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G = C42:6Dic5order 320 = 26·5

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

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C42:6Dic5, C20.38C42, (C4xC20):22C4, C10.19C4wrC2, C4:Dic5:12C4, C4.Dic5:7C4, C20.58(C4:C4), (C2xC20).58Q8, C4.8(C4xDic5), (C2xC42).7D5, C5:4(C42:6C4), (C2xC4).162D20, (C2xC20).478D4, C4.20(C4:Dic5), (C2xC4).42Dic10, C2.3(D20:4C4), (C22xC10).176D4, (C22xC4).412D10, C23.72(C5:D4), C4.44(D10:C4), C20.106(C22:C4), C4.22(C10.D4), C22.8(C23.D5), (C22xC20).532C22, C23.21D10.1C2, C22.36(D10:C4), C2.3(C10.10C42), C10.20(C2.C42), C22.12(C10.D4), (C2xC4xC20).15C2, (C2xC4).98(C4xD5), (C2xC10).61(C4:C4), (C2xC20).390(C2xC4), (C2xC4).71(C2xDic5), (C2xC4.Dic5).1C2, (C2xC4).231(C5:D4), (C2xC10).113(C22:C4), SmallGroup(320,81)

Series: Derived Chief Lower central Upper central

C1C20 — C42:6Dic5
C1C5C10C2xC10C22xC10C22xC20C23.21D10 — C42:6Dic5
C5C10C20 — C42:6Dic5
C1C2xC4C22xC4C2xC42

Generators and relations for C42:6Dic5
 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, C2xC4, C2xC4, C23, C10, C10, C10, C42, C42, C22:C4, C4:C4, C2xC8, M4(2), C22xC4, C22xC4, Dic5, C20, C20, C2xC10, C2xC10, C2xC42, C42:C2, C2xM4(2), C5:2C8, C2xDic5, C2xC20, C2xC20, C22xC10, C42:6C4, C2xC5:2C8, C4.Dic5, C4.Dic5, C4xDic5, C4:Dic5, C23.D5, C4xC20, C4xC20, C22xC20, C22xC20, C2xC4.Dic5, C23.21D10, C2xC4xC20, C42:6Dic5
Quotients: C1, C2, C4, C22, C2xC4, D4, Q8, D5, C42, C22:C4, C4:C4, Dic5, D10, C2.C42, C4wrC2, Dic10, C4xD5, D20, C2xDic5, C5:D4, C42:6C4, C4xDic5, C10.D4, C4:Dic5, D10:C4, C23.D5, D20:4C4, C10.10C42, C42:6Dic5

Smallest permutation representation of C42:6Dic5
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 2A2B2C2D2E4A4B4C4D4E···4N4O4P4Q4R5A5B8A8B8C8D10A···10N20A···20AV
order12222244444···4444455888810···1020···20
size11112211112···22020202022202020202···22···2

92 irreducible representations

dim11111112222222222222
type+++++-++-+-+
imageC1C2C2C2C4C4C4D4Q8D4D5Dic5D10C4wrC2Dic10C4xD5D20C5:D4C5:D4D20:4C4
kernelC42:6Dic5C2xC4.Dic5C23.21D10C2xC4xC20C4.Dic5C4:Dic5C4xC20C2xC20C2xC20C22xC10C2xC42C42C22xC4C10C2xC4C2xC4C2xC4C2xC4C23C2
# reps111144421124284844432

Matrix representation of C42:6Dic5 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] >;

C42:6Dic5 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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