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## G = C6.18D24order 288 = 25·32

### 7th non-split extension by C6 of D24 acting via D24/D12=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C12 — C6.18D24
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — C6×C12 — C3×C4⋊Dic3 — C6.18D24
 Lower central C32 — C3×C6 — C3×C12 — C6.18D24
 Upper central C1 — C22 — C2×C4

Generators and relations for C6.18D24
G = < a,b,c | a6=b24=1, c2=a3, bab-1=cac-1=a-1, cbc-1=b-1 >

Subgroups: 298 in 85 conjugacy classes, 42 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 [×5], C12 [×4], C12 [×3], C2×C6 [×2], C2×C6, C4⋊C4 [×2], C2×C8, C3×C6 [×3], C3⋊C8 [×2], C24 [×2], C2×Dic3 [×5], C2×C12 [×2], C2×C12 [×2], C2.D8, C3×Dic3, C3⋊Dic3, C3×C12 [×2], C62, C2×C3⋊C8, C4⋊Dic3, C4⋊Dic3 [×3], C3×C4⋊C4, C2×C24, C3×C3⋊C8 [×2], C6×Dic3, C2×C3⋊Dic3, C6×C12, C6.Q16, C241C4, C6×C3⋊C8, C3×C4⋊Dic3, C12⋊Dic3, C6.18D24
Quotients: C1, C2 [×3], C4 [×2], C22, S3 [×2], C2×C4, D4, Q8, Dic3 [×2], D6 [×2], C4⋊C4, D8, Q16, Dic6 [×2], C4×S3, D12, C2×Dic3, C3⋊D4, C2.D8, S32, D24, Dic12, Dic3⋊C4, C4⋊Dic3, D4⋊S3, C3⋊Q16, S3×Dic3, C3⋊D12, C322Q8, C6.Q16, C241C4, C3⋊D24, C323Q16, Dic3⋊Dic3, C6.18D24

Smallest permutation representation of C6.18D24
On 96 points
Generators in S96
(1 25 9 33 17 41)(2 42 18 34 10 26)(3 27 11 35 19 43)(4 44 20 36 12 28)(5 29 13 37 21 45)(6 46 22 38 14 30)(7 31 15 39 23 47)(8 48 24 40 16 32)(49 93 65 85 57 77)(50 78 58 86 66 94)(51 95 67 87 59 79)(52 80 60 88 68 96)(53 73 69 89 61 81)(54 82 62 90 70 74)(55 75 71 91 63 83)(56 84 64 92 72 76)
(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 84 33 72)(2 83 34 71)(3 82 35 70)(4 81 36 69)(5 80 37 68)(6 79 38 67)(7 78 39 66)(8 77 40 65)(9 76 41 64)(10 75 42 63)(11 74 43 62)(12 73 44 61)(13 96 45 60)(14 95 46 59)(15 94 47 58)(16 93 48 57)(17 92 25 56)(18 91 26 55)(19 90 27 54)(20 89 28 53)(21 88 29 52)(22 87 30 51)(23 86 31 50)(24 85 32 49)

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

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

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

48 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 4A 4B 4C 4D 4E 4F 6A ··· 6F 6G 6H 6I 8A 8B 8C 8D 12A 12B 12C 12D 12E ··· 12J 12K 12L 12M 12N 24A ··· 24H order 1 2 2 2 3 3 3 4 4 4 4 4 4 6 ··· 6 6 6 6 8 8 8 8 12 12 12 12 12 ··· 12 12 12 12 12 24 ··· 24 size 1 1 1 1 2 2 4 2 2 12 12 36 36 2 ··· 2 4 4 4 6 6 6 6 2 2 2 2 4 ··· 4 12 12 12 12 6 ··· 6

48 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 type + + + + + + - + - + + - - + + - + + - - - + + - image C1 C2 C2 C2 C4 S3 S3 Q8 D4 Dic3 D6 D8 Q16 Dic6 C4×S3 D12 C3⋊D4 D24 Dic12 S32 D4⋊S3 C3⋊Q16 S3×Dic3 C32⋊2Q8 C3⋊D12 C3⋊D24 C32⋊3Q16 kernel C6.18D24 C6×C3⋊C8 C3×C4⋊Dic3 C12⋊Dic3 C3×C3⋊C8 C2×C3⋊C8 C4⋊Dic3 C3×C12 C62 C3⋊C8 C2×C12 C3×C6 C3×C6 C12 C12 C2×C6 C2×C6 C6 C6 C2×C4 C6 C6 C4 C4 C22 C2 C2 # reps 1 1 1 1 4 1 1 1 1 2 2 2 2 4 2 2 2 4 4 1 1 1 1 1 1 2 2

Matrix representation of C6.18D24 in GL6(𝔽73)

 72 0 0 0 0 0 0 72 0 0 0 0 0 0 1 72 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 63 0 0 0 0 0 59 51 0 0 0 0 0 0 0 72 0 0 0 0 72 0 0 0 0 0 0 0 59 7 0 0 0 0 66 66
,
 55 68 0 0 0 0 65 18 0 0 0 0 0 0 0 46 0 0 0 0 46 0 0 0 0 0 0 0 72 1 0 0 0 0 0 1

G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,1,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[63,59,0,0,0,0,0,51,0,0,0,0,0,0,0,72,0,0,0,0,72,0,0,0,0,0,0,0,59,66,0,0,0,0,7,66],[55,65,0,0,0,0,68,18,0,0,0,0,0,0,0,46,0,0,0,0,46,0,0,0,0,0,0,0,72,0,0,0,0,0,1,1] >;

C6.18D24 in GAP, Magma, Sage, TeX

C_6._{18}D_{24}
% in TeX

G:=Group("C6.18D24");
// GroupNames label

G:=SmallGroup(288,223);
// by ID

G=gap.SmallGroup(288,223);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,56,85,148,422,100,1356,9414]);
// Polycyclic

G:=Group<a,b,c|a^6=b^24=1,c^2=a^3,b*a*b^-1=c*a*c^-1=a^-1,c*b*c^-1=b^-1>;
// generators/relations

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