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## G = C6×C24⋊C2order 288 = 25·32

### Direct product of C6 and C24⋊C2

Series: Derived Chief Lower central Upper central

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

Generators and relations for C6×C24⋊C2
G = < a,b,c | a6=b24=c2=1, ab=ba, ac=ca, cbc=b11 >

Subgroups: 410 in 147 conjugacy classes, 66 normal (34 characteristic)
C1, C2, C2 [×2], C2 [×2], C3 [×2], C3, C4 [×2], C4 [×2], C22, C22 [×4], S3 [×2], C6 [×2], C6 [×4], C6 [×5], C8 [×2], C2×C4, C2×C4, D4 [×3], Q8 [×3], C23, C32, Dic3 [×2], C12 [×4], C12 [×4], D6 [×4], C2×C6 [×2], C2×C6 [×5], C2×C8, SD16 [×4], C2×D4, C2×Q8, C3×S3 [×2], C3×C6, C3×C6 [×2], C24 [×4], C24 [×2], Dic6 [×2], Dic6, D12 [×2], D12, C2×Dic3, C2×C12 [×2], C2×C12 [×2], C3×D4 [×3], C3×Q8 [×3], C22×S3, C22×C6, C2×SD16, C3×Dic3 [×2], C3×C12 [×2], S3×C6 [×4], C62, C24⋊C2 [×4], C2×C24 [×2], C2×C24, C3×SD16 [×4], C2×Dic6, C2×D12, C6×D4, C6×Q8, C3×C24 [×2], C3×Dic6 [×2], C3×Dic6, C3×D12 [×2], C3×D12, C6×Dic3, C6×C12, S3×C2×C6, C2×C24⋊C2, C6×SD16, C3×C24⋊C2 [×4], C6×C24, C6×Dic6, C6×D12, C6×C24⋊C2
Quotients: C1, C2 [×7], C3, C22 [×7], S3, C6 [×7], D4 [×2], C23, D6 [×3], C2×C6 [×7], SD16 [×2], C2×D4, C3×S3, D12 [×2], C3×D4 [×2], C22×S3, C22×C6, C2×SD16, S3×C6 [×3], C24⋊C2 [×2], C3×SD16 [×2], C2×D12, C6×D4, C3×D12 [×2], S3×C2×C6, C2×C24⋊C2, C6×SD16, C3×C24⋊C2 [×2], C6×D12, C6×C24⋊C2

Smallest permutation representation of C6×C24⋊C2
On 96 points
Generators in S96
(1 28 9 36 17 44)(2 29 10 37 18 45)(3 30 11 38 19 46)(4 31 12 39 20 47)(5 32 13 40 21 48)(6 33 14 41 22 25)(7 34 15 42 23 26)(8 35 16 43 24 27)(49 96 65 88 57 80)(50 73 66 89 58 81)(51 74 67 90 59 82)(52 75 68 91 60 83)(53 76 69 92 61 84)(54 77 70 93 62 85)(55 78 71 94 63 86)(56 79 72 95 64 87)
(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 81)(3 92)(4 79)(5 90)(6 77)(7 88)(8 75)(9 86)(10 73)(11 84)(12 95)(13 82)(14 93)(15 80)(16 91)(17 78)(18 89)(19 76)(20 87)(21 74)(22 85)(23 96)(24 83)(25 54)(26 65)(27 52)(28 63)(29 50)(30 61)(31 72)(32 59)(33 70)(34 57)(35 68)(36 55)(37 66)(38 53)(39 64)(40 51)(41 62)(42 49)(43 60)(44 71)(45 58)(46 69)(47 56)(48 67)

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

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

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

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 type + + + + + + + + + + + + image C1 C2 C2 C2 C2 C3 C6 C6 C6 C6 S3 D4 D4 D6 D6 SD16 C3×S3 D12 C3×D4 D12 C3×D4 S3×C6 S3×C6 C24⋊C2 C3×SD16 C3×D12 C3×D12 C3×C24⋊C2 kernel C6×C24⋊C2 C3×C24⋊C2 C6×C24 C6×Dic6 C6×D12 C2×C24⋊C2 C24⋊C2 C2×C24 C2×Dic6 C2×D12 C2×C24 C3×C12 C62 C24 C2×C12 C3×C6 C2×C8 C12 C12 C2×C6 C2×C6 C8 C2×C4 C6 C6 C4 C22 C2 # reps 1 4 1 1 1 2 8 2 2 2 1 1 1 2 1 4 2 2 2 2 2 4 2 8 8 4 4 16

Matrix representation of C6×C24⋊C2 in GL4(𝔽73) generated by

 9 0 0 0 0 9 0 0 0 0 9 0 0 0 0 9
,
 9 0 0 0 56 65 0 0 0 0 56 40 0 0 0 43
,
 10 10 0 0 12 63 0 0 0 0 44 47 0 0 66 29
G:=sub<GL(4,GF(73))| [9,0,0,0,0,9,0,0,0,0,9,0,0,0,0,9],[9,56,0,0,0,65,0,0,0,0,56,0,0,0,40,43],[10,12,0,0,10,63,0,0,0,0,44,66,0,0,47,29] >;

C6×C24⋊C2 in GAP, Magma, Sage, TeX

C_6\times C_{24}\rtimes C_2
% in TeX

G:=Group("C6xC24:C2");
// GroupNames label

G:=SmallGroup(288,673);
// by ID

G=gap.SmallGroup(288,673);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,590,142,2524,102,9414]);
// Polycyclic

G:=Group<a,b,c|a^6=b^24=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^11>;
// generators/relations

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