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## G = C3×D24⋊C2order 288 = 25·32

### Direct product of C3 and D24⋊C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C3×D24⋊C2
 Chief series C1 — C3 — C6 — C12 — C3×C12 — S3×C12 — C3×Q8⋊3S3 — C3×D24⋊C2
 Lower central C3 — C6 — C12 — C3×D24⋊C2
 Upper central C1 — C6 — C12 — C3×Q16

Generators and relations for C3×D24⋊C2
G = < a,b,c,d | a3=b24=c2=d2=1, ab=ba, ac=ca, ad=da, cbc=b-1, dbd=b17, dcd=b4c >

Subgroups: 346 in 133 conjugacy classes, 54 normal (34 characteristic)
C1, C2, C2 [×3], C3 [×2], C3, C4, C4 [×3], C22 [×3], S3 [×3], C6 [×2], C6 [×4], C8, C8, C2×C4 [×3], D4 [×4], Q8 [×2], C32, Dic3, C12 [×2], C12 [×8], D6, D6 [×2], C2×C6 [×3], C2×C8, D8, SD16 [×2], Q16, C4○D4 [×2], C3×S3 [×3], C3×C6, C3⋊C8, C24 [×2], C24 [×2], C4×S3, C4×S3 [×2], D12 [×2], D12 [×2], C2×C12 [×3], C3×D4 [×4], C3×Q8 [×4], C3×Q8 [×2], C4○D8, C3×Dic3, C3×C12, C3×C12 [×2], S3×C6, S3×C6 [×2], S3×C8, D24, Q82S3 [×2], C2×C24, C3×D8, C3×SD16 [×2], C3×Q16 [×2], C3×Q16, Q83S3 [×2], C3×C4○D4 [×2], C3×C3⋊C8, C3×C24, S3×C12, S3×C12 [×2], C3×D12 [×2], C3×D12 [×2], Q8×C32 [×2], D24⋊C2, C3×C4○D8, S3×C24, C3×D24, C3×Q82S3 [×2], C32×Q16, C3×Q83S3 [×2], C3×D24⋊C2
Quotients: C1, C2 [×7], C3, C22 [×7], S3, C6 [×7], D4 [×2], C23, D6 [×3], C2×C6 [×7], C2×D4, C3×S3, C3×D4 [×2], C22×S3, C22×C6, C4○D8, S3×C6 [×3], S3×D4, C6×D4, S3×C2×C6, D24⋊C2, C3×C4○D8, C3×S3×D4, C3×D24⋊C2

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

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

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

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

63 conjugacy classes

 class 1 2A 2B 2C 2D 3A 3B 3C 3D 3E 4A 4B 4C 4D 4E 6A 6B 6C 6D 6E 6F 6G 6H 6I 6J 6K 8A 8B 8C 8D 12A 12B 12C 12D 12E 12F 12G ··· 12M 12N ··· 12S 24A 24B 24C 24D 24E ··· 24J 24K 24L 24M 24N order 1 2 2 2 2 3 3 3 3 3 4 4 4 4 4 6 6 6 6 6 6 6 6 6 6 6 8 8 8 8 12 12 12 12 12 12 12 ··· 12 12 ··· 12 24 24 24 24 24 ··· 24 24 24 24 24 size 1 1 6 12 12 1 1 2 2 2 2 3 3 4 4 1 1 2 2 2 6 6 12 12 12 12 2 2 6 6 2 2 3 3 3 3 4 ··· 4 8 ··· 8 2 2 2 2 4 ··· 4 6 6 6 6

63 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C3 C6 C6 C6 C6 C6 S3 D4 D4 D6 D6 C3×S3 C3×D4 C3×D4 C4○D8 S3×C6 S3×C6 C3×C4○D8 S3×D4 D24⋊C2 C3×S3×D4 C3×D24⋊C2 kernel C3×D24⋊C2 S3×C24 C3×D24 C3×Q8⋊2S3 C32×Q16 C3×Q8⋊3S3 D24⋊C2 S3×C8 D24 Q8⋊2S3 C3×Q16 Q8⋊3S3 C3×Q16 C3×Dic3 S3×C6 C24 C3×Q8 Q16 Dic3 D6 C32 C8 Q8 C3 C6 C3 C2 C1 # reps 1 1 1 2 1 2 2 2 2 4 2 4 1 1 1 1 2 2 2 2 4 2 4 8 1 2 2 4

Matrix representation of C3×D24⋊C2 in GL4(𝔽7) generated by

 2 0 0 0 0 2 0 0 0 0 2 0 0 0 0 2
,
 0 3 0 6 5 6 6 1 2 2 1 3 4 6 6 4
,
 4 2 4 6 4 5 1 2 1 1 3 4 6 1 2 2
,
 3 1 5 6 4 3 6 4 1 4 5 6 3 5 4 3
G:=sub<GL(4,GF(7))| [2,0,0,0,0,2,0,0,0,0,2,0,0,0,0,2],[0,5,2,4,3,6,2,6,0,6,1,6,6,1,3,4],[4,4,1,6,2,5,1,1,4,1,3,2,6,2,4,2],[3,4,1,3,1,3,4,5,5,6,5,4,6,4,6,3] >;

C3×D24⋊C2 in GAP, Magma, Sage, TeX

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

G:=Group("C3xD24:C2");
// GroupNames label

G:=SmallGroup(288,690);
// by ID

G=gap.SmallGroup(288,690);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,701,344,1094,303,268,1271,648,102,9414]);
// Polycyclic

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

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