Copied to
clipboard

G = Dic5.20C24order 320 = 26·5

20th non-split extension by Dic5 of C24 acting via C24/C23=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Dic5.20C24, C5:C8.4C23, Q8.F5:5C2, (C2xQ8).8F5, D5:C8:3C22, Q8.12(C2xF5), D20.12(C2xC4), (C2xD20).15C4, D5:M4(2):8C2, C5:2(Q8oM4(2)), (Q8xC10).10C4, Q8:2D5.3C4, C4.F5:10C22, C4.29(C22xF5), C2.13(C23xF5), C20.29(C22xC4), C10.12(C23xC4), (C4xD5).52C23, D10.5(C22xC4), C22.22(C22xF5), C22.F5.5C22, Q8:2D5.16C22, Dic5.48(C22xC4), (C2xDic5).365C23, (C2xC4).46(C2xF5), (C2xC20).73(C2xC4), (C4xD5).34(C2xC4), (C5xQ8).12(C2xC4), (C2xC4xD5).219C22, (C2xQ8:2D5).14C2, (C22xD5).63(C2xC4), (C2xC10).102(C22xC4), SmallGroup(320,1598)

Series: Derived Chief Lower central Upper central

C1C10 — Dic5.20C24
C1C5C10Dic5C5:C8D5:C8Q8.F5 — Dic5.20C24
C5C10 — Dic5.20C24
C1C2C2xQ8

Generators and relations for Dic5.20C24
 G = < a,b,c,d,e,f | a10=f2=1, b2=d2=e2=a5, c2=b, bab-1=a-1, cac-1=a3, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, fcf=a5c, ede-1=a5d, df=fd, ef=fe >

Subgroups: 794 in 258 conjugacy classes, 136 normal (13 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2xC4, C2xC4, D4, Q8, C23, D5, C10, C10, C2xC8, M4(2), C22xC4, C2xD4, C2xQ8, C4oD4, Dic5, C20, D10, D10, C2xC10, C2xM4(2), C8oD4, C2xC4oD4, C5:C8, C4xD5, D20, C2xDic5, C2xC20, C5xQ8, C22xD5, Q8oM4(2), D5:C8, C4.F5, C22.F5, C2xC4xD5, C2xD20, Q8:2D5, Q8xC10, D5:M4(2), Q8.F5, C2xQ8:2D5, Dic5.20C24
Quotients: C1, C2, C4, C22, C2xC4, C23, C22xC4, C24, F5, C23xC4, C2xF5, Q8oM4(2), C22xF5, C23xF5, Dic5.20C24

Smallest permutation representation of Dic5.20C24
On 80 points
Generators in S80
(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 16 6 11)(2 15 7 20)(3 14 8 19)(4 13 9 18)(5 12 10 17)(21 34 26 39)(22 33 27 38)(23 32 28 37)(24 31 29 36)(25 40 30 35)(41 54 46 59)(42 53 47 58)(43 52 48 57)(44 51 49 56)(45 60 50 55)(61 74 66 79)(62 73 67 78)(63 72 68 77)(64 71 69 76)(65 80 70 75)
(1 58 16 42 6 53 11 47)(2 55 15 45 7 60 20 50)(3 52 14 48 8 57 19 43)(4 59 13 41 9 54 18 46)(5 56 12 44 10 51 17 49)(21 72 34 68 26 77 39 63)(22 79 33 61 27 74 38 66)(23 76 32 64 28 71 37 69)(24 73 31 67 29 78 36 62)(25 80 40 70 30 75 35 65)
(1 31 6 36)(2 32 7 37)(3 33 8 38)(4 34 9 39)(5 35 10 40)(11 24 16 29)(12 25 17 30)(13 26 18 21)(14 27 19 22)(15 28 20 23)(41 77 46 72)(42 78 47 73)(43 79 48 74)(44 80 49 75)(45 71 50 76)(51 70 56 65)(52 61 57 66)(53 62 58 67)(54 63 59 68)(55 64 60 69)
(1 24 6 29)(2 25 7 30)(3 26 8 21)(4 27 9 22)(5 28 10 23)(11 36 16 31)(12 37 17 32)(13 38 18 33)(14 39 19 34)(15 40 20 35)(41 66 46 61)(42 67 47 62)(43 68 48 63)(44 69 49 64)(45 70 50 65)(51 76 56 71)(52 77 57 72)(53 78 58 73)(54 79 59 74)(55 80 60 75)
(41 46)(42 47)(43 48)(44 49)(45 50)(51 56)(52 57)(53 58)(54 59)(55 60)(61 66)(62 67)(63 68)(64 69)(65 70)(71 76)(72 77)(73 78)(74 79)(75 80)

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

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

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

44 conjugacy classes

class 1 2A2B2C···2H4A···4F4G4H4I 5 8A···8P10A10B10C20A···20F
order1222···24···444458···810101020···20
size11210···102···25510410···104448···8

44 irreducible representations

dim111111144448
type++++++++
imageC1C2C2C2C4C4C4F5C2xF5C2xF5Q8oM4(2)Dic5.20C24
kernelDic5.20C24D5:M4(2)Q8.F5C2xQ8:2D5C2xD20Q8:2D5Q8xC10C2xQ8C2xC4Q8C5C1
# reps168168213422

Matrix representation of Dic5.20C24 in GL8(F41)

61000000
400000000
180010000
18040340000
000040000
000004000
000000400
000000040
,
635000000
4035000000
183100000
18534400000
000032000
000003200
000000320
000000032
,
11539100000
11010390000
35180360000
311830300000
00000010
000053619
000032000
0000221905
,
400000000
040000000
004000000
000400000
00000900
00009000
0000437940
000000032
,
400000000
040000000
004000000
000400000
00000100
000040000
000053619
0000801840
,
10000000
01000000
00100000
00010000
00001000
00000100
000000400
0000833040

G:=sub<GL(8,GF(41))| [6,40,18,18,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,1,34,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40],[6,40,18,18,0,0,0,0,35,35,3,5,0,0,0,0,0,0,1,34,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,32],[11,11,35,31,0,0,0,0,5,0,18,18,0,0,0,0,39,10,0,30,0,0,0,0,10,39,36,30,0,0,0,0,0,0,0,0,0,5,32,22,0,0,0,0,0,36,0,19,0,0,0,0,1,1,0,0,0,0,0,0,0,9,0,5],[40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,9,4,0,0,0,0,0,9,0,37,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,40,32],[40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,40,5,8,0,0,0,0,1,0,36,0,0,0,0,0,0,0,1,18,0,0,0,0,0,0,9,40],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,8,0,0,0,0,0,1,0,33,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40] >;

Dic5.20C24 in GAP, Magma, Sage, TeX

{\rm Dic}_5._{20}C_2^4
% in TeX

G:=Group("Dic5.20C2^4");
// GroupNames label

G:=SmallGroup(320,1598);
// by ID

G=gap.SmallGroup(320,1598);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,232,387,184,1123,102,6278,818]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^10=f^2=1,b^2=d^2=e^2=a^5,c^2=b,b*a*b^-1=a^-1,c*a*c^-1=a^3,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,c*d=d*c,c*e=e*c,f*c*f=a^5*c,e*d*e^-1=a^5*d,d*f=f*d,e*f=f*e>;
// generators/relations

׿
x
:
Z
F
o
wr
Q
<