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G = D245S3order 288 = 25·32

5th semidirect product of D24 and S3 acting via S3/C3=C2

metabelian, supersoluble, monomial

Aliases: D245S3, D12.3D6, C24.25D6, Dic125S3, Dic6.3D6, C8.17S32, C6.33(S3×D4), (C3×D24)⋊11C2, C327(C4○D8), D12⋊S33C2, C32(D83S3), C3⋊Dic3.43D4, C32(D24⋊C2), (C3×Dic12)⋊11C2, Dic6⋊S33C2, C12.75(C22×S3), (C3×C12).55C23, (C3×C24).25C22, (C3×D12).8C22, C2.10(D6⋊D6), (C3×Dic6).8C22, C324C8.23C22, (C8×C3⋊S3)⋊2C2, C4.70(C2×S32), (C2×C3⋊S3).44D4, (C3×C6).39(C2×D4), (C4×C3⋊S3).69C22, SmallGroup(288,458)

Series: Derived Chief Lower central Upper central

C1C3×C12 — D245S3
C1C3C32C3×C6C3×C12C3×D12D12⋊S3 — D245S3
C32C3×C6C3×C12 — D245S3
C1C2C4C8

Generators and relations for D245S3
 G = < a,b,c,d | a24=b2=c3=d2=1, bab=a-1, ac=ca, dad=a17, bc=cb, dbd=a4b, dcd=c-1 >

Subgroups: 562 in 135 conjugacy classes, 38 normal (26 characteristic)
C1, C2, C2 [×3], C3 [×2], C3, C4, C4 [×3], C22 [×3], S3 [×6], C6 [×2], C6 [×3], C8, C8, C2×C4 [×3], D4 [×4], Q8 [×2], C32, Dic3 [×5], C12 [×2], C12 [×3], D6 [×5], C2×C6 [×2], C2×C8, D8, SD16 [×2], Q16, C4○D4 [×2], C3×S3 [×2], C3⋊S3, C3×C6, C3⋊C8 [×3], C24 [×2], C24, Dic6 [×2], C4×S3 [×5], D12 [×2], D12 [×2], C2×Dic3 [×2], C3⋊D4 [×2], C3×D4 [×2], C3×Q8 [×2], C4○D8, C3×Dic3 [×2], C3⋊Dic3, C3×C12, S3×C6 [×2], C2×C3⋊S3, S3×C8 [×3], D24, Dic12, D4.S3 [×2], Q82S3 [×2], C3×D8, C3×Q16, D42S3 [×2], Q83S3 [×2], C324C8, C3×C24, S3×Dic3 [×2], C3⋊D12 [×2], C3×Dic6 [×2], C3×D12 [×2], C4×C3⋊S3, D83S3, D24⋊C2, Dic6⋊S3 [×2], C3×D24, C3×Dic12, C8×C3⋊S3, D12⋊S3 [×2], D245S3
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], D4 [×2], C23, D6 [×6], C2×D4, C22×S3 [×2], C4○D8, S32, S3×D4 [×2], C2×S32, D83S3, D24⋊C2, D6⋊D6, D245S3

Smallest permutation representation of D245S3
On 48 points
Generators in S48
(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)
(1 38)(2 37)(3 36)(4 35)(5 34)(6 33)(7 32)(8 31)(9 30)(10 29)(11 28)(12 27)(13 26)(14 25)(15 48)(16 47)(17 46)(18 45)(19 44)(20 43)(21 42)(22 41)(23 40)(24 39)
(1 9 17)(2 10 18)(3 11 19)(4 12 20)(5 13 21)(6 14 22)(7 15 23)(8 16 24)(25 41 33)(26 42 34)(27 43 35)(28 44 36)(29 45 37)(30 46 38)(31 47 39)(32 48 40)
(2 18)(3 11)(5 21)(6 14)(8 24)(9 17)(12 20)(15 23)(25 29)(26 46)(27 39)(28 32)(30 42)(31 35)(33 45)(34 38)(36 48)(37 41)(40 44)(43 47)

