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G = SD16xF5order 320 = 26·5

Direct product of SD16 and F5

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: SD16xF5, C5:(C4xSD16), C8:5(C2xF5), C40:4(C2xC4), Q8:D5:1C4, (C8xF5):4C2, (Q8xF5):1C2, Q8:1(C2xF5), C40:C2:3C4, D4.D5:3C4, C40:C4:4C2, D4.3(C2xF5), (D4xF5).1C2, C2.19(D4xF5), Q8:F5:1C2, (C5xSD16):3C4, D20.1(C2xC4), C10.18(C4xD4), (C2xF5).11D4, C4:F5.3C22, C4.5(C22xF5), Dic10:1(C2xC4), D5.3(C4oD8), D10.65(C2xD4), D20:C4.1C2, C20.5(C22xC4), D5.2(C2xSD16), (D5xSD16).2C2, (D4xD5).7C22, (Q8xD5).4C22, D5:C8.11C22, (C4xD5).27C23, (C8xD5).27C22, (C4xF5).11C22, Dic5.3(C4oD4), (C5xQ8):1(C2xC4), C5:2C8:13(C2xC4), (C5xD4).3(C2xC4), SmallGroup(320,1072)

Series: Derived Chief Lower central Upper central

C1C20 — SD16xF5
C1C5C10D10C4xD5C4xF5D4xF5 — SD16xF5
C5C10C20 — SD16xF5
C1C2C4SD16

Generators and relations for SD16xF5
 G = < a,b,c,d | a8=b2=c5=d4=1, bab=a3, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c3 >

Subgroups: 546 in 122 conjugacy classes, 44 normal (42 characteristic)
C1, C2, C2, C4, C4, C22, C5, C8, C8, C2xC4, D4, D4, Q8, Q8, C23, D5, D5, C10, C10, C42, C22:C4, C4:C4, C2xC8, SD16, SD16, C22xC4, C2xD4, C2xQ8, Dic5, Dic5, C20, C20, F5, F5, D10, D10, C2xC10, C4xC8, D4:C4, Q8:C4, C4.Q8, C4xD4, C4xQ8, C2xSD16, C5:2C8, C40, C5:C8, Dic10, Dic10, C4xD5, C4xD5, D20, C5:D4, C5xD4, C5xQ8, C2xF5, C2xF5, C22xD5, C4xSD16, C8xD5, C40:C2, D4.D5, Q8:D5, C5xSD16, D5:C8, C4xF5, C4xF5, C4:F5, C4:F5, C22:F5, D4xD5, Q8xD5, C22xF5, C8xF5, C40:C4, D20:C4, Q8:F5, D5xSD16, D4xF5, Q8xF5, SD16xF5
Quotients: C1, C2, C4, C22, C2xC4, D4, C23, SD16, C22xC4, C2xD4, C4oD4, F5, C4xD4, C2xSD16, C4oD8, C2xF5, C4xSD16, C22xF5, D4xF5, SD16xF5

Smallest permutation representation of SD16xF5
On 40 points
Generators in S40
(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)
(2 4)(3 7)(6 8)(9 11)(10 14)(13 15)(17 19)(18 22)(21 23)(25 27)(26 30)(29 31)(33 37)(34 40)(36 38)
(1 24 32 12 35)(2 17 25 13 36)(3 18 26 14 37)(4 19 27 15 38)(5 20 28 16 39)(6 21 29 9 40)(7 22 30 10 33)(8 23 31 11 34)
(1 5)(2 6)(3 7)(4 8)(9 17 29 36)(10 18 30 37)(11 19 31 38)(12 20 32 39)(13 21 25 40)(14 22 26 33)(15 23 27 34)(16 24 28 35)

G:=sub<Sym(40)| (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), (2,4)(3,7)(6,8)(9,11)(10,14)(13,15)(17,19)(18,22)(21,23)(25,27)(26,30)(29,31)(33,37)(34,40)(36,38), (1,24,32,12,35)(2,17,25,13,36)(3,18,26,14,37)(4,19,27,15,38)(5,20,28,16,39)(6,21,29,9,40)(7,22,30,10,33)(8,23,31,11,34), (1,5)(2,6)(3,7)(4,8)(9,17,29,36)(10,18,30,37)(11,19,31,38)(12,20,32,39)(13,21,25,40)(14,22,26,33)(15,23,27,34)(16,24,28,35)>;

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), (2,4)(3,7)(6,8)(9,11)(10,14)(13,15)(17,19)(18,22)(21,23)(25,27)(26,30)(29,31)(33,37)(34,40)(36,38), (1,24,32,12,35)(2,17,25,13,36)(3,18,26,14,37)(4,19,27,15,38)(5,20,28,16,39)(6,21,29,9,40)(7,22,30,10,33)(8,23,31,11,34), (1,5)(2,6)(3,7)(4,8)(9,17,29,36)(10,18,30,37)(11,19,31,38)(12,20,32,39)(13,21,25,40)(14,22,26,33)(15,23,27,34)(16,24,28,35) );

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)], [(2,4),(3,7),(6,8),(9,11),(10,14),(13,15),(17,19),(18,22),(21,23),(25,27),(26,30),(29,31),(33,37),(34,40),(36,38)], [(1,24,32,12,35),(2,17,25,13,36),(3,18,26,14,37),(4,19,27,15,38),(5,20,28,16,39),(6,21,29,9,40),(7,22,30,10,33),(8,23,31,11,34)], [(1,5),(2,6),(3,7),(4,8),(9,17,29,36),(10,18,30,37),(11,19,31,38),(12,20,32,39),(13,21,25,40),(14,22,26,33),(15,23,27,34),(16,24,28,35)]])

35 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G4H4I4J···4N 5 8A8B8C···8H10A10B20A20B40A40B
order1222224444444444···45888···8101020204040
size114552024555510101020···2042210···1041681688

35 irreducible representations

dim1111111111112222444488
type++++++++++++++
imageC1C2C2C2C2C2C2C2C4C4C4C4D4C4oD4SD16C4oD8F5C2xF5C2xF5C2xF5D4xF5SD16xF5
kernelSD16xF5C8xF5C40:C4D20:C4Q8:F5D5xSD16D4xF5Q8xF5C40:C2D4.D5Q8:D5C5xSD16C2xF5Dic5F5D5SD16C8D4Q8C2C1
# reps1111111122222244111112

Matrix representation of SD16xF5 in GL6(F41)

15150000
26150000
0040000
0004000
0000400
0000040
,
4000000
010000
0040000
0004000
0000400
0000040
,
100000
010000
0000040
0010040
0001040
0000140
,
3200000
0320000
000010
001000
000001
000100

G:=sub<GL(6,GF(41))| [15,26,0,0,0,0,15,15,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[40,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,40,40,40,40],[32,0,0,0,0,0,0,32,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,1,0] >;

SD16xF5 in GAP, Magma, Sage, TeX

{\rm SD}_{16}\times F_5
% in TeX

G:=Group("SD16xF5");
// GroupNames label

G:=SmallGroup(320,1072);
// by ID

G=gap.SmallGroup(320,1072);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,219,184,851,438,102,6278,1595]);
// Polycyclic

G:=Group<a,b,c,d|a^8=b^2=c^5=d^4=1,b*a*b=a^3,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^3>;
// generators/relations

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