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## G = C3×C2.D24order 288 = 25·32

### Direct product of C3 and C2.D24

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C3×C2.D24
 Chief series C1 — C3 — C6 — C2×C6 — C2×C12 — C6×C12 — C6×D12 — C3×C2.D24
 Lower central C3 — C6 — C12 — C3×C2.D24
 Upper central C1 — C2×C6 — C2×C12 — C2×C24

Generators and relations for C3×C2.D24
G = < a,b,c,d | a3=b2=c24=1, d2=b, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=bc-1 >

Subgroups: 346 in 111 conjugacy classes, 46 normal (42 characteristic)
C1, C2 [×3], C2 [×2], C3 [×2], C3, C4 [×2], C4, C22, C22 [×4], S3 [×2], C6 [×6], C6 [×5], C8, C2×C4, C2×C4, D4 [×3], C23, C32, Dic3, C12 [×4], C12 [×3], D6 [×4], C2×C6 [×2], C2×C6 [×5], C4⋊C4, C2×C8, C2×D4, C3×S3 [×2], C3×C6 [×3], C24 [×4], D12 [×2], D12, C2×Dic3, C2×C12 [×2], C2×C12 [×2], C3×D4 [×3], C22×S3, C22×C6, D4⋊C4, C3×Dic3, C3×C12 [×2], S3×C6 [×4], C62, C4⋊Dic3, C3×C4⋊C4, C2×C24 [×2], C2×C24, C2×D12, C6×D4, C3×C24, C3×D12 [×2], C3×D12, C6×Dic3, C6×C12, S3×C2×C6, C2.D24, C3×D4⋊C4, C3×C4⋊Dic3, C6×C24, C6×D12, C3×C2.D24
Quotients: C1, C2 [×3], C3, C4 [×2], C22, S3, C6 [×3], C2×C4, D4 [×2], C12 [×2], D6, C2×C6, C22⋊C4, D8, SD16, C3×S3, C4×S3, D12, C3⋊D4, C2×C12, C3×D4 [×2], D4⋊C4, S3×C6, C24⋊C2, D24, D6⋊C4, C3×C22⋊C4, C3×D8, C3×SD16, S3×C12, C3×D12, C3×C3⋊D4, C2.D24, C3×D4⋊C4, C3×C24⋊C2, C3×D24, C3×D6⋊C4, C3×C2.D24

Smallest permutation representation of C3×C2.D24
On 96 points
Generators in S96
(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 41 33)(26 42 34)(27 43 35)(28 44 36)(29 45 37)(30 46 38)(31 47 39)(32 48 40)(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 81 89)(74 82 90)(75 83 91)(76 84 92)(77 85 93)(78 86 94)(79 87 95)(80 88 96)
(1 29)(2 30)(3 31)(4 32)(5 33)(6 34)(7 35)(8 36)(9 37)(10 38)(11 39)(12 40)(13 41)(14 42)(15 43)(16 44)(17 45)(18 46)(19 47)(20 48)(21 25)(22 26)(23 27)(24 28)(49 74)(50 75)(51 76)(52 77)(53 78)(54 79)(55 80)(56 81)(57 82)(58 83)(59 84)(60 85)(61 86)(62 87)(63 88)(64 89)(65 90)(66 91)(67 92)(68 93)(69 94)(70 95)(71 96)(72 73)
(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 55 29 80)(2 79 30 54)(3 53 31 78)(4 77 32 52)(5 51 33 76)(6 75 34 50)(7 49 35 74)(8 73 36 72)(9 71 37 96)(10 95 38 70)(11 69 39 94)(12 93 40 68)(13 67 41 92)(14 91 42 66)(15 65 43 90)(16 89 44 64)(17 63 45 88)(18 87 46 62)(19 61 47 86)(20 85 48 60)(21 59 25 84)(22 83 26 58)(23 57 27 82)(24 81 28 56)

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

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

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

90 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A 3B 3C 3D 3E 4A 4B 4C 4D 6A ··· 6F 6G ··· 6O 6P 6Q 6R 6S 8A 8B 8C 8D 12A ··· 12P 12Q 12R 12S 12T 24A ··· 24AF order 1 2 2 2 2 2 3 3 3 3 3 4 4 4 4 6 ··· 6 6 ··· 6 6 6 6 6 8 8 8 8 12 ··· 12 12 12 12 12 24 ··· 24 size 1 1 1 1 12 12 1 1 2 2 2 2 2 12 12 1 ··· 1 2 ··· 2 12 12 12 12 2 2 2 2 2 ··· 2 12 12 12 12 2 ··· 2

90 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + + + + image C1 C2 C2 C2 C3 C4 C6 C6 C6 C12 S3 D4 D4 D6 D8 SD16 C3×S3 C4×S3 C3⋊D4 C3×D4 D12 C3×D4 S3×C6 C24⋊C2 D24 C3×D8 C3×SD16 S3×C12 C3×C3⋊D4 C3×D12 C3×C24⋊C2 C3×D24 kernel C3×C2.D24 C3×C4⋊Dic3 C6×C24 C6×D12 C2.D24 C3×D12 C4⋊Dic3 C2×C24 C2×D12 D12 C2×C24 C3×C12 C62 C2×C12 C3×C6 C3×C6 C2×C8 C12 C12 C12 C2×C6 C2×C6 C2×C4 C6 C6 C6 C6 C4 C4 C22 C2 C2 # reps 1 1 1 1 2 4 2 2 2 8 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 8 8

Matrix representation of C3×C2.D24 in GL3(𝔽73) generated by

 64 0 0 0 64 0 0 0 64
,
 72 0 0 0 1 0 0 0 1
,
 46 0 0 0 21 51 0 0 7
,
 27 0 0 0 35 10 0 9 38
G:=sub<GL(3,GF(73))| [64,0,0,0,64,0,0,0,64],[72,0,0,0,1,0,0,0,1],[46,0,0,0,21,0,0,51,7],[27,0,0,0,35,9,0,10,38] >;

C3×C2.D24 in GAP, Magma, Sage, TeX

C_3\times C_2.D_{24}
% in TeX

G:=Group("C3xC2.D24");
// GroupNames label

G:=SmallGroup(288,255);
// by ID

G=gap.SmallGroup(288,255);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-3,197,260,1683,136,9414]);
// Polycyclic

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

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