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G = C248D6order 288 = 25·32

8th semidirect product of C24 and D6 acting via D6/C3=C22

metabelian, supersoluble, monomial

Aliases: C248D6, (C3×D4)⋊4D6, (C3×D8)⋊4S3, D82(C3⋊S3), C242S35C2, C24⋊S35C2, C35(D8⋊S3), C6.119(S3×D4), (C32×D8)⋊7C2, C327D86C2, (C3×C24)⋊12C22, C3⋊Dic3.65D4, C12.D63C2, C329SD165C2, C3219(C8⋊C22), (C3×C12).92C23, C12.88(C22×S3), (D4×C32)⋊8C22, C324Q87C22, C324C810C22, C12⋊S3.16C22, C82(C2×C3⋊S3), D42(C2×C3⋊S3), (D4×C3⋊S3)⋊4C2, C2.16(D4×C3⋊S3), (C2×C3⋊S3).65D4, C4.2(C22×C3⋊S3), (C3×C6).240(C2×D4), (C4×C3⋊S3).23C22, SmallGroup(288,768)

Series: Derived Chief Lower central Upper central

C1C3×C12 — C248D6
C1C3C32C3×C6C3×C12C4×C3⋊S3D4×C3⋊S3 — C248D6
C32C3×C6C3×C12 — C248D6
C1C2C4D8

Generators and relations for C248D6
 G = < a,b,c | a24=b6=c2=1, bab-1=a7, cac=a11, cbc=b-1 >

Subgroups: 972 in 204 conjugacy classes, 55 normal (27 characteristic)
C1, C2, C2 [×4], C3 [×4], C4, C4 [×2], C22 [×6], S3 [×8], C6 [×4], C6 [×8], C8, C8, C2×C4 [×2], D4 [×2], D4 [×3], Q8, C23, C32, Dic3 [×8], C12 [×4], D6 [×16], C2×C6 [×8], M4(2), D8, D8, SD16 [×2], C2×D4, C4○D4, C3⋊S3 [×2], C3×C6, C3×C6 [×2], C3⋊C8 [×4], C24 [×4], Dic6 [×4], C4×S3 [×4], D12 [×4], C2×Dic3 [×4], C3⋊D4 [×8], C3×D4 [×8], C22×S3 [×4], C8⋊C22, C3⋊Dic3, C3⋊Dic3, C3×C12, C2×C3⋊S3, C2×C3⋊S3 [×3], C62 [×2], C8⋊S3 [×4], C24⋊C2 [×4], D4⋊S3 [×4], D4.S3 [×4], C3×D8 [×4], S3×D4 [×4], D42S3 [×4], C324C8, C3×C24, C324Q8, C4×C3⋊S3, C12⋊S3, C2×C3⋊Dic3, C327D4 [×2], D4×C32 [×2], C22×C3⋊S3, D8⋊S3 [×4], C24⋊S3, C242S3, C327D8, C329SD16, C32×D8, D4×C3⋊S3, C12.D6, C248D6
Quotients: C1, C2 [×7], C22 [×7], S3 [×4], D4 [×2], C23, D6 [×12], C2×D4, C3⋊S3, C22×S3 [×4], C8⋊C22, C2×C3⋊S3 [×3], S3×D4 [×4], C22×C3⋊S3, D8⋊S3 [×4], D4×C3⋊S3, C248D6

Smallest permutation representation of C248D6
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 38 53)(2 45 54 8 39 60)(3 28 55 15 40 67)(4 35 56 22 41 50)(5 42 57)(6 25 58 12 43 64)(7 32 59 19 44 71)(9 46 61)(10 29 62 16 47 68)(11 36 63 23 48 51)(13 26 65)(14 33 66 20 27 72)(17 30 69)(18 37 70 24 31 52)(21 34 49)
(1 65)(2 52)(3 63)(4 50)(5 61)(6 72)(7 59)(8 70)(9 57)(10 68)(11 55)(12 66)(13 53)(14 64)(15 51)(16 62)(17 49)(18 60)(19 71)(20 58)(21 69)(22 56)(23 67)(24 54)(25 27)(26 38)(28 36)(29 47)(30 34)(31 45)(33 43)(35 41)(37 39)(40 48)(42 46)

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

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

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

39 conjugacy classes

class 1 2A2B2C2D2E3A3B3C3D4A4B4C6A6B6C6D6E···6L8A8B12A12B12C12D24A···24H
order122222333344466666···6881212121224···24
size1144183622222183622228···843644444···4

39 irreducible representations

dim1111111122222444
type+++++++++++++++
imageC1C2C2C2C2C2C2C2S3D4D4D6D6C8⋊C22S3×D4D8⋊S3
kernelC248D6C24⋊S3C242S3C327D8C329SD16C32×D8D4×C3⋊S3C12.D6C3×D8C3⋊Dic3C2×C3⋊S3C24C3×D4C32C6C3
# reps1111111141148148

Matrix representation of C248D6 in GL6(𝔽73)

110000
7200000
00005162
00001162
0011425162
0031421162
,
110000
7200000
00727200
001000
00727211
0010720
,
0720000
7200000
001100
0007200
00117272
0007201

G:=sub<GL(6,GF(73))| [1,72,0,0,0,0,1,0,0,0,0,0,0,0,0,0,11,31,0,0,0,0,42,42,0,0,51,11,51,11,0,0,62,62,62,62],[1,72,0,0,0,0,1,0,0,0,0,0,0,0,72,1,72,1,0,0,72,0,72,0,0,0,0,0,1,72,0,0,0,0,1,0],[0,72,0,0,0,0,72,0,0,0,0,0,0,0,1,0,1,0,0,0,1,72,1,72,0,0,0,0,72,0,0,0,0,0,72,1] >;

C248D6 in GAP, Magma, Sage, TeX

C_{24}\rtimes_8D_6
% in TeX

G:=Group("C24:8D6");
// GroupNames label

G:=SmallGroup(288,768);
// by ID

G=gap.SmallGroup(288,768);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,422,135,346,185,80,2693,9414]);
// Polycyclic

G:=Group<a,b,c|a^24=b^6=c^2=1,b*a*b^-1=a^7,c*a*c=a^11,c*b*c=b^-1>;
// generators/relations

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