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G = C6xC24:C2order 288 = 25·32

Direct product of C6 and C24:C2

direct product, metabelian, supersoluble, monomial

Aliases: C6xC24:C2, C24:28D6, C12.67D12, C62.85D4, C8:8(S3xC6), (C2xC24):7C6, C24:9(C2xC6), C6.9(C6xD4), (C6xC24):12C2, (C2xC24):13S3, C6:1(C3xSD16), C3:1(C6xSD16), (C3xC6):6SD16, C4.6(C3xD12), (C2xDic6):5C6, Dic6:3(C2xC6), D12.6(C2xC6), (C2xD12).4C6, C6.97(C2xD12), C12.29(C3xD4), C2.11(C6xD12), (C2xC6).73D12, (C3xC24):27C22, (C6xDic6):29C2, (C6xD12).19C2, (C2xC12).441D6, (C3xC12).131D4, C32:13(C2xSD16), C12.28(C22xC6), C22.12(C3xD12), (C6xC12).321C22, (C3xC12).160C23, C12.215(C22xS3), (C3xDic6):36C22, (C3xD12).45C22, (C2xC8):5(C3xS3), C4.26(S3xC2xC6), (C2xC4).78(S3xC6), (C2xC6).20(C3xD4), (C3xC6).179(C2xD4), (C2xC12).106(C2xC6), SmallGroup(288,673)

Series: Derived Chief Lower central Upper central

C1C12 — C6xC24:C2
C1C3C6C12C3xC12C3xD12C6xD12 — C6xC24:C2
C3C6C12 — C6xC24:C2
C1C2xC6C2xC12C2xC24

Generators and relations for C6xC24: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, C2, C3, C3, C4, C4, C22, C22, S3, C6, C6, C6, C8, C2xC4, C2xC4, D4, Q8, C23, C32, Dic3, C12, C12, D6, C2xC6, C2xC6, C2xC8, SD16, C2xD4, C2xQ8, C3xS3, C3xC6, C3xC6, C24, C24, Dic6, Dic6, D12, D12, C2xDic3, C2xC12, C2xC12, C3xD4, C3xQ8, C22xS3, C22xC6, C2xSD16, C3xDic3, C3xC12, S3xC6, C62, C24:C2, C2xC24, C2xC24, C3xSD16, C2xDic6, C2xD12, C6xD4, C6xQ8, C3xC24, C3xDic6, C3xDic6, C3xD12, C3xD12, C6xDic3, C6xC12, S3xC2xC6, C2xC24:C2, C6xSD16, C3xC24:C2, C6xC24, C6xDic6, C6xD12, C6xC24:C2
Quotients: C1, C2, C3, C22, S3, C6, D4, C23, D6, C2xC6, SD16, C2xD4, C3xS3, D12, C3xD4, C22xS3, C22xC6, C2xSD16, S3xC6, C24:C2, C3xSD16, C2xD12, C6xD4, C3xD12, S3xC2xC6, C2xC24:C2, C6xSD16, C3xC24:C2, C6xD12, C6xC24:C2

Smallest permutation representation of C6xC24:C2
On 96 points
Generators in S96
(1 68 9 52 17 60)(2 69 10 53 18 61)(3 70 11 54 19 62)(4 71 12 55 20 63)(5 72 13 56 21 64)(6 49 14 57 22 65)(7 50 15 58 23 66)(8 51 16 59 24 67)(25 76 41 92 33 84)(26 77 42 93 34 85)(27 78 43 94 35 86)(28 79 44 95 36 87)(29 80 45 96 37 88)(30 81 46 73 38 89)(31 82 47 74 39 90)(32 83 48 75 40 91)
(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 41)(2 28)(3 39)(4 26)(5 37)(6 48)(7 35)(8 46)(9 33)(10 44)(11 31)(12 42)(13 29)(14 40)(15 27)(16 38)(17 25)(18 36)(19 47)(20 34)(21 45)(22 32)(23 43)(24 30)(49 75)(50 86)(51 73)(52 84)(53 95)(54 82)(55 93)(56 80)(57 91)(58 78)(59 89)(60 76)(61 87)(62 74)(63 85)(64 96)(65 83)(66 94)(67 81)(68 92)(69 79)(70 90)(71 77)(72 88)

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

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

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

90 conjugacy classes

class 1 2A2B2C2D2E3A3B3C3D3E4A4B4C4D6A···6F6G···6O6P6Q6R6S8A8B8C8D12A···12P12Q12R12S12T24A···24AF
order1222223333344446···66···66666888812···121212121224···24
size11111212112222212121···12···21212121222222···2121212122···2

90 irreducible representations

dim1111111111222222222222222222
type++++++++++++
imageC1C2C2C2C2C3C6C6C6C6S3D4D4D6D6SD16C3xS3D12C3xD4D12C3xD4S3xC6S3xC6C24:C2C3xSD16C3xD12C3xD12C3xC24:C2
kernelC6xC24:C2C3xC24:C2C6xC24C6xDic6C6xD12C2xC24:C2C24:C2C2xC24C2xDic6C2xD12C2xC24C3xC12C62C24C2xC12C3xC6C2xC8C12C12C2xC6C2xC6C8C2xC4C6C6C4C22C2
# reps14111282221112142222242884416

Matrix representation of C6xC24:C2 in GL4(F73) generated by

9000
0900
0090
0009
,
9000
566500
005640
00043
,
101000
126300
004447
006629
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] >;

C6xC24: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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