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G = C3×C24⋊C4order 288 = 25·32

Direct product of C3 and C24⋊C4

direct product, metacyclic, supersoluble, monomial

Aliases: C3×C24⋊C4, C245C12, C247Dic3, C3⋊C84C12, (C3×C24)⋊13C4, C6.4(C4×C12), C83(C3×Dic3), C4.21(S3×C12), (C2×C24).24C6, (C6×C24).25C2, (C2×C24).36S3, C12.112(C4×S3), C12.26(C2×C12), C327(C8⋊C4), (C2×C12).455D6, C62.69(C2×C4), (C3×C6).14C42, (C4×Dic3).6C6, (C6×Dic3).6C4, C6.20(C4×Dic3), C2.4(Dic3×C12), C4.12(C6×Dic3), C6.15(C8⋊S3), C6.2(C3×M4(2)), C22.10(S3×C12), (C2×Dic3).3C12, C12.69(C2×Dic3), (C3×C6).13M4(2), (C6×C12).333C22, (Dic3×C12).16C2, (C3×C3⋊C8)⋊10C4, C32(C3×C8⋊C4), (C6×C3⋊C8).22C2, (C2×C3⋊C8).10C6, (C2×C8).8(C3×S3), C2.2(C3×C8⋊S3), (C2×C4).92(S3×C6), (C2×C6).80(C4×S3), (C2×C6).14(C2×C12), (C2×C12).122(C2×C6), (C3×C12).106(C2×C4), SmallGroup(288,249)

Series: Derived Chief Lower central Upper central

C1C6 — C3×C24⋊C4
C1C3C6C12C2×C12C6×C12Dic3×C12 — C3×C24⋊C4
C3C6 — C3×C24⋊C4
C1C2×C12C2×C24

Generators and relations for C3×C24⋊C4
 G = < a,b,c | a3=b24=c4=1, ab=ba, ac=ca, cbc-1=b5 >

Subgroups: 154 in 91 conjugacy classes, 58 normal (34 characteristic)
C1, C2, C2 [×2], C3 [×2], C3, C4 [×2], C4 [×2], C22, C6 [×2], C6 [×4], C6 [×3], C8 [×2], C8 [×2], C2×C4, C2×C4 [×2], C32, Dic3 [×2], C12 [×4], C12 [×4], 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], C8⋊C4, C3×Dic3 [×2], 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, C24⋊C4, C3×C8⋊C4, C6×C3⋊C8, Dic3×C12, C6×C24, C3×C24⋊C4
Quotients: C1, C2 [×3], C3, C4 [×6], C22, S3, C6 [×3], C2×C4 [×3], Dic3 [×2], C12 [×6], D6, C2×C6, C42, M4(2) [×2], C3×S3, C4×S3 [×2], C2×Dic3, C2×C12 [×3], C8⋊C4, C3×Dic3 [×2], S3×C6, C8⋊S3 [×2], C4×Dic3, C4×C12, C3×M4(2) [×2], S3×C12 [×2], C6×Dic3, C24⋊C4, C3×C8⋊C4, C3×C8⋊S3 [×2], Dic3×C12, C3×C24⋊C4

Smallest permutation representation of C3×C24⋊C4
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 42 79 53)(2 47 80 58)(3 28 81 63)(4 33 82 68)(5 38 83 49)(6 43 84 54)(7 48 85 59)(8 29 86 64)(9 34 87 69)(10 39 88 50)(11 44 89 55)(12 25 90 60)(13 30 91 65)(14 35 92 70)(15 40 93 51)(16 45 94 56)(17 26 95 61)(18 31 96 66)(19 36 73 71)(20 41 74 52)(21 46 75 57)(22 27 76 62)(23 32 77 67)(24 37 78 72)

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

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

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

108 conjugacy classes

class 1 2A2B2C3A3B3C3D3E4A4B4C4D4E4F4G4H6A···6F6G···6O8A8B8C8D8E8F8G8H12A···12H12I···12T12U···12AB24A···24AF24AG···24AN
order122233333444444446···66···68888888812···1212···1212···1224···2424···24
size111111222111166661···12···2222266661···12···26···62···26···6

108 irreducible representations

dim1111111111111122222222222222
type+++++-+
imageC1C2C2C2C3C4C4C4C6C6C6C12C12C12S3Dic3D6M4(2)C3×S3C4×S3C4×S3C3×Dic3S3×C6C8⋊S3C3×M4(2)S3×C12S3×C12C3×C8⋊S3
kernelC3×C24⋊C4C6×C3⋊C8Dic3×C12C6×C24C24⋊C4C3×C3⋊C8C3×C24C6×Dic3C2×C3⋊C8C4×Dic3C2×C24C3⋊C8C24C2×Dic3C2×C24C24C2×C12C3×C6C2×C8C12C2×C6C8C2×C4C6C6C4C22C2
# reps11112444222888121422242884416

Matrix representation of C3×C24⋊C4 in GL4(𝔽73) generated by

8000
0800
00640
00064
,
64000
0800
0071
00017
,
0100
72000
00129
001072
G:=sub<GL(4,GF(73))| [8,0,0,0,0,8,0,0,0,0,64,0,0,0,0,64],[64,0,0,0,0,8,0,0,0,0,7,0,0,0,1,17],[0,72,0,0,1,0,0,0,0,0,1,10,0,0,29,72] >;

C3×C24⋊C4 in GAP, Magma, Sage, TeX

C_3\times C_{24}\rtimes C_4
% in TeX

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

G:=SmallGroup(288,249);
// by ID

G=gap.SmallGroup(288,249);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-3,84,701,176,102,9414]);
// 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^5>;
// generators/relations

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