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## G = S3×C2×C24order 288 = 25·32

### Direct product of C2×C24 and S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — S3×C2×C24
 Chief series C1 — C3 — C6 — C12 — C3×C12 — S3×C12 — S3×C2×C12 — S3×C2×C24
 Lower central C3 — S3×C2×C24
 Upper central C1 — C2×C24

Generators and relations for S3×C2×C24
G = < a,b,c,d | a2=b24=c3=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 282 in 163 conjugacy classes, 98 normal (38 characteristic)
C1, C2, C2 [×2], C2 [×4], C3 [×2], C3, C4 [×2], C4 [×2], C22, C22 [×6], S3 [×4], C6 [×2], C6 [×4], C6 [×7], C8 [×2], C8 [×2], C2×C4, C2×C4 [×5], C23, C32, Dic3 [×2], C12 [×4], C12 [×4], D6 [×6], C2×C6 [×2], C2×C6 [×7], C2×C8, C2×C8 [×5], C22×C4, C3×S3 [×4], C3×C6, C3×C6 [×2], C3⋊C8 [×2], C24 [×4], C24 [×4], C4×S3 [×4], C2×Dic3, C2×C12 [×2], C2×C12 [×6], C22×S3, C22×C6, C22×C8, C3×Dic3 [×2], C3×C12 [×2], S3×C6 [×6], C62, S3×C8 [×4], C2×C3⋊C8, C2×C24 [×2], C2×C24 [×6], S3×C2×C4, C22×C12, C3×C3⋊C8 [×2], C3×C24 [×2], S3×C12 [×4], C6×Dic3, C6×C12, S3×C2×C6, S3×C2×C8, C22×C24, S3×C24 [×4], C6×C3⋊C8, C6×C24, S3×C2×C12, S3×C2×C24
Quotients: C1, C2 [×7], C3, C4 [×4], C22 [×7], S3, C6 [×7], C8 [×4], C2×C4 [×6], C23, C12 [×4], D6 [×3], C2×C6 [×7], C2×C8 [×6], C22×C4, C3×S3, C24 [×4], C4×S3 [×2], C2×C12 [×6], C22×S3, C22×C6, C22×C8, S3×C6 [×3], S3×C8 [×2], C2×C24 [×6], S3×C2×C4, C22×C12, S3×C12 [×2], S3×C2×C6, S3×C2×C8, C22×C24, S3×C24 [×2], S3×C2×C12, S3×C2×C24

Smallest permutation representation of S3×C2×C24
On 96 points
Generators in S96
(1 30)(2 31)(3 32)(4 33)(5 34)(6 35)(7 36)(8 37)(9 38)(10 39)(11 40)(12 41)(13 42)(14 43)(15 44)(16 45)(17 46)(18 47)(19 48)(20 25)(21 26)(22 27)(23 28)(24 29)(49 76)(50 77)(51 78)(52 79)(53 80)(54 81)(55 82)(56 83)(57 84)(58 85)(59 86)(60 87)(61 88)(62 89)(63 90)(64 91)(65 92)(66 93)(67 94)(68 95)(69 96)(70 73)(71 74)(72 75)
(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 9 17)(2 10 18)(3 11 19)(4 12 20)(5 13 21)(6 14 22)(7 15 23)(8 16 24)(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 65 57)(50 66 58)(51 67 59)(52 68 60)(53 69 61)(54 70 62)(55 71 63)(56 72 64)(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 76)(2 77)(3 78)(4 79)(5 80)(6 81)(7 82)(8 83)(9 84)(10 85)(11 86)(12 87)(13 88)(14 89)(15 90)(16 91)(17 92)(18 93)(19 94)(20 95)(21 96)(22 73)(23 74)(24 75)(25 68)(26 69)(27 70)(28 71)(29 72)(30 49)(31 50)(32 51)(33 52)(34 53)(35 54)(36 55)(37 56)(38 57)(39 58)(40 59)(41 60)(42 61)(43 62)(44 63)(45 64)(46 65)(47 66)(48 67)

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

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

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

144 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C 3D 3E 4A 4B 4C 4D 4E 4F 4G 4H 6A ··· 6F 6G ··· 6O 6P ··· 6W 8A ··· 8H 8I ··· 8P 12A ··· 12H 12I ··· 12T 12U ··· 12AB 24A ··· 24P 24Q ··· 24AN 24AO ··· 24BD order 1 2 2 2 2 2 2 2 3 3 3 3 3 4 4 4 4 4 4 4 4 6 ··· 6 6 ··· 6 6 ··· 6 8 ··· 8 8 ··· 8 12 ··· 12 12 ··· 12 12 ··· 12 24 ··· 24 24 ··· 24 24 ··· 24 size 1 1 1 1 3 3 3 3 1 1 2 2 2 1 1 1 1 3 3 3 3 1 ··· 1 2 ··· 2 3 ··· 3 1 ··· 1 3 ··· 3 1 ··· 1 2 ··· 2 3 ··· 3 1 ··· 1 2 ··· 2 3 ··· 3

144 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + image C1 C2 C2 C2 C2 C3 C4 C4 C4 C6 C6 C6 C6 C8 C12 C12 C12 C24 S3 D6 D6 C3×S3 C4×S3 C4×S3 S3×C6 S3×C6 S3×C8 S3×C12 S3×C12 S3×C24 kernel S3×C2×C24 S3×C24 C6×C3⋊C8 C6×C24 S3×C2×C12 S3×C2×C8 S3×C12 C6×Dic3 S3×C2×C6 S3×C8 C2×C3⋊C8 C2×C24 S3×C2×C4 S3×C6 C4×S3 C2×Dic3 C22×S3 D6 C2×C24 C24 C2×C12 C2×C8 C12 C2×C6 C8 C2×C4 C6 C4 C22 C2 # reps 1 4 1 1 1 2 4 2 2 8 2 2 2 16 8 4 4 32 1 2 1 2 2 2 4 2 8 4 4 16

Matrix representation of S3×C2×C24 in GL4(𝔽73) generated by

 1 0 0 0 0 72 0 0 0 0 1 0 0 0 0 1
,
 21 0 0 0 0 49 0 0 0 0 70 0 0 0 0 70
,
 1 0 0 0 0 1 0 0 0 0 64 0 0 0 8 8
,
 1 0 0 0 0 1 0 0 0 0 72 7 0 0 0 1
G:=sub<GL(4,GF(73))| [1,0,0,0,0,72,0,0,0,0,1,0,0,0,0,1],[21,0,0,0,0,49,0,0,0,0,70,0,0,0,0,70],[1,0,0,0,0,1,0,0,0,0,64,8,0,0,0,8],[1,0,0,0,0,1,0,0,0,0,72,0,0,0,7,1] >;

S3×C2×C24 in GAP, Magma, Sage, TeX

S_3\times C_2\times C_{24}
% in TeX

G:=Group("S3xC2xC24");
// GroupNames label

G:=SmallGroup(288,670);
// by ID

G=gap.SmallGroup(288,670);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,142,102,9414]);
// Polycyclic

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

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