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G = C325SD16order 144 = 24·32

3rd semidirect product of C32 and SD16 acting via SD16/C4=C22

metabelian, supersoluble, monomial

Aliases: Dic61S3, C6.14D12, C12.13D6, C325SD16, C3⋊C83S3, C4.3S32, (C3×C6).10D4, C32(C24⋊C2), (C3×Dic6)⋊2C2, C6.3(C3⋊D4), C12⋊S3.2C2, C31(Q82S3), (C3×C12).5C22, C2.6(C3⋊D12), (C3×C3⋊C8)⋊3C2, SmallGroup(144,60)

Series: Derived Chief Lower central Upper central

C1C3×C12 — C325SD16
C1C3C32C3×C6C3×C12C3×Dic6 — C325SD16
C32C3×C6C3×C12 — C325SD16
C1C2C4

Generators and relations for C325SD16
 G = < a,b,c,d | a3=b3=c8=d2=1, ab=ba, cac-1=dad=a-1, bc=cb, dbd=b-1, dcd=c3 >

36C2
2C3
6C4
18C22
2C6
12S3
12S3
12S3
12S3
3C8
3Q8
9D4
2C12
2Dic3
6C12
6D6
6D6
6D6
6D6
4C3⋊S3
9SD16
3C24
3D12
3D12
3C3×Q8
6D12
2C2×C3⋊S3
2C3×Dic3
3C24⋊C2
3Q82S3

Character table of C325SD16

 class 12A2B3A3B3C4A4B6A6B6C8A8B12A12B12C12D12E12F12G24A24B24C24D
 size 1136224212224662244412126666
ρ1111111111111111111111111    trivial
ρ21111111-1111-1-111111-1-1-1-1-1-1    linear of order 2
ρ311-111111111-1-11111111-1-1-1-1    linear of order 2
ρ411-11111-11111111111-1-11111    linear of order 2
ρ5220-12-12-2-12-10022-1-1-1110000    orthogonal lifted from D6
ρ62202-1-1202-1-1-2-2-1-12-1-1001111    orthogonal lifted from D6
ρ7220-12-122-12-10022-1-1-1-1-10000    orthogonal lifted from S3
ρ82202-1-1202-1-122-1-12-1-100-1-1-1-1    orthogonal lifted from S3
ρ9220222-2022200-2-2-2-2-2000000    orthogonal lifted from D4
ρ102202-1-1-202-1-10011-211003-3-33    orthogonal lifted from D12
ρ112202-1-1-202-1-10011-21100-333-3    orthogonal lifted from D12
ρ122-2022200-2-2-2--2-20000000-2-2--2--2    complex lifted from SD16
ρ132-2022200-2-2-2-2--20000000--2--2-2-2    complex lifted from SD16
ρ14220-12-1-20-12-100-2-2111--3-30000    complex lifted from C3⋊D4
ρ15220-12-1-20-12-100-2-2111-3--30000    complex lifted from C3⋊D4
ρ162-202-1-100-211--2-2-3303-30087ζ3285ζ328587ζ385ζ38583ζ38ζ3883ζ328ζ328    complex lifted from C24⋊C2
ρ172-202-1-100-211-2--23-30-330083ζ38ζ3883ζ328ζ32887ζ3285ζ328587ζ385ζ385    complex lifted from C24⋊C2
ρ182-202-1-100-211-2--2-3303-30083ζ328ζ32883ζ38ζ3887ζ385ζ38587ζ3285ζ3285    complex lifted from C24⋊C2
ρ192-202-1-100-211--2-23-30-330087ζ385ζ38587ζ3285ζ328583ζ328ζ32883ζ38ζ38    complex lifted from C24⋊C2
ρ204-40-24-2002-420000000000000    orthogonal lifted from Q82S3
ρ21440-2-21-40-2-2100222-1-1000000    orthogonal lifted from C3⋊D12
ρ22440-2-2140-2-2100-2-2-211000000    orthogonal lifted from S32
ρ234-40-2-210022-100-23230-33000000    orthogonal faithful
ρ244-40-2-210022-10023-2303-3000000    orthogonal faithful

Permutation representations of C325SD16
On 24 points - transitive group 24T233
Generators in S24
(1 13 24)(2 17 14)(3 15 18)(4 19 16)(5 9 20)(6 21 10)(7 11 22)(8 23 12)
(1 13 24)(2 14 17)(3 15 18)(4 16 19)(5 9 20)(6 10 21)(7 11 22)(8 12 23)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)
(1 3)(2 6)(5 7)(9 22)(10 17)(11 20)(12 23)(13 18)(14 21)(15 24)(16 19)

G:=sub<Sym(24)| (1,13,24)(2,17,14)(3,15,18)(4,19,16)(5,9,20)(6,21,10)(7,11,22)(8,23,12), (1,13,24)(2,14,17)(3,15,18)(4,16,19)(5,9,20)(6,10,21)(7,11,22)(8,12,23), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24), (1,3)(2,6)(5,7)(9,22)(10,17)(11,20)(12,23)(13,18)(14,21)(15,24)(16,19)>;

G:=Group( (1,13,24)(2,17,14)(3,15,18)(4,19,16)(5,9,20)(6,21,10)(7,11,22)(8,23,12), (1,13,24)(2,14,17)(3,15,18)(4,16,19)(5,9,20)(6,10,21)(7,11,22)(8,12,23), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24), (1,3)(2,6)(5,7)(9,22)(10,17)(11,20)(12,23)(13,18)(14,21)(15,24)(16,19) );

G=PermutationGroup([(1,13,24),(2,17,14),(3,15,18),(4,19,16),(5,9,20),(6,21,10),(7,11,22),(8,23,12)], [(1,13,24),(2,14,17),(3,15,18),(4,16,19),(5,9,20),(6,10,21),(7,11,22),(8,12,23)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24)], [(1,3),(2,6),(5,7),(9,22),(10,17),(11,20),(12,23),(13,18),(14,21),(15,24),(16,19)])

G:=TransitiveGroup(24,233);

C325SD16 is a maximal subgroup of
S3×C24⋊C2  C241D6  Dic12⋊S3  D6.3D12  D1218D6  D12.27D6  Dic6.29D6  Dic63D6  Dic6⋊D6  Dic6.20D6  D125D6  S3×Q82S3  Dic6.10D6  Dic6.22D6  D12.14D6  C6.D36  C18.D12  He33SD16  He35SD16  C3315SD16  C3317SD16  C3318SD16
C325SD16 is a maximal quotient of
C6.17D24  C6.Dic12  C12.Dic6  C6.D36  C18.D12  He34SD16  C3315SD16  C3317SD16  C3318SD16

Matrix representation of C325SD16 in GL6(𝔽73)

100000
010000
001000
000100
00007272
000010
,
100000
010000
0007200
0017200
000010
000001
,
660000
6760000
0072000
0007200
000010
00007272
,
010000
100000
000100
001000
000010
00007272

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,1,0,0,0,0,72,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,72,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[6,67,0,0,0,0,6,6,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,72,0,0,0,0,0,72],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,72,0,0,0,0,0,72] >;

C325SD16 in GAP, Magma, Sage, TeX

C_3^2\rtimes_5{\rm SD}_{16}
% in TeX

G:=Group("C3^2:5SD16");
// GroupNames label

G:=SmallGroup(144,60);
// by ID

G=gap.SmallGroup(144,60);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-3,48,73,31,218,50,490,3461]);
// Polycyclic

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

Export

Subgroup lattice of C325SD16 in TeX
Character table of C325SD16 in TeX

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