Copied to
clipboard

G = C3⋊D72  order 432 = 24·33

The semidirect product of C3 and D72 acting via D72/D36=C2

metabelian, supersoluble, monomial

Aliases: C3⋊2D72, D36⋊1S3, C6.14D36, C36.32D6, C12.11D18, C32.2D24, C3⋊C8⋊3D9, (C3×C9)⋊1D8, C12.46S32, C9⋊1(D4⋊S3), (C3×D36)⋊1C2, C4.10(S3×D9), (C3×C18).3D4, C36⋊S3⋊1C2, (C3×C6).31D12, (C3×C12).71D6, C18.3(C3⋊D4), (C3×C36).3C22, C6.3(C3⋊D12), C3.1(C3⋊D24), C2.6(C3⋊D36), (C9×C3⋊C8)⋊1C2, (C3×C3⋊C8).1S3, SmallGroup(432,64)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3×C36 — C3⋊D72
Chief series
C1 — C3 — C9 — C3×C9 — C3×C18 — C3×C36 — C3×D36 — C3⋊D72
Lower central
C3×C9 — C3×C18 — C3×C36 — C3⋊D72
Upper central
C1 — C2 — C4

Generators and relations for C3⋊D72
 G = < a,b,c | a3=b72=c2=1, bab-1=cac=a-1, cbc=b-1 >

Subgroups: 808 in 82 conjugacy classes, 25 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C22, S3, C6, C6, C8, D4, C9, C9, C32, C12, C12, D6, C2×C6, D8, D9, C18, C18, C3×S3, C3⋊S3, C3×C6, C3⋊C8, C24, D12, C3×D4, C3×C9, C36, C36, D18, C3×C12, S3×C6, C2×C3⋊S3, D24, D4⋊S3, C3×D9, C9⋊S3, C3×C18, C72, D36, D36, C3×C3⋊C8, C3×D12, C12⋊S3, C3×C36, C6×D9, C2×C9⋊S3, D72, C3⋊D24, C9×C3⋊C8, C3×D36, C36⋊S3, C3⋊D72
Quotients: C1, C2, C22, S3, D4, D6, D8, D9, D12, C3⋊D4, D18, S32, D24, D4⋊S3, D36, C3⋊D12, S3×D9, D72, C3⋊D24, C3⋊D36, C3⋊D72

Smallest permutation representation of C3⋊D72
►On 72 points
Generators in S72
(1 25 49)(2 50 26)(3 27 51)(4 52 28)(5 29 53)(6 54 30)(7 31 55)(8 56 32)(9 33 57)(10 58 34)(11 35 59)(12 60 36)(13 37 61)(14 62 38)(15 39 63)(16 64 40)(17 41 65)(18 66 42)(19 43 67)(20 68 44)(21 45 69)(22 70 46)(23 47 71)(24 72 48)

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

60 conjugacy classes

class 1 2A2B2C3A3B3C 4 6A6B6C6D6E8A8B9A9B9C9D9E9F12A12B12C12D12E18A18B18C18D18E18F24A24B24C24D36A···36F36G···36L72A···72L
order12223334666668899999912121212121818181818182424242436···3636···3672···72
size113610822422243636662224442244422244466662···24···46···6

60 irreducible representations

dim111122222222222224444444
type+++++++++++++++++++++++
imageC1C2C2C2S3S3D4D6D6D8D9C3⋊D4D12D18D24D36D72S32D4⋊S3C3⋊D12S3×D9C3⋊D24C3⋊D36C3⋊D72
kernelC3⋊D72C9×C3⋊C8C3×D36C36⋊S3D36C3×C3⋊C8C3×C18C36C3×C12C3×C9C3⋊C8C18C3×C6C12C32C6C3C12C9C6C4C3C2C1
# reps1111111112322346121113236

Matrix representation of C3⋊D72 ►in GL6(𝔽73)

100000
010000
0007200
0017200
000010
000001
,
23550000
1850000
0007200
0072000
00007045
00002842
,
66140000
770000
000100
001000
0000423
00004531

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,72,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[23,18,0,0,0,0,55,5,0,0,0,0,0,0,0,72,0,0,0,0,72,0,0,0,0,0,0,0,70,28,0,0,0,0,45,42],[66,7,0,0,0,0,14,7,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,42,45,0,0,0,0,3,31] >;
 

C3⋊D72 in GAP, Magma, Sage, TeX 

C_3\rtimes D_{72}
 
% in TeX
 
G:=Group("C3:D72");
 
// GroupNames label
 
G:=SmallGroup(432,64);
 
// by ID
 
G=gap.SmallGroup(432,64);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,85,92,254,58,571,10085,292,14118]);
 
// Polycyclic
 
G:=Group<a,b,c|a^3=b^72=c^2=1,b*a*b^-1=c*a*c=a^-1,c*b*c=b^-1>;
 
// generators/relations
 

׿
×
⋊
ℤ
𝔽
○
≀
ℚ
◁