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G = C12.98(C4×S3)  order 288 = 25·32

10th non-split extension by C12 of C4×S3 acting via C4×S3/D6=C2

metabelian, supersoluble, monomial

Aliases: C3⋊C84Dic3, C12.98(C4×S3), C31(C24⋊C4), C2.4(Dic32), (C3×C6).9C42, C6.4(C4×Dic3), C6.1(C8⋊S3), C324(C8⋊C4), (C2×C12).294D6, C62.24(C2×C4), C4.21(S3×Dic3), (C3×C6).4M4(2), C12.32(C2×Dic3), (C6×C12).199C22, C2.1(C12.31D6), C22.8(C6.D6), (C3×C3⋊C8)⋊7C4, (C2×C3⋊C8).8S3, (C6×C3⋊C8).19C2, (C2×C6).26(C4×S3), (C2×C4).127(S32), (C3×C12).84(C2×C4), (C2×C3⋊Dic3).7C4, (C4×C3⋊Dic3).10C2, SmallGroup(288,203)

Series: Derived Chief Lower central Upper central

C1C3×C6 — C12.98(C4×S3)
C1C3C32C3×C6C3×C12C6×C12C6×C3⋊C8 — C12.98(C4×S3)
C32C3×C6 — C12.98(C4×S3)
C1C2×C4

Generators and relations for C12.98(C4×S3)
 G = < a,b,c,d | a12=c3=1, b4=a6, d2=a9, bab-1=a5, ac=ca, ad=da, bc=cb, dbd-1=a6b, dcd-1=c-1 >

Subgroups: 266 in 95 conjugacy classes, 48 normal (10 characteristic)
C1, C2, C2 [×2], C3 [×2], C3, C4 [×2], C4 [×2], C22, C6 [×6], C6 [×3], C8 [×4], C2×C4, C2×C4 [×2], C32, Dic3 [×8], C12 [×4], C12 [×2], C2×C6 [×2], C2×C6, C42, C2×C8 [×2], C3×C6, C3×C6 [×2], C3⋊C8 [×4], C24 [×4], C2×Dic3 [×6], C2×C12 [×2], C2×C12, C8⋊C4, C3⋊Dic3 [×2], C3×C12 [×2], C62, C2×C3⋊C8 [×2], C4×Dic3 [×3], C2×C24 [×2], C3×C3⋊C8 [×4], C2×C3⋊Dic3 [×2], C6×C12, C24⋊C4 [×2], C6×C3⋊C8 [×2], C4×C3⋊Dic3, C12.98(C4×S3)
Quotients: C1, C2 [×3], C4 [×6], C22, S3 [×2], C2×C4 [×3], Dic3 [×4], D6 [×2], C42, M4(2) [×2], C4×S3 [×4], C2×Dic3 [×2], C8⋊C4, S32, C8⋊S3 [×4], C4×Dic3 [×2], S3×Dic3 [×2], C6.D6, C24⋊C4 [×2], C12.31D6 [×2], Dic32, C12.98(C4×S3)

Smallest permutation representation of C12.98(C4×S3)
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 27 38 92 7 33 44 86)(2 32 39 85 8 26 45 91)(3 25 40 90 9 31 46 96)(4 30 41 95 10 36 47 89)(5 35 42 88 11 29 48 94)(6 28 43 93 12 34 37 87)(13 53 76 61 19 59 82 67)(14 58 77 66 20 52 83 72)(15 51 78 71 21 57 84 65)(16 56 79 64 22 50 73 70)(17 49 80 69 23 55 74 63)(18 54 81 62 24 60 75 68)
(1 9 5)(2 10 6)(3 11 7)(4 12 8)(13 21 17)(14 22 18)(15 23 19)(16 24 20)(25 29 33)(26 30 34)(27 31 35)(28 32 36)(37 45 41)(38 46 42)(39 47 43)(40 48 44)(49 53 57)(50 54 58)(51 55 59)(52 56 60)(61 65 69)(62 66 70)(63 67 71)(64 68 72)(73 81 77)(74 82 78)(75 83 79)(76 84 80)(85 89 93)(86 90 94)(87 91 95)(88 92 96)
(1 72 10 69 7 66 4 63)(2 61 11 70 8 67 5 64)(3 62 12 71 9 68 6 65)(13 29 22 26 19 35 16 32)(14 30 23 27 20 36 17 33)(15 31 24 28 21 25 18 34)(37 57 46 54 43 51 40 60)(38 58 47 55 44 52 41 49)(39 59 48 56 45 53 42 50)(73 91 82 88 79 85 76 94)(74 92 83 89 80 86 77 95)(75 93 84 90 81 87 78 96)

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

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

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

60 conjugacy classes

class 1 2A2B2C3A3B3C4A4B4C4D4E4F4G4H6A···6F6G6H6I8A···8H12A···12H12I12J12K12L24A···24P
order1222333444444446···66668···812···121212121224···24
size11112241111181818182···24446···62···244446···6

60 irreducible representations

dim1111122222224444
type++++-++-+
imageC1C2C2C4C4S3Dic3D6M4(2)C4×S3C4×S3C8⋊S3S32S3×Dic3C6.D6C12.31D6
kernelC12.98(C4×S3)C6×C3⋊C8C4×C3⋊Dic3C3×C3⋊C8C2×C3⋊Dic3C2×C3⋊C8C3⋊C8C2×C12C3×C6C12C2×C6C6C2×C4C4C22C2
# reps12184242444161214

Matrix representation of C12.98(C4×S3) in GL6(𝔽73)

2700000
0270000
001000
000100
0000072
0000172
,
010000
4600000
0046000
0004600
000001
000010
,
100000
010000
00727200
001000
000010
000001
,
49710000
19240000
001000
00727200
000010
000001

G:=sub<GL(6,GF(73))| [27,0,0,0,0,0,0,27,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,72,72],[0,46,0,0,0,0,1,0,0,0,0,0,0,0,46,0,0,0,0,0,0,46,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,1,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[49,19,0,0,0,0,71,24,0,0,0,0,0,0,1,72,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

C12.98(C4×S3) in GAP, Magma, Sage, TeX

C_{12}._{98}(C_4\times S_3)
% in TeX

G:=Group("C12.98(C4xS3)");
// GroupNames label

G:=SmallGroup(288,203);
// by ID

G=gap.SmallGroup(288,203);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,28,253,92,100,1356,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^12=c^3=1,b^4=a^6,d^2=a^9,b*a*b^-1=a^5,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=a^6*b,d*c*d^-1=c^-1>;
// generators/relations

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