Copied to
clipboard

G = D72⋊C2order 288 = 25·32

6th semidirect product of D72 and C2 acting faithfully

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C83D18, D726C2, Q83D18, C723C22, SD161D9, D18.7D4, D4.3D18, C24.29D6, D362C22, C36.5C23, Dic9.9D4, D4⋊D93C2, (D4×D9)⋊3C2, C9⋊C82C22, C8⋊D91C2, C93(C8⋊C22), C3.(Q83D6), (C3×D4).5D6, C6.93(S3×D4), C2.19(D4×D9), Q82D92C2, Q83D91C2, (C9×SD16)⋊1C2, C18.31(C2×D4), (C3×Q8).25D6, (Q8×C9)⋊2C22, C4.5(C22×D9), (C3×SD16).1S3, (C4×D9).2C22, (D4×C9).3C22, C12.44(C22×S3), SmallGroup(288,124)

Series: Derived Chief Lower central Upper central

C1C36 — D72⋊C2
C1C3C9C18C36C4×D9D4×D9 — D72⋊C2
C9C18C36 — D72⋊C2
C1C2C4SD16

Generators and relations for D72⋊C2
 G = < a,b,c | a72=b2=c2=1, bab=a-1, cac=a19, bc=cb >

Subgroups: 604 in 102 conjugacy classes, 34 normal (all characteristic)
C1, C2, C2 [×4], C3, C4, C4 [×2], C22 [×6], S3 [×3], C6, C6, C8, C8, C2×C4 [×2], D4, D4 [×4], Q8, C23, C9, Dic3, C12, C12, D6 [×5], C2×C6, M4(2), D8 [×2], SD16, SD16, C2×D4, C4○D4, D9 [×3], C18, C18, C3⋊C8, C24, C4×S3 [×2], D12 [×3], C3⋊D4, C3×D4, C3×Q8, C22×S3, C8⋊C22, Dic9, C36, C36, D18, D18 [×4], C2×C18, C8⋊S3, D24, D4⋊S3, Q82S3, C3×SD16, S3×D4, Q83S3, C9⋊C8, C72, C4×D9, C4×D9, D36 [×2], D36, C9⋊D4, D4×C9, Q8×C9, C22×D9, Q83D6, C8⋊D9, D72, D4⋊D9, Q82D9, C9×SD16, D4×D9, Q83D9, D72⋊C2
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D6 [×3], C2×D4, D9, C22×S3, C8⋊C22, D18 [×3], S3×D4, C22×D9, Q83D6, D4×D9, D72⋊C2

Smallest permutation representation of D72⋊C2
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 9)(2 8)(3 7)(4 6)(10 72)(11 71)(12 70)(13 69)(14 68)(15 67)(16 66)(17 65)(18 64)(19 63)(20 62)(21 61)(22 60)(23 59)(24 58)(25 57)(26 56)(27 55)(28 54)(29 53)(30 52)(31 51)(32 50)(33 49)(34 48)(35 47)(36 46)(37 45)(38 44)(39 43)(40 42)
(2 20)(3 39)(4 58)(6 24)(7 43)(8 62)(10 28)(11 47)(12 66)(14 32)(15 51)(16 70)(18 36)(19 55)(22 40)(23 59)(26 44)(27 63)(30 48)(31 67)(34 52)(35 71)(38 56)(42 60)(46 64)(50 68)(54 72)

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

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

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

39 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C6A6B8A8B9A9B9C12A12B18A18B18C18D18E18F24A24B36A36B36C36D36E36F72A···72F
order122222344466889991212181818181818242436363636363672···72
size114183636224182843622248222888444448884···4

39 irreducible representations

dim11111111222222222244444
type+++++++++++++++++++++++
imageC1C2C2C2C2C2C2C2S3D4D4D6D6D6D9D18D18D18C8⋊C22S3×D4Q83D6D4×D9D72⋊C2
kernelD72⋊C2C8⋊D9D72D4⋊D9Q82D9C9×SD16D4×D9Q83D9C3×SD16Dic9D18C24C3×D4C3×Q8SD16C8D4Q8C9C6C3C2C1
# reps11111111111111333311236

Matrix representation of D72⋊C2 in GL8(𝔽73)

004530000
0070420000
2870000000
331000000
000028287071
0000444430
0000633905
00001001
,
2870000000
4245000000
004530000
0031280000
00001000
000007200
00004954125
0000720072
,
10000000
01000000
007200000
000720000
00001000
000007200
000019247248
00000101

G:=sub<GL(8,GF(73))| [0,0,28,3,0,0,0,0,0,0,70,31,0,0,0,0,45,70,0,0,0,0,0,0,3,42,0,0,0,0,0,0,0,0,0,0,28,44,63,1,0,0,0,0,28,44,39,0,0,0,0,0,70,3,0,0,0,0,0,0,71,0,5,1],[28,42,0,0,0,0,0,0,70,45,0,0,0,0,0,0,0,0,45,31,0,0,0,0,0,0,3,28,0,0,0,0,0,0,0,0,1,0,49,72,0,0,0,0,0,72,54,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,25,72],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,1,0,19,0,0,0,0,0,0,72,24,1,0,0,0,0,0,0,72,0,0,0,0,0,0,0,48,1] >;

D72⋊C2 in GAP, Magma, Sage, TeX

D_{72}\rtimes C_2
% in TeX

G:=Group("D72:C2");
// GroupNames label

G:=SmallGroup(288,124);
// by ID

G=gap.SmallGroup(288,124);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,422,135,100,346,185,80,6725,292,9414]);
// Polycyclic

G:=Group<a,b,c|a^72=b^2=c^2=1,b*a*b=a^-1,c*a*c=a^19,b*c=c*b>;
// generators/relations

׿
×
𝔽