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## G = C3×C24⋊1C4order 288 = 25·32

### Direct product of C3 and C24⋊1C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C3×C24⋊1C4
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C6×C12 — C3×C4⋊Dic3 — C3×C24⋊1C4
 Lower central C3 — C6 — C12 — C3×C24⋊1C4
 Upper central C1 — C2×C6 — C2×C12 — C2×C24

Generators and relations for C3×C241C4
G = < a,b,c | a3=b24=c4=1, ab=ba, ac=ca, cbc-1=b-1 >

Subgroups: 186 in 83 conjugacy classes, 50 normal (38 characteristic)
C1, C2 [×3], C3 [×2], C3, C4 [×2], C4 [×2], C22, C6 [×6], C6 [×3], C8 [×2], C2×C4, C2×C4 [×2], C32, Dic3 [×2], C12 [×4], C12 [×4], C2×C6 [×2], C2×C6, C4⋊C4 [×2], C2×C8, C3×C6 [×3], C24 [×4], C24 [×2], C2×Dic3 [×2], C2×C12 [×2], C2×C12 [×3], C2.D8, C3×Dic3 [×2], C3×C12 [×2], C62, C4⋊Dic3 [×2], C3×C4⋊C4 [×2], C2×C24 [×2], C2×C24, C3×C24 [×2], C6×Dic3 [×2], C6×C12, C241C4, C3×C2.D8, C3×C4⋊Dic3 [×2], C6×C24, C3×C241C4
Quotients: C1, C2 [×3], C3, C4 [×2], C22, S3, C6 [×3], C2×C4, D4, Q8, Dic3 [×2], C12 [×2], D6, C2×C6, C4⋊C4, D8, Q16, C3×S3, Dic6, D12, C2×Dic3, C2×C12, C3×D4, C3×Q8, C2.D8, C3×Dic3 [×2], S3×C6, D24, Dic12, C4⋊Dic3, C3×C4⋊C4, C3×D8, C3×Q16, C3×Dic6, C3×D12, C6×Dic3, C241C4, C3×C2.D8, C3×D24, C3×Dic12, C3×C4⋊Dic3, C3×C241C4

Smallest permutation representation of C3×C241C4
On 96 points
Generators in S96
(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)(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 54 88 43)(2 53 89 42)(3 52 90 41)(4 51 91 40)(5 50 92 39)(6 49 93 38)(7 72 94 37)(8 71 95 36)(9 70 96 35)(10 69 73 34)(11 68 74 33)(12 67 75 32)(13 66 76 31)(14 65 77 30)(15 64 78 29)(16 63 79 28)(17 62 80 27)(18 61 81 26)(19 60 82 25)(20 59 83 48)(21 58 84 47)(22 57 85 46)(23 56 86 45)(24 55 87 44)

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

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

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

90 conjugacy classes

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

90 irreducible representations

 dim 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 C3 C4 C6 C6 C12 S3 Q8 D4 Dic3 D6 D8 Q16 C3×S3 Dic6 C3×Q8 D12 C3×D4 C3×Dic3 S3×C6 D24 Dic12 C3×D8 C3×Q16 C3×Dic6 C3×D12 C3×D24 C3×Dic12 kernel C3×C24⋊1C4 C3×C4⋊Dic3 C6×C24 C24⋊1C4 C3×C24 C4⋊Dic3 C2×C24 C24 C2×C24 C3×C12 C62 C24 C2×C12 C3×C6 C3×C6 C2×C8 C12 C12 C2×C6 C2×C6 C8 C2×C4 C6 C6 C6 C6 C4 C22 C2 C2 # reps 1 2 1 2 4 4 2 8 1 1 1 2 1 2 2 2 2 2 2 2 4 2 4 4 4 4 4 4 8 8

Matrix representation of C3×C241C4 in GL3(𝔽73) generated by

 8 0 0 0 64 0 0 0 64
,
 1 0 0 0 7 0 0 0 21
,
 27 0 0 0 0 1 0 1 0
G:=sub<GL(3,GF(73))| [8,0,0,0,64,0,0,0,64],[1,0,0,0,7,0,0,0,21],[27,0,0,0,0,1,0,1,0] >;

C3×C241C4 in GAP, Magma, Sage, TeX

C_3\times C_{24}\rtimes_1C_4
% in TeX

G:=Group("C3xC24:1C4");
// GroupNames label

G:=SmallGroup(288,252);
// by ID

G=gap.SmallGroup(288,252);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-3,84,365,512,2524,102,9414]);
// Polycyclic

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

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