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G = C3×C3⋊Q16order 144 = 24·32

Direct product of C3 and C3⋊Q16

direct product, metabelian, supersoluble, monomial

Aliases: C3×C3⋊Q16, C326Q16, C12.36D6, Dic6.2C6, C3⋊C8.C6, C4.4(S3×C6), C32(C3×Q16), C12.4(C2×C6), C6.10(C3×D4), (C3×C6).31D4, (C3×Q8).5C6, Q8.3(C3×S3), (C3×Q8).10S3, C6.32(C3⋊D4), (C3×Dic6).3C2, (Q8×C32).1C2, (C3×C12).11C22, (C3×C3⋊C8).2C2, C2.7(C3×C3⋊D4), SmallGroup(144,83)

Series: Derived Chief Lower central Upper central

C1C12 — C3×C3⋊Q16
C1C3C6C12C3×C12C3×Dic6 — C3×C3⋊Q16
C3C6C12 — C3×C3⋊Q16
C1C6C12C3×Q8

Generators and relations for C3×C3⋊Q16
 G = < a,b,c,d | a3=b3=c8=1, d2=c4, ab=ba, ac=ca, ad=da, cbc-1=b-1, bd=db, dcd-1=c-1 >

2C3
2C4
6C4
2C6
3Q8
3C8
2C12
2C12
2C12
2C12
2C12
2Dic3
6C12
3Q16
2C3×Q8
3C24
3C3×Q8
2C3×C12
2C3×Dic3
3C3×Q16

Smallest permutation representation of C3×C3⋊Q16
On 48 points
Generators in S48
(1 35 46)(2 36 47)(3 37 48)(4 38 41)(5 39 42)(6 40 43)(7 33 44)(8 34 45)(9 30 19)(10 31 20)(11 32 21)(12 25 22)(13 26 23)(14 27 24)(15 28 17)(16 29 18)
(1 35 46)(2 47 36)(3 37 48)(4 41 38)(5 39 42)(6 43 40)(7 33 44)(8 45 34)(9 30 19)(10 20 31)(11 32 21)(12 22 25)(13 26 23)(14 24 27)(15 28 17)(16 18 29)
(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)
(1 23 5 19)(2 22 6 18)(3 21 7 17)(4 20 8 24)(9 35 13 39)(10 34 14 38)(11 33 15 37)(12 40 16 36)(25 43 29 47)(26 42 30 46)(27 41 31 45)(28 48 32 44)

G:=sub<Sym(48)| (1,35,46)(2,36,47)(3,37,48)(4,38,41)(5,39,42)(6,40,43)(7,33,44)(8,34,45)(9,30,19)(10,31,20)(11,32,21)(12,25,22)(13,26,23)(14,27,24)(15,28,17)(16,29,18), (1,35,46)(2,47,36)(3,37,48)(4,41,38)(5,39,42)(6,43,40)(7,33,44)(8,45,34)(9,30,19)(10,20,31)(11,32,21)(12,22,25)(13,26,23)(14,24,27)(15,28,17)(16,18,29), (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), (1,23,5,19)(2,22,6,18)(3,21,7,17)(4,20,8,24)(9,35,13,39)(10,34,14,38)(11,33,15,37)(12,40,16,36)(25,43,29,47)(26,42,30,46)(27,41,31,45)(28,48,32,44)>;

G:=Group( (1,35,46)(2,36,47)(3,37,48)(4,38,41)(5,39,42)(6,40,43)(7,33,44)(8,34,45)(9,30,19)(10,31,20)(11,32,21)(12,25,22)(13,26,23)(14,27,24)(15,28,17)(16,29,18), (1,35,46)(2,47,36)(3,37,48)(4,41,38)(5,39,42)(6,43,40)(7,33,44)(8,45,34)(9,30,19)(10,20,31)(11,32,21)(12,22,25)(13,26,23)(14,24,27)(15,28,17)(16,18,29), (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), (1,23,5,19)(2,22,6,18)(3,21,7,17)(4,20,8,24)(9,35,13,39)(10,34,14,38)(11,33,15,37)(12,40,16,36)(25,43,29,47)(26,42,30,46)(27,41,31,45)(28,48,32,44) );

G=PermutationGroup([(1,35,46),(2,36,47),(3,37,48),(4,38,41),(5,39,42),(6,40,43),(7,33,44),(8,34,45),(9,30,19),(10,31,20),(11,32,21),(12,25,22),(13,26,23),(14,27,24),(15,28,17),(16,29,18)], [(1,35,46),(2,47,36),(3,37,48),(4,41,38),(5,39,42),(6,43,40),(7,33,44),(8,45,34),(9,30,19),(10,20,31),(11,32,21),(12,22,25),(13,26,23),(14,24,27),(15,28,17),(16,18,29)], [(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)], [(1,23,5,19),(2,22,6,18),(3,21,7,17),(4,20,8,24),(9,35,13,39),(10,34,14,38),(11,33,15,37),(12,40,16,36),(25,43,29,47),(26,42,30,46),(27,41,31,45),(28,48,32,44)])

C3×C3⋊Q16 is a maximal subgroup of
D12.11D6  Dic6.9D6  Dic6.10D6  Dic6.22D6  D12.13D6  D12.14D6  D12.15D6  C3×S3×Q16  He36Q16  Dic18.C6  He37Q16  C32.CSU2(𝔽3)  C32⋊CSU2(𝔽3)  C322CSU2(𝔽3)
C3×C3⋊Q16 is a maximal quotient of
He36Q16  Dic18.C6

36 conjugacy classes

class 1  2 3A3B3C3D3E4A4B4C6A6B6C6D6E8A8B12A12B12C···12M12N12O24A24B24C24D
order12333334446666688121212···12121224242424
size111122224121122266224···412126666

36 irreducible representations

dim11111111222222222244
type+++++++--
imageC1C2C2C2C3C6C6C6S3D4D6Q16C3×S3C3⋊D4C3×D4S3×C6C3×Q16C3×C3⋊D4C3⋊Q16C3×C3⋊Q16
kernelC3×C3⋊Q16C3×C3⋊C8C3×Dic6Q8×C32C3⋊Q16C3⋊C8Dic6C3×Q8C3×Q8C3×C6C12C32Q8C6C6C4C3C2C3C1
# reps11112222111222224412

Matrix representation of C3×C3⋊Q16 in GL4(𝔽7) generated by

2000
0200
0020
0002
,
4446
1563
5561
3454
,
5005
5410
1200
0105
,
2125
2022
6415
6544
G:=sub<GL(4,GF(7))| [2,0,0,0,0,2,0,0,0,0,2,0,0,0,0,2],[4,1,5,3,4,5,5,4,4,6,6,5,6,3,1,4],[5,5,1,0,0,4,2,1,0,1,0,0,5,0,0,5],[2,2,6,6,1,0,4,5,2,2,1,4,5,2,5,4] >;

C3×C3⋊Q16 in GAP, Magma, Sage, TeX

C_3\times C_3\rtimes Q_{16}
% in TeX

G:=Group("C3xC3:Q16");
// GroupNames label

G:=SmallGroup(144,83);
// by ID

G=gap.SmallGroup(144,83);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-2,-3,144,169,151,867,441,69,3461]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^3=c^8=1,d^2=c^4,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

Export

Subgroup lattice of C3×C3⋊Q16 in TeX

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