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G = D24.S3order 288 = 25·32

2nd non-split extension by D24 of S3 acting via S3/C3=C2

metabelian, supersoluble, monomial

Aliases: D24.2S3, C24.15D6, C322SD32, Dic123S3, C8.14S32, (C3×C6).8D8, C6.9(D4⋊S3), C32(D8.S3), (C3×D24).2C2, (C3×C12).23D4, C24.S32C2, (C3×Dic12)⋊5C2, C32(C8.6D6), (C3×C24).8C22, C4.2(D6⋊S3), C12.22(C3⋊D4), C2.4(C322D8), SmallGroup(288,195)

Series: Derived Chief Lower central Upper central

C1C3×C24 — D24.S3
C1C3C32C3×C6C3×C12C3×C24C3×D24 — D24.S3
C32C3×C6C3×C12C3×C24 — D24.S3
C1C2C4C8

Generators and relations for D24.S3
 G = < a,b,c,d | a24=b2=c3=1, d2=a12, bab=a-1, ac=ca, dad-1=a7, bc=cb, dbd-1=a15b, dcd-1=c-1 >

Subgroups: 238 in 57 conjugacy classes, 22 normal (all characteristic)
C1, C2, C2, C3 [×2], C3, C4, C4, C22, S3, C6 [×2], C6 [×2], C8, D4, Q8, C32, Dic3, C12 [×2], C12 [×2], D6, C2×C6, C16, D8, Q16, C3×S3, C3×C6, C24 [×2], C24, Dic6, D12, C3×D4, C3×Q8, SD32, C3×Dic3, C3×C12, S3×C6, C3⋊C16 [×3], D24, Dic12, C3×D8, C3×Q16, C3×C24, C3×Dic6, C3×D12, D8.S3, C8.6D6, C24.S3, C3×D24, C3×Dic12, D24.S3
Quotients: C1, C2 [×3], C22, S3 [×2], D4, D6 [×2], D8, C3⋊D4 [×2], SD32, S32, D4⋊S3 [×2], D6⋊S3, D8.S3, C8.6D6, C322D8, D24.S3

Smallest permutation representation of D24.S3
On 96 points
Generators in S96
(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)(73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96)
(1 57)(2 56)(3 55)(4 54)(5 53)(6 52)(7 51)(8 50)(9 49)(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 89)(26 88)(27 87)(28 86)(29 85)(30 84)(31 83)(32 82)(33 81)(34 80)(35 79)(36 78)(37 77)(38 76)(39 75)(40 74)(41 73)(42 96)(43 95)(44 94)(45 93)(46 92)(47 91)(48 90)
(1 17 9)(2 18 10)(3 19 11)(4 20 12)(5 21 13)(6 22 14)(7 23 15)(8 24 16)(25 33 41)(26 34 42)(27 35 43)(28 36 44)(29 37 45)(30 38 46)(31 39 47)(32 40 48)(49 57 65)(50 58 66)(51 59 67)(52 60 68)(53 61 69)(54 62 70)(55 63 71)(56 64 72)(73 89 81)(74 90 82)(75 91 83)(76 92 84)(77 93 85)(78 94 86)(79 95 87)(80 96 88)
(1 45 13 33)(2 28 14 40)(3 35 15 47)(4 42 16 30)(5 25 17 37)(6 32 18 44)(7 39 19 27)(8 46 20 34)(9 29 21 41)(10 36 22 48)(11 43 23 31)(12 26 24 38)(49 94 61 82)(50 77 62 89)(51 84 63 96)(52 91 64 79)(53 74 65 86)(54 81 66 93)(55 88 67 76)(56 95 68 83)(57 78 69 90)(58 85 70 73)(59 92 71 80)(60 75 72 87)

G:=sub<Sym(96)| (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)(73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96), (1,57)(2,56)(3,55)(4,54)(5,53)(6,52)(7,51)(8,50)(9,49)(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)(25,89)(26,88)(27,87)(28,86)(29,85)(30,84)(31,83)(32,82)(33,81)(34,80)(35,79)(36,78)(37,77)(38,76)(39,75)(40,74)(41,73)(42,96)(43,95)(44,94)(45,93)(46,92)(47,91)(48,90), (1,17,9)(2,18,10)(3,19,11)(4,20,12)(5,21,13)(6,22,14)(7,23,15)(8,24,16)(25,33,41)(26,34,42)(27,35,43)(28,36,44)(29,37,45)(30,38,46)(31,39,47)(32,40,48)(49,57,65)(50,58,66)(51,59,67)(52,60,68)(53,61,69)(54,62,70)(55,63,71)(56,64,72)(73,89,81)(74,90,82)(75,91,83)(76,92,84)(77,93,85)(78,94,86)(79,95,87)(80,96,88), (1,45,13,33)(2,28,14,40)(3,35,15,47)(4,42,16,30)(5,25,17,37)(6,32,18,44)(7,39,19,27)(8,46,20,34)(9,29,21,41)(10,36,22,48)(11,43,23,31)(12,26,24,38)(49,94,61,82)(50,77,62,89)(51,84,63,96)(52,91,64,79)(53,74,65,86)(54,81,66,93)(55,88,67,76)(56,95,68,83)(57,78,69,90)(58,85,70,73)(59,92,71,80)(60,75,72,87)>;

