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G = C323Q16order 144 = 24·32

2nd semidirect product of C32 and Q16 acting via Q16/C4=C22

metabelian, supersoluble, monomial

Aliases: C323Q16, C32Dic12, C12.15D6, C6.15D12, Dic6.1S3, C3⋊C8.S3, C4.4S32, (C3×C6).12D4, C31(C3⋊Q16), C6.4(C3⋊D4), (C3×C12).7C22, (C3×Dic6).2C2, C2.7(C3⋊D12), C324Q8.2C2, (C3×C3⋊C8).1C2, SmallGroup(144,62)

Series: Derived Chief Lower central Upper central

C1C3×C12 — C323Q16
C1C3C32C3×C6C3×C12C3×Dic6 — C323Q16
C32C3×C6C3×C12 — C323Q16
C1C2C4

Generators and relations for C323Q16
 G = < a,b,c,d | a3=b3=c8=1, d2=c4, ab=ba, cac-1=a-1, ad=da, bc=cb, dbd-1=b-1, dcd-1=c-1 >

2C3
6C4
18C4
2C6
3C8
3Q8
9Q8
2C12
2Dic3
6Dic3
6Dic3
6Dic3
6Dic3
6C12
9Q16
3C24
3Dic6
3Dic6
3C3×Q8
6Dic6
2C3×Dic3
2C3⋊Dic3
3Dic12
3C3⋊Q16

Character table of C323Q16

 class 123A3B3C4A4B4C6A6B6C8A8B12A12B12C12D12E12F12G24A24B24C24D
 size 1122421236224662244412126666
ρ1111111111111111111111111    trivial
ρ21111111-1111-1-11111111-1-1-1-1    linear of order 2
ρ3111111-11111-1-111111-1-1-1-1-1-1    linear of order 2
ρ4111111-1-11111111111-1-11111    linear of order 2
ρ5222-1-1200-12-1-2-2-1-12-1-1001111    orthogonal lifted from D6
ρ622-12-12202-1-10022-1-1-1-1-10000    orthogonal lifted from S3
ρ722-12-12-202-1-10022-1-1-1110000    orthogonal lifted from D6
ρ822222-20022200-2-2-2-2-2000000    orthogonal lifted from D4
ρ9222-1-1200-12-122-1-12-1-100-1-1-1-1    orthogonal lifted from S3
ρ10222-1-1-200-12-10011-21100-333-3    orthogonal lifted from D12
ρ11222-1-1-200-12-10011-211003-3-33    orthogonal lifted from D12
ρ122-2222000-2-2-2-220000000-2-222    symplectic lifted from Q16, Schur index 2
ρ132-2222000-2-2-22-2000000022-2-2    symplectic lifted from Q16, Schur index 2
ρ142-22-1-10001-21-22-330-3300ζ87ζ328785ζ32ζ83ζ328ζ328ζ83ζ3838ζ3ζ87ζ385ζ385    symplectic lifted from Dic12, Schur index 2
ρ152-22-1-10001-212-23-303-300ζ83ζ3838ζ3ζ87ζ385ζ385ζ87ζ328785ζ32ζ83ζ328ζ328    symplectic lifted from Dic12, Schur index 2
ρ162-22-1-10001-212-2-330-3300ζ87ζ385ζ385ζ83ζ3838ζ3ζ83ζ328ζ328ζ87ζ328785ζ32    symplectic lifted from Dic12, Schur index 2
ρ172-22-1-10001-21-223-303-300ζ83ζ328ζ328ζ87ζ328785ζ32ζ87ζ385ζ385ζ83ζ3838ζ3    symplectic lifted from Dic12, Schur index 2
ρ1822-12-1-2002-1-100-2-2111-3--30000    complex lifted from C3⋊D4
ρ1922-12-1-2002-1-100-2-2111--3-30000    complex lifted from C3⋊D4
ρ2044-2-21400-2-2100-2-2-211000000    orthogonal lifted from S32
ρ2144-2-21-400-2-2100222-1-1000000    orthogonal lifted from C3⋊D12
ρ224-4-24-2000-4220000000000000    symplectic lifted from C3⋊Q16, Schur index 2
ρ234-4-2-2100022-10023-230-33000000    symplectic faithful, Schur index 2
ρ244-4-2-2100022-100-232303-3000000    symplectic faithful, Schur index 2

Smallest permutation representation of C323Q16
On 48 points
Generators in S48
(1 33 25)(2 26 34)(3 35 27)(4 28 36)(5 37 29)(6 30 38)(7 39 31)(8 32 40)(9 46 20)(10 21 47)(11 48 22)(12 23 41)(13 42 24)(14 17 43)(15 44 18)(16 19 45)
(1 25 33)(2 26 34)(3 27 35)(4 28 36)(5 29 37)(6 30 38)(7 31 39)(8 32 40)(9 46 20)(10 47 21)(11 48 22)(12 41 23)(13 42 24)(14 43 17)(15 44 18)(16 45 19)
(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 9 5 13)(2 16 6 12)(3 15 7 11)(4 14 8 10)(17 32 21 28)(18 31 22 27)(19 30 23 26)(20 29 24 25)(33 46 37 42)(34 45 38 41)(35 44 39 48)(36 43 40 47)

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

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

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

C323Q16 is a maximal subgroup of
S3×Dic12  C24.3D6  Dic12⋊S3  D6.1D12  D12.27D6  D12.29D6  Dic6.29D6  Dic6.19D6  Dic6.D6  D12.22D6  D12.8D6  S3×C3⋊Q16  Dic6.9D6  D12.24D6  D12.15D6  C3⋊Dic36  C9⋊Dic12  He32Q16  He33Q16  C337Q16  C338Q16  C339Q16
C323Q16 is a maximal quotient of
C6.Dic12  C12.73D12  C6.18D24  C3⋊Dic36  C9⋊Dic12  He33Q16  C337Q16  C338Q16  C339Q16

Matrix representation of C323Q16 in GL6(𝔽73)

100000
010000
001000
000100
0000072
0000172
,
100000
010000
00727200
001000
000010
000001
,
0410000
16410000
0072000
0007200
000001
000010
,
19130000
62540000
001000
00727200
000010
000001

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,72,72],[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],[0,16,0,0,0,0,41,41,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[19,62,0,0,0,0,13,54,0,0,0,0,0,0,1,72,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

C323Q16 in GAP, Magma, Sage, TeX

C_3^2\rtimes_3Q_{16}
% in TeX

G:=Group("C3^2:3Q16");
// GroupNames label

G:=SmallGroup(144,62);
// by ID

G=gap.SmallGroup(144,62);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-3,48,73,79,218,50,490,3461]);
// Polycyclic

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

Export

Subgroup lattice of C323Q16 in TeX
Character table of C323Q16 in TeX

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