic5order 320 = 26·5

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

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C426Dic5, C20.38C42, (C4×C20)⋊22C4, C10.19C4≀C2, C4⋊Dic512C4, C4.Dic57C4, C20.58(C4⋊C4), (C2×C20).58Q8, C4.8(C4×Dic5), (C2×C42).7D5, C54(C426C4), (C2×C4).162D20, (C2×C20).478D4, C4.20(C4⋊Dic5), (C2×C4).42Dic10, C2.3(D204C4), (C22×C10).176D4, (C22×C4).412D10, C23.72(C5⋊D4), C4.44(D10⋊C4), C20.106(C22⋊C4), C4.22(C10.D4), C22.8(C23.D5), (C22×C20).532C22, C23.21D10.1C2, C22.36(D10⋊C4), C2.3(C10.10C42), C10.20(C2.C42), C22.12(C10.D4), (C2×C4×C20).15C2, (C2×C4).98(C4×D5), (C2×C10).61(C4⋊C4), (C2×C20).390(C2×C4), (C2×C4).71(C2×Dic5), (C2×C4.Dic5).1C2, (C2×C4).231(C5⋊D4), (C2×C10).113(C22⋊C4), SmallGroup(320,81)

Series: Derived Chief Lower central Upper central

C1C20 — C426Dic5
C1C5C10C2×C10C22×C10C22×C20C23.21D10 — C426Dic5
C5C10C20 — C426Dic5
C1C2×C4C22×C4C2×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 [×2], C2 [×2], C4 [×4], C4 [×6], C22 [×3], C22 [×2], C5, C8 [×2], C2×C4 [×6], C2×C4 [×8], C23, C10, C10 [×2], C10 [×2], C42 [×2], C42 [×2], C22⋊C4, C4⋊C4 [×2], C2×C8, M4(2) [×3], C22×C4, C22×C4, Dic5 [×2], C20 [×4], C20 [×4], C2×C10 [×3], C2×C10 [×2], C2×C42, C42⋊C2, C2×M4(2), C52C8 [×2], C2×Dic5 [×2], C2×C20 [×6], C2×C20 [×6], C22×C10, C426C4, C2×C52C8, C4.Dic5 [×2], C4.Dic5, C4×Dic5, C4⋊Dic5 [×2], C23.D5, C4×C20 [×2], C4×C20, C22×C20, C22×C20, C2×C4.Dic5, C23.21D10, C2×C4×C20, C426Dic5
Quotients: C1, C2 [×3], C4 [×6], C22, C2×C4 [×3], D4 [×3], Q8, D5, C42, C22⋊C4 [×3], C4⋊C4 [×3], Dic5 [×2], D10, C2.C42, C4≀C2 [×2], Dic10, C4×D5 [×2], D20, C2×Dic5, C5⋊D4 [×2], C426C4, C4×Dic5, C10.D4 [×2], C4⋊Dic5, D10⋊C4 [×2], C23.D5, D204C4 [×2], 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 49)(12 50)(13 41)(14 42)(15 43)(16 44)(17 45)(18 46)(19 47)(20 48)(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 12)(2 46 40 13)(3 47 31 14)(4 48 32 15)(5 49 33 16)(6 50 34 17)(7 41 35 18)(8 42 36 19)(9 43 37 20)(10 44 38 11)(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 70 16 65)(12 69 17 64)(13 68 18 63)(14 67 19 62)(15 66 20 61)(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)(11,49)(12,50)(13,41)(14,42)(15,43)(16,44)(17,45)(18,46)(19,47)(20,48)(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), (1,45,39,12)(2,46,40,13)(3,47,31,14)(4,48,32,15)(5,49,33,16)(6,50,34,17)(7,41,35,18)(8,42,36,19)(9,43,37,20)(10,44,38,11)(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)(11,70,16,65)(12,69,17,64)(13,68,18,63)(14,67,19,62)(15,66,20,61)(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)(11,49)(12,50)(13,41)(14,42)(15,43)(16,44)(17,45)(18,46)(19,47)(20,48)(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), (1,45,39,12)(2,46,40,13)(3,47,31,14)(4,48,32,15)(5,49,33,16)(6,50,34,17)(7,41,35,18)(8,42,36,19)(9,43,37,20)(10,44,38,11)(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)(11,70,16,65)(12,69,17,64)(13,68,18,63)(14,67,19,62)(15,66,20,61)(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),(11,49),(12,50),(13,41),(14,42),(15,43),(16,44),(17,45),(18,46),(19,47),(20,48),(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)], [(1,45,39,12),(2,46,40,13),(3,47,31,14),(4,48,32,15),(5,49,33,16),(6,50,34,17),(7,41,35,18),(8,42,36,19),(9,43,37,20),(10,44,38,11),(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),(11,70,16,65),(12,69,17,64),(13,68,18,63),(14,67,19,62),(15,66,20,61),(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 2A2B2C2D2E4A4B4C4D4E···4N4O4P4Q4R5A5B8A8B8C8D10A···10N20A···20AV
order12222244444···4444455888810···1020···20
size11112211112···22020202022202020202···22···2

92 irreducible representations

dim11111112222222222222
type+++++-++-+-+
imageC1C2C2C2C4C4C4D4Q8D4D5Dic5D10C4≀C2Dic10C4×D5D20C5⋊D4C5⋊D4D204C4
kernelC426Dic5C2×C4.Dic5C23.21D10C2×C4×C20C4.Dic5C4⋊Dic5C4×C20C2×C20C2×C20C22×C10C2×C42C42C22×C4C10C2×C4C2×C4C2×C4C2×C4C23C2
# reps111144421124284844432

Matrix representation of C426Dic5 in GL4(𝔽41) generated by

402200
03200
0010
00032
,
92900
03200
00320
0009
,
1000
0100
00230
00025
,
161500
242500
0001
00400
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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