G:=sub<Sym(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), (1,38)(2,37)(3,36)(4,35)(5,34)(6,33)(7,32)(8,31)(9,30)(10,29)(11,28)(12,27)(13,26)(14,25)(15,48)(16,47)(17,46)(18,45)(19,44)(20,43)(21,42)(22,41)(23,40)(24,39), (1,9,17)(2,10,18)(3,11,19)(4,12,20)(5,13,21)(6,14,22)(7,15,23)(8,16,24)(25,41,33)(26,42,34)(27,43,35)(28,44,36)(29,45,37)(30,46,38)(31,47,39)(32,48,40), (2,18)(3,11)(5,21)(6,14)(8,24)(9,17)(12,20)(15,23)(25,29)(26,46)(27,39)(28,32)(30,42)(31,35)(33,45)(34,38)(36,48)(37,41)(40,44)(43,47)>;

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), (1,38)(2,37)(3,36)(4,35)(5,34)(6,33)(7,32)(8,31)(9,30)(10,29)(11,28)(12,27)(13,26)(14,25)(15,48)(16,47)(17,46)(18,45)(19,44)(20,43)(21,42)(22,41)(23,40)(24,39), (1,9,17)(2,10,18)(3,11,19)(4,12,20)(5,13,21)(6,14,22)(7,15,23)(8,16,24)(25,41,33)(26,42,34)(27,43,35)(28,44,36)(29,45,37)(30,46,38)(31,47,39)(32,48,40), (2,18)(3,11)(5,21)(6,14)(8,24)(9,17)(12,20)(15,23)(25,29)(26,46)(27,39)(28,32)(30,42)(31,35)(33,45)(34,38)(36,48)(37,41)(40,44)(43,47) );

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)], [(1,38),(2,37),(3,36),(4,35),(5,34),(6,33),(7,32),(8,31),(9,30),(10,29),(11,28),(12,27),(13,26),(14,25),(15,48),(16,47),(17,46),(18,45),(19,44),(20,43),(21,42),(22,41),(23,40),(24,39)], [(1,9,17),(2,10,18),(3,11,19),(4,12,20),(5,13,21),(6,14,22),(7,15,23),(8,16,24),(25,41,33),(26,42,34),(27,43,35),(28,44,36),(29,45,37),(30,46,38),(31,47,39),(32,48,40)], [(2,18),(3,11),(5,21),(6,14),(8,24),(9,17),(12,20),(15,23),(25,29),(26,46),(27,39),(28,32),(30,42),(31,35),(33,45),(34,38),(36,48),(37,41),(40,44),(43,47)])

36 conjugacy classes

class 1 2A2B2C2D3A3B3C4A4B4C4D4E6A6B6C6D6E8A8B8C8D12A12B12C12D12E12F24A···24H
order122223334444466666888812121212121224···24
size1112121822429912122242424221818444424244···4

36 irreducible representations

dim111111222222224444444
type++++++++++++++++-+
imageC1C2C2C2C2C2S3S3D4D4D6D6D6C4○D8S32S3×D4C2×S32D83S3D24⋊C2D6⋊D6D245S3
kernelD245S3Dic6⋊S3C3×D24C3×Dic12C8×C3⋊S3D12⋊S3D24Dic12C3⋊Dic3C2×C3⋊S3C24Dic6D12C32C8C6C4C3C3C2C1
# reps121112111122241212224

Matrix representation of D245S3 in GL6(𝔽73)

2200000
18100000
000100
0072100
000010
000001
,
51390000
55220000
0017200
0007200
000010
000001
,
100000
010000
001000
000100
0000721
0000720
,
100000
3720000
000100
001000
000001
000010

G:=sub<GL(6,GF(73))| [22,18,0,0,0,0,0,10,0,0,0,0,0,0,0,72,0,0,0,0,1,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[51,55,0,0,0,0,39,22,0,0,0,0,0,0,1,0,0,0,0,0,72,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,1,0],[1,3,0,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

D245S3 in GAP, Magma, Sage, TeX

D_{24}\rtimes_5S_3
% in TeX

G:=Group("D24:5S3");
// GroupNames label

G:=SmallGroup(288,458);
// by ID

G=gap.SmallGroup(288,458);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,120,254,135,142,675,346,80,1356,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^24=b^2=c^3=d^2=1,b*a*b=a^-1,a*c=c*a,d*a*d=a^17,b*c=c*b,d*b*d=a^4*b,d*c*d=c^-1>;
// generators/relations

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