Copied to
clipboard

G = C36.D4order 288 = 25·32

7th non-split extension by C36 of D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C36.7D4, C23.2Dic9, (C6×D4).1S3, (C2×D4).1D9, (C2×C4).3D18, (D4×C18).1C2, (C2×C12).45D6, C92(C4.D4), C4.Dic93C2, C4.12(C9⋊D4), C12.5(C3⋊D4), (C22×C18).2C4, C3.(C12.D4), (C2×C36).23C22, (C22×C6).6Dic3, C22.2(C2×Dic9), C18.13(C22⋊C4), C2.3(C18.D4), C6.14(C6.D4), (C2×C18).29(C2×C4), (C2×C6).33(C2×Dic3), SmallGroup(288,39)

Series: Derived Chief Lower central Upper central

C1C2×C18 — C36.D4
C1C3C9C18C36C2×C36C4.Dic9 — C36.D4
C9C18C2×C18 — C36.D4
C1C2C2×C4C2×D4

Generators and relations for C36.D4
 G = < a,b,c | a36=1, b4=a18, c2=a9, bab-1=a-1, cac-1=a17, cbc-1=a27b3 >

Subgroups: 196 in 69 conjugacy classes, 30 normal (16 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], C3⋊C8 [×2], C2×C12, C3×D4 [×2], C22×C6 [×2], C4.D4, C36 [×2], C2×C18, C2×C18 [×4], C4.Dic3 [×2], C6×D4, C9⋊C8 [×2], C2×C36, D4×C9 [×2], C22×C18 [×2], C12.D4, C4.Dic9 [×2], D4×C18, C36.D4
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C2×C4, D4 [×2], Dic3 [×2], D6, C22⋊C4, D9, C2×Dic3, C3⋊D4 [×2], C4.D4, Dic9 [×2], D18, C6.D4, C2×Dic9, C9⋊D4 [×2], C12.D4, C18.D4, C36.D4

Smallest permutation representation of C36.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 68 10 59 19 50 28 41)(2 67 11 58 20 49 29 40)(3 66 12 57 21 48 30 39)(4 65 13 56 22 47 31 38)(5 64 14 55 23 46 32 37)(6 63 15 54 24 45 33 72)(7 62 16 53 25 44 34 71)(8 61 17 52 26 43 35 70)(9 60 18 51 27 42 36 69)
(1 59 10 68 19 41 28 50)(2 40 11 49 20 58 29 67)(3 57 12 66 21 39 30 48)(4 38 13 47 22 56 31 65)(5 55 14 64 23 37 32 46)(6 72 15 45 24 54 33 63)(7 53 16 62 25 71 34 44)(8 70 17 43 26 52 35 61)(9 51 18 60 27 69 36 42)

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

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

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

51 conjugacy classes

class 1 2A2B2C2D 3 4A4B6A6B6C6D6E6F6G8A8B8C8D9A9B9C12A12B18A···18I18J···18U36A···36F
order1222234466666668888999121218···1818···1836···36
size11244222222444436363636222442···24···44···4

51 irreducible representations

dim1111222222222444
type++++++-++-+
imageC1C2C2C4S3D4D6Dic3D9C3⋊D4D18Dic9C9⋊D4C4.D4C12.D4C36.D4
kernelC36.D4C4.Dic9D4×C18C22×C18C6×D4C36C2×C12C22×C6C2×D4C12C2×C4C23C4C9C3C1
# reps12141212343612126

Matrix representation of C36.D4 in GL4(𝔽73) generated by

553400
111800
271804
5718690
,
10063
00172
440072
01072
,
10630
00721
01720
440720
G:=sub<GL(4,GF(73))| [55,11,27,57,34,18,18,18,0,0,0,69,0,0,4,0],[1,0,44,0,0,0,0,1,0,1,0,0,63,72,72,72],[1,0,0,44,0,0,1,0,63,72,72,72,0,1,0,0] >;

C36.D4 in GAP, Magma, Sage, TeX

C_{36}.D_4
% in TeX

G:=Group("C36.D4");
// GroupNames label

G:=SmallGroup(288,39);
// by ID

G=gap.SmallGroup(288,39);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,28,141,219,100,675,6725,292,9414]);
// Polycyclic

G:=Group<a,b,c|a^36=1,b^4=a^18,c^2=a^9,b*a*b^-1=a^-1,c*a*c^-1=a^17,c*b*c^-1=a^27*b^3>;
// generators/relations

׿
×
𝔽