Copied to
clipboard

G = C9×C4.D4order 288 = 25·32

Direct product of C9 and C4.D4

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Aliases: C9×C4.D4, C23.C36, C36.58D4, M4(2)⋊3C18, C4.9(D4×C9), (C6×D4).7C6, (D4×C18).8C2, (C2×D4).2C18, C12.67(C3×D4), (C9×M4(2))⋊9C2, (C22×C6).3C12, (C22×C18).1C4, C22.3(C2×C36), (C2×C36).58C22, (C3×M4(2)).3C6, C18.22(C22⋊C4), C3.(C3×C4.D4), (C2×C4).1(C2×C18), C2.4(C9×C22⋊C4), (C3×C4.D4).C3, (C2×C6).24(C2×C12), (C2×C18).20(C2×C4), (C2×C12).60(C2×C6), C6.22(C3×C22⋊C4), SmallGroup(288,50)

Series: Derived Chief Lower central Upper central

C1C22 — C9×C4.D4
C1C2C6C12C2×C12C2×C36C9×M4(2) — C9×C4.D4
C1C2C22 — C9×C4.D4
C1C18C2×C36 — C9×C4.D4

Generators and relations for C9×C4.D4
 G = < a,b,c,d | a9=b4=1, c4=b2, d2=b, ab=ba, ac=ca, ad=da, cbc-1=b-1, bd=db, dcd-1=b-1c3 >

Subgroups: 126 in 69 conjugacy classes, 36 normal (18 characteristic)
C1, C2, C2 [×3], C3, C4 [×2], C22, C22 [×4], C6, C6 [×3], C8 [×2], C2×C4, D4 [×2], C23 [×2], C9, C12 [×2], C2×C6, C2×C6 [×4], M4(2) [×2], C2×D4, C18, C18 [×3], C24 [×2], C2×C12, C3×D4 [×2], C22×C6 [×2], C4.D4, C36 [×2], C2×C18, C2×C18 [×4], C3×M4(2) [×2], C6×D4, C72 [×2], C2×C36, D4×C9 [×2], C22×C18 [×2], C3×C4.D4, C9×M4(2) [×2], D4×C18, C9×C4.D4
Quotients: C1, C2 [×3], C3, C4 [×2], C22, C6 [×3], C2×C4, D4 [×2], C9, C12 [×2], C2×C6, C22⋊C4, C18 [×3], C2×C12, C3×D4 [×2], C4.D4, C36 [×2], C2×C18, C3×C22⋊C4, C2×C36, D4×C9 [×2], C3×C4.D4, C9×C22⋊C4, C9×C4.D4

Smallest permutation representation of C9×C4.D4
On 72 points
Generators in S72
(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)
(1 59 46 40)(2 60 47 41)(3 61 48 42)(4 62 49 43)(5 63 50 44)(6 55 51 45)(7 56 52 37)(8 57 53 38)(9 58 54 39)(10 29 24 67)(11 30 25 68)(12 31 26 69)(13 32 27 70)(14 33 19 71)(15 34 20 72)(16 35 21 64)(17 36 22 65)(18 28 23 66)
(1 66 59 23 46 28 40 18)(2 67 60 24 47 29 41 10)(3 68 61 25 48 30 42 11)(4 69 62 26 49 31 43 12)(5 70 63 27 50 32 44 13)(6 71 55 19 51 33 45 14)(7 72 56 20 52 34 37 15)(8 64 57 21 53 35 38 16)(9 65 58 22 54 36 39 17)
(1 28 59 23 46 66 40 18)(2 29 60 24 47 67 41 10)(3 30 61 25 48 68 42 11)(4 31 62 26 49 69 43 12)(5 32 63 27 50 70 44 13)(6 33 55 19 51 71 45 14)(7 34 56 20 52 72 37 15)(8 35 57 21 53 64 38 16)(9 36 58 22 54 65 39 17)

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

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

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

99 conjugacy classes

class 1 2A2B2C2D3A3B4A4B6A6B6C6D6E6F6G6H8A8B8C8D9A···9F12A12B12C12D18A···18F18G···18L18M···18X24A···24H36A···36L72A···72X
order1222233446666666688889···91212121218···1818···1818···1824···2436···3672···72
size1124411221122444444441···122221···12···24···44···42···24···4

99 irreducible representations

dim111111111111222444
type+++++
imageC1C2C2C3C4C6C6C9C12C18C18C36D4C3×D4D4×C9C4.D4C3×C4.D4C9×C4.D4
kernelC9×C4.D4C9×M4(2)D4×C18C3×C4.D4C22×C18C3×M4(2)C6×D4C4.D4C22×C6M4(2)C2×D4C23C36C12C4C9C3C1
# reps121244268126242412126

Matrix representation of C9×C4.D4 in GL6(𝔽73)

400000
040000
001000
000100
000010
000001
,
100000
010000
00727100
001100
0004601
002727720
,
010000
7200000
00270710
00460172
0001460
0000270
,
010000
100000
00270710
000011
0001460
0010270

G:=sub<GL(6,GF(73))| [4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,1,0,27,0,0,71,1,46,27,0,0,0,0,0,72,0,0,0,0,1,0],[0,72,0,0,0,0,1,0,0,0,0,0,0,0,27,46,0,0,0,0,0,0,1,0,0,0,71,1,46,27,0,0,0,72,0,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,27,0,0,1,0,0,0,0,1,0,0,0,71,1,46,27,0,0,0,1,0,0] >;

C9×C4.D4 in GAP, Magma, Sage, TeX

C_9\times C_4.D_4
% in TeX

G:=Group("C9xC4.D4");
// GroupNames label

G:=SmallGroup(288,50);
// by ID

G=gap.SmallGroup(288,50);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-2,168,197,268,4371,2951,242]);
// Polycyclic

G:=Group<a,b,c,d|a^9=b^4=1,c^4=b^2,d^2=b,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=b^-1*c^3>;
// generators/relations

׿
×
𝔽