Copied to
clipboard

G = D4×C13⋊C4order 416 = 25·13

Direct product of D4 and C13⋊C4

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

Aliases: D4×C13⋊C4, D523C4, D26.12C23, C13⋊(C4×D4), C52⋊(C2×C4), D26⋊(C2×C4), C13⋊D4⋊C4, (D4×C13)⋊3C4, Dic13⋊(C2×C4), C52⋊C42C2, D13.2(C2×D4), (D4×D13).3C2, D13.D43C2, C26.8(C22×C4), D13.2(C4○D4), (C4×D13).12C22, (C22×D13).16C22, C41(C2×C13⋊C4), (C2×C26)⋊(C2×C4), (C4×C13⋊C4)⋊3C2, C221(C2×C13⋊C4), (C22×C13⋊C4)⋊1C2, C2.9(C22×C13⋊C4), (C2×C13⋊C4).3C22, SmallGroup(416,206)

Series: Derived Chief Lower central Upper central

C1C26 — D4×C13⋊C4
C1C13D13D26C2×C13⋊C4C22×C13⋊C4 — D4×C13⋊C4
C13C26 — D4×C13⋊C4
C1C2D4

Generators and relations for D4×C13⋊C4
 G = < a,b,c,d | a4=b2=c13=d4=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c5 >

Subgroups: 748 in 94 conjugacy classes, 38 normal (20 characteristic)
C1, C2, C2 [×6], C4, C4 [×6], C22 [×2], C22 [×7], C2×C4 [×9], D4, D4 [×3], C23 [×2], C13, C42, C22⋊C4 [×2], C4⋊C4, C22×C4 [×2], C2×D4, D13 [×2], D13 [×2], C26, C26 [×2], C4×D4, Dic13, C52, C13⋊C4 [×2], C13⋊C4 [×3], D26, D26 [×2], D26 [×4], C2×C26 [×2], C4×D13, D52, C13⋊D4 [×2], D4×C13, C2×C13⋊C4 [×2], C2×C13⋊C4 [×2], C2×C13⋊C4 [×4], C22×D13 [×2], C4×C13⋊C4, C52⋊C4, D13.D4 [×2], D4×D13, C22×C13⋊C4 [×2], D4×C13⋊C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], C23, C22×C4, C2×D4, C4○D4, C4×D4, C13⋊C4, C2×C13⋊C4 [×3], C22×C13⋊C4, D4×C13⋊C4

Smallest permutation representation of D4×C13⋊C4
On 52 points
Generators in S52
(1 27 14 40)(2 28 15 41)(3 29 16 42)(4 30 17 43)(5 31 18 44)(6 32 19 45)(7 33 20 46)(8 34 21 47)(9 35 22 48)(10 36 23 49)(11 37 24 50)(12 38 25 51)(13 39 26 52)
(27 40)(28 41)(29 42)(30 43)(31 44)(32 45)(33 46)(34 47)(35 48)(36 49)(37 50)(38 51)(39 52)
(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)
(2 9 13 6)(3 4 12 11)(5 7 10 8)(15 22 26 19)(16 17 25 24)(18 20 23 21)(28 35 39 32)(29 30 38 37)(31 33 36 34)(41 48 52 45)(42 43 51 50)(44 46 49 47)

G:=sub<Sym(52)| (1,27,14,40)(2,28,15,41)(3,29,16,42)(4,30,17,43)(5,31,18,44)(6,32,19,45)(7,33,20,46)(8,34,21,47)(9,35,22,48)(10,36,23,49)(11,37,24,50)(12,38,25,51)(13,39,26,52), (27,40)(28,41)(29,42)(30,43)(31,44)(32,45)(33,46)(34,47)(35,48)(36,49)(37,50)(38,51)(39,52), (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), (2,9,13,6)(3,4,12,11)(5,7,10,8)(15,22,26,19)(16,17,25,24)(18,20,23,21)(28,35,39,32)(29,30,38,37)(31,33,36,34)(41,48,52,45)(42,43,51,50)(44,46,49,47)>;

G:=Group( (1,27,14,40)(2,28,15,41)(3,29,16,42)(4,30,17,43)(5,31,18,44)(6,32,19,45)(7,33,20,46)(8,34,21,47)(9,35,22,48)(10,36,23,49)(11,37,24,50)(12,38,25,51)(13,39,26,52), (27,40)(28,41)(29,42)(30,43)(31,44)(32,45)(33,46)(34,47)(35,48)(36,49)(37,50)(38,51)(39,52), (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), (2,9,13,6)(3,4,12,11)(5,7,10,8)(15,22,26,19)(16,17,25,24)(18,20,23,21)(28,35,39,32)(29,30,38,37)(31,33,36,34)(41,48,52,45)(42,43,51,50)(44,46,49,47) );

G=PermutationGroup([(1,27,14,40),(2,28,15,41),(3,29,16,42),(4,30,17,43),(5,31,18,44),(6,32,19,45),(7,33,20,46),(8,34,21,47),(9,35,22,48),(10,36,23,49),(11,37,24,50),(12,38,25,51),(13,39,26,52)], [(27,40),(28,41),(29,42),(30,43),(31,44),(32,45),(33,46),(34,47),(35,48),(36,49),(37,50),(38,51),(39,52)], [(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)], [(2,9,13,6),(3,4,12,11),(5,7,10,8),(15,22,26,19),(16,17,25,24),(18,20,23,21),(28,35,39,32),(29,30,38,37),(31,33,36,34),(41,48,52,45),(42,43,51,50),(44,46,49,47)])

35 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E4F···4L13A13B13C26A26B26C26D···26I52A52B52C
order12222222444444···413131326262626···26525252
size11221313262621313131326···264444448···8888

35 irreducible representations

dim111111111224448
type+++++++++++
imageC1C2C2C2C2C2C4C4C4D4C4○D4C13⋊C4C2×C13⋊C4C2×C13⋊C4D4×C13⋊C4
kernelD4×C13⋊C4C4×C13⋊C4C52⋊C4D13.D4D4×D13C22×C13⋊C4D52C13⋊D4D4×C13C13⋊C4D13D4C4C22C1
# reps111212242223363

Matrix representation of D4×C13⋊C4 in GL6(𝔽53)

010000
5200000
001000
000100
000010
000001
,
5200000
010000
001000
000100
000010
000001
,
100000
010000
004145752
0051511513
0029381451
002812020
,
3000000
0300000
0017472044
003401121
009421138
0020322738

G:=sub<GL(6,GF(53))| [0,52,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[52,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,41,51,29,28,0,0,45,51,38,1,0,0,7,15,14,20,0,0,52,13,51,20],[30,0,0,0,0,0,0,30,0,0,0,0,0,0,17,3,9,20,0,0,47,40,42,32,0,0,20,11,11,27,0,0,44,21,38,38] >;

D4×C13⋊C4 in GAP, Magma, Sage, TeX

D_4\times C_{13}\rtimes C_4
% in TeX

G:=Group("D4xC13:C4");
// GroupNames label

G:=SmallGroup(416,206);
// by ID

G=gap.SmallGroup(416,206);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-13,48,188,9221,1751]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^2=c^13=d^4=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^5>;
// generators/relations

׿
×
𝔽