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)(73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96), (1,57)(2,56)(3,55)(4,54)(5,53)(6,52)(7,51)(8,50)(9,49)(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)(25,89)(26,88)(27,87)(28,86)(29,85)(30,84)(31,83)(32,82)(33,81)(34,80)(35,79)(36,78)(37,77)(38,76)(39,75)(40,74)(41,73)(42,96)(43,95)(44,94)(45,93)(46,92)(47,91)(48,90), (1,17,9)(2,18,10)(3,19,11)(4,20,12)(5,21,13)(6,22,14)(7,23,15)(8,24,16)(25,33,41)(26,34,42)(27,35,43)(28,36,44)(29,37,45)(30,38,46)(31,39,47)(32,40,48)(49,57,65)(50,58,66)(51,59,67)(52,60,68)(53,61,69)(54,62,70)(55,63,71)(56,64,72)(73,89,81)(74,90,82)(75,91,83)(76,92,84)(77,93,85)(78,94,86)(79,95,87)(80,96,88), (1,45,13,33)(2,28,14,40)(3,35,15,47)(4,42,16,30)(5,25,17,37)(6,32,18,44)(7,39,19,27)(8,46,20,34)(9,29,21,41)(10,36,22,48)(11,43,23,31)(12,26,24,38)(49,94,61,82)(50,77,62,89)(51,84,63,96)(52,91,64,79)(53,74,65,86)(54,81,66,93)(55,88,67,76)(56,95,68,83)(57,78,69,90)(58,85,70,73)(59,92,71,80)(60,75,72,87) );

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),(73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96)], [(1,57),(2,56),(3,55),(4,54),(5,53),(6,52),(7,51),(8,50),(9,49),(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),(25,89),(26,88),(27,87),(28,86),(29,85),(30,84),(31,83),(32,82),(33,81),(34,80),(35,79),(36,78),(37,77),(38,76),(39,75),(40,74),(41,73),(42,96),(43,95),(44,94),(45,93),(46,92),(47,91),(48,90)], [(1,17,9),(2,18,10),(3,19,11),(4,20,12),(5,21,13),(6,22,14),(7,23,15),(8,24,16),(25,33,41),(26,34,42),(27,35,43),(28,36,44),(29,37,45),(30,38,46),(31,39,47),(32,40,48),(49,57,65),(50,58,66),(51,59,67),(52,60,68),(53,61,69),(54,62,70),(55,63,71),(56,64,72),(73,89,81),(74,90,82),(75,91,83),(76,92,84),(77,93,85),(78,94,86),(79,95,87),(80,96,88)], [(1,45,13,33),(2,28,14,40),(3,35,15,47),(4,42,16,30),(5,25,17,37),(6,32,18,44),(7,39,19,27),(8,46,20,34),(9,29,21,41),(10,36,22,48),(11,43,23,31),(12,26,24,38),(49,94,61,82),(50,77,62,89),(51,84,63,96),(52,91,64,79),(53,74,65,86),(54,81,66,93),(55,88,67,76),(56,95,68,83),(57,78,69,90),(58,85,70,73),(59,92,71,80),(60,75,72,87)])

33 conjugacy classes

class 1 2A2B3A3B3C4A4B6A6B6C6D6E8A8B12A12B12C12D12E12F16A16B16C16D24A···24H
order1223334466666881212121212121616161624···24
size112422422422424242244442424181818184···4

33 irreducible representations

dim111122222224444444
type+++++++++++--+
imageC1C2C2C2S3S3D4D6D8C3⋊D4SD32S32D4⋊S3D6⋊S3D8.S3C8.6D6C322D8D24.S3
kernelD24.S3C24.S3C3×D24C3×Dic12D24Dic12C3×C12C24C3×C6C12C32C8C6C4C3C3C2C1
# reps111111122441212224

Matrix representation of D24.S3 in GL6(𝔽97)

9070000
90900000
00969600
001000
000010
000001
,
48920000
92490000
00969600
000100
000010
000001
,
100000
010000
001000
000100
0000961
0000960
,
21250000
25760000
0096000
0009600
000001
000010

G:=sub<GL(6,GF(97))| [90,90,0,0,0,0,7,90,0,0,0,0,0,0,96,1,0,0,0,0,96,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[48,92,0,0,0,0,92,49,0,0,0,0,0,0,96,0,0,0,0,0,96,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,96,96,0,0,0,0,1,0],[21,25,0,0,0,0,25,76,0,0,0,0,0,0,96,0,0,0,0,0,0,96,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

D24.S3 in GAP, Magma, Sage, TeX

D_{24}.S_3
% in TeX

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

G:=SmallGroup(288,195);
// by ID

G=gap.SmallGroup(288,195);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,85,120,254,135,142,675,346,80,1356,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^24=b^2=c^3=1,d^2=a^12,b*a*b=a^-1,a*c=c*a,d*a*d^-1=a^7,b*c=c*b,d*b*d^-1=a^15*b,d*c*d^-1=c^-1>;
// generators/relations

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