Copied to
clipboard

G = C325D8order 144 = 24·32

2nd semidirect product of C32 and D8 acting via D8/C8=C2

metabelian, supersoluble, monomial

Aliases: C241S3, C31D24, C325D8, C6.8D12, C12.47D6, C81(C3⋊S3), (C3×C24)⋊1C2, (C3×C6).23D4, C12⋊S31C2, C2.4(C12⋊S3), (C3×C12).33C22, C4.9(C2×C3⋊S3), SmallGroup(144,88)

Series: Derived Chief Lower central Upper central

C1C3×C12 — C325D8
C1C3C32C3×C6C3×C12C12⋊S3 — C325D8
C32C3×C6C3×C12 — C325D8
C1C2C4C8

Generators and relations for C325D8
 G = < a,b,c,d | a3=b3=c8=d2=1, ab=ba, ac=ca, dad=a-1, bc=cb, dbd=b-1, dcd=c-1 >

Subgroups: 354 in 66 conjugacy classes, 27 normal (9 characteristic)
C1, C2, C2 [×2], C3 [×4], C4, C22 [×2], S3 [×8], C6 [×4], C8, D4 [×2], C32, C12 [×4], D6 [×8], D8, C3⋊S3 [×2], C3×C6, C24 [×4], D12 [×8], C3×C12, C2×C3⋊S3 [×2], D24 [×4], C3×C24, C12⋊S3 [×2], C325D8
Quotients: C1, C2 [×3], C22, S3 [×4], D4, D6 [×4], D8, C3⋊S3, D12 [×4], C2×C3⋊S3, D24 [×4], C12⋊S3, C325D8

Smallest permutation representation of C325D8
On 72 points
Generators in S72
(1 25 47)(2 26 48)(3 27 41)(4 28 42)(5 29 43)(6 30 44)(7 31 45)(8 32 46)(9 33 51)(10 34 52)(11 35 53)(12 36 54)(13 37 55)(14 38 56)(15 39 49)(16 40 50)(17 68 57)(18 69 58)(19 70 59)(20 71 60)(21 72 61)(22 65 62)(23 66 63)(24 67 64)
(1 40 18)(2 33 19)(3 34 20)(4 35 21)(5 36 22)(6 37 23)(7 38 24)(8 39 17)(9 59 48)(10 60 41)(11 61 42)(12 62 43)(13 63 44)(14 64 45)(15 57 46)(16 58 47)(25 50 69)(26 51 70)(27 52 71)(28 53 72)(29 54 65)(30 55 66)(31 56 67)(32 49 68)
(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 8)(2 7)(3 6)(4 5)(9 67)(10 66)(11 65)(12 72)(13 71)(14 70)(15 69)(16 68)(17 40)(18 39)(19 38)(20 37)(21 36)(22 35)(23 34)(24 33)(25 46)(26 45)(27 44)(28 43)(29 42)(30 41)(31 48)(32 47)(49 58)(50 57)(51 64)(52 63)(53 62)(54 61)(55 60)(56 59)

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

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

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

C325D8 is a maximal subgroup of
C3⋊D48  C24.49D6  C325D16  C6.D24  C327D16  C3210SD32  S3×D24  C241D6  D6.3D12  C24.78D6  C243D6  D8×C3⋊S3  C247D6  C24.28D6  He34D8  C721S3  C338D8  C3312D8
C325D8 is a maximal quotient of
C325D16  C6.D24  C325Q32  C241Dic3  C62.84D4  C721S3  He35D8  C338D8  C3312D8

39 conjugacy classes

class 1 2A2B2C3A3B3C3D 4 6A6B6C6D8A8B12A···12H24A···24P
order12223333466668812···1224···24
size113636222222222222···22···2

39 irreducible representations

dim111222222
type+++++++++
imageC1C2C2S3D4D6D8D12D24
kernelC325D8C3×C24C12⋊S3C24C3×C6C12C32C6C3
# reps1124142816

Matrix representation of C325D8 in GL4(𝔽73) generated by

727200
1000
0001
007272
,
0100
727200
0001
007272
,
506800
55500
001868
00523
,
23500
555000
00235
005550
G:=sub<GL(4,GF(73))| [72,1,0,0,72,0,0,0,0,0,0,72,0,0,1,72],[0,72,0,0,1,72,0,0,0,0,0,72,0,0,1,72],[50,5,0,0,68,55,0,0,0,0,18,5,0,0,68,23],[23,55,0,0,5,50,0,0,0,0,23,55,0,0,5,50] >;

C325D8 in GAP, Magma, Sage, TeX

C_3^2\rtimes_5D_8
% in TeX

G:=Group("C3^2:5D8");
// GroupNames label

G:=SmallGroup(144,88);
// by ID

G=gap.SmallGroup(144,88);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-3,73,79,218,50,964,3461]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^3=c^8=d^2=1,a*b=b*a,a*c=c*a,d*a*d=a^-1,b*c=c*b,d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations

׿
×
𝔽