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G = Dic3×C24order 288 = 25·32

Direct product of C24 and Dic3

direct product, metacyclic, supersoluble, monomial, A-group

Aliases: Dic3×C24, C244C12, C3⋊C85C12, C32(C4×C24), C328(C4×C8), (C3×C24)⋊10C4, C6.24(S3×C8), C2.2(S3×C24), C6.3(C4×C12), C6.3(C2×C24), C4.20(S3×C12), (C2×C24).23C6, (C6×C24).19C2, (C2×C24).38S3, C12.111(C4×S3), C12.25(C2×C12), (C2×C12).453D6, C62.67(C2×C4), C22.8(S3×C12), (C3×C6).13C42, C2.2(Dic3×C12), C4.11(C6×Dic3), C6.19(C4×Dic3), (C6×Dic3).15C4, (C2×Dic3).7C12, (C4×Dic3).10C6, C12.68(C2×Dic3), (C6×C12).331C22, (Dic3×C12).21C2, (C3×C3⋊C8)⋊11C4, (C6×C3⋊C8).24C2, (C2×C3⋊C8).12C6, (C2×C8).10(C3×S3), (C3×C6).29(C2×C8), (C2×C6).78(C4×S3), (C2×C4).90(S3×C6), (C2×C6).12(C2×C12), (C2×C12).120(C2×C6), (C3×C12).105(C2×C4), SmallGroup(288,247)

Series: Derived Chief Lower central Upper central

C1C3 — Dic3×C24
C1C3C6C12C2×C12C6×C12Dic3×C12 — Dic3×C24
C3 — Dic3×C24
C1C2×C24

Generators and relations for Dic3×C24
 G = < a,b,c | a24=b6=1, c2=b3, ab=ba, ac=ca, cbc-1=b-1 >

Subgroups: 154 in 99 conjugacy classes, 66 normal (34 characteristic)
C1, C2, C2 [×2], C3 [×2], C3, C4 [×2], C4 [×4], C22, C6 [×2], C6 [×4], C6 [×3], C8 [×2], C8 [×2], C2×C4, C2×C4 [×2], C32, Dic3 [×4], C12 [×4], C12 [×6], C2×C6 [×2], C2×C6, C42, C2×C8, C2×C8, C3×C6, C3×C6 [×2], C3⋊C8 [×2], C24 [×4], C24 [×4], C2×Dic3 [×2], C2×C12 [×2], C2×C12 [×3], C4×C8, C3×Dic3 [×4], C3×C12 [×2], C62, C2×C3⋊C8, C4×Dic3, C4×C12, C2×C24 [×2], C2×C24 [×2], C3×C3⋊C8 [×2], C3×C24 [×2], C6×Dic3 [×2], C6×C12, C8×Dic3, C4×C24, C6×C3⋊C8, Dic3×C12, C6×C24, Dic3×C24
Quotients: C1, C2 [×3], C3, C4 [×6], C22, S3, C6 [×3], C8 [×4], C2×C4 [×3], Dic3 [×2], C12 [×6], D6, C2×C6, C42, C2×C8 [×2], C3×S3, C24 [×4], C4×S3 [×2], C2×Dic3, C2×C12 [×3], C4×C8, C3×Dic3 [×2], S3×C6, S3×C8 [×2], C4×Dic3, C4×C12, C2×C24 [×2], S3×C12 [×2], C6×Dic3, C8×Dic3, C4×C24, S3×C24 [×2], Dic3×C12, Dic3×C24

Smallest permutation representation of Dic3×C24
On 96 points
Generators in S96
(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 9 92 17 76)(2 85 10 93 18 77)(3 86 11 94 19 78)(4 87 12 95 20 79)(5 88 13 96 21 80)(6 89 14 73 22 81)(7 90 15 74 23 82)(8 91 16 75 24 83)(25 60 41 52 33 68)(26 61 42 53 34 69)(27 62 43 54 35 70)(28 63 44 55 36 71)(29 64 45 56 37 72)(30 65 46 57 38 49)(31 66 47 58 39 50)(32 67 48 59 40 51)
(1 61 92 34)(2 62 93 35)(3 63 94 36)(4 64 95 37)(5 65 96 38)(6 66 73 39)(7 67 74 40)(8 68 75 41)(9 69 76 42)(10 70 77 43)(11 71 78 44)(12 72 79 45)(13 49 80 46)(14 50 81 47)(15 51 82 48)(16 52 83 25)(17 53 84 26)(18 54 85 27)(19 55 86 28)(20 56 87 29)(21 57 88 30)(22 58 89 31)(23 59 90 32)(24 60 91 33)

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

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

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

144 conjugacy classes

class 1 2A2B2C3A3B3C3D3E4A4B4C4D4E···4L6A···6F6G···6O8A···8H8I···8P12A···12H12I···12T12U···12AJ24A···24P24Q···24AN24AO···24BD
order12223333344444···46···66···68···88···812···1212···1212···1224···2424···2424···24
size11111122211113···31···12···21···13···31···12···23···31···12···23···3

144 irreducible representations

dim1111111111111111222222222222
type+++++-+
imageC1C2C2C2C3C4C4C4C6C6C6C8C12C12C12C24S3Dic3D6C3×S3C4×S3C4×S3C3×Dic3S3×C6S3×C8S3×C12S3×C12S3×C24
kernelDic3×C24C6×C3⋊C8Dic3×C12C6×C24C8×Dic3C3×C3⋊C8C3×C24C6×Dic3C2×C3⋊C8C4×Dic3C2×C24C3×Dic3C3⋊C8C24C2×Dic3Dic3C2×C24C24C2×C12C2×C8C12C2×C6C8C2×C4C6C4C22C2
# reps1111244422216888321212224284416

Matrix representation of Dic3×C24 in GL4(𝔽73) generated by

7000
05200
00490
00049
,
1000
07200
00640
00198
,
72000
04600
00663
004067
G:=sub<GL(4,GF(73))| [7,0,0,0,0,52,0,0,0,0,49,0,0,0,0,49],[1,0,0,0,0,72,0,0,0,0,64,19,0,0,0,8],[72,0,0,0,0,46,0,0,0,0,6,40,0,0,63,67] >;

Dic3×C24 in GAP, Magma, Sage, TeX

{\rm Dic}_3\times C_{24}
% in TeX

G:=Group("Dic3xC24");
// GroupNames label

G:=SmallGroup(288,247);
// by ID

G=gap.SmallGroup(288,247);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-3,84,176,102,9414]);
// Polycyclic

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

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