Copied to
clipboard

G = C322C16order 144 = 24·32

The semidirect product of C32 and C16 acting via C16/C4=C4

metabelian, soluble, monomial, A-group

Aliases: C322C16, (C3×C6).2C8, (C3×C12).1C4, C2.(C322C8), C4.2(C32⋊C4), C324C8.3C2, SmallGroup(144,51)

Series: Derived Chief Lower central Upper central

C1C32 — C322C16
C1C32C3×C6C3×C12C324C8 — C322C16
C32 — C322C16
C1C4

Generators and relations for C322C16
 G = < a,b,c | a3=b3=c16=1, cbc-1=ab=ba, cac-1=a-1b >

2C3
2C3
2C6
2C6
9C8
2C12
2C12
9C16
6C3⋊C8
6C3⋊C8

Character table of C322C16

 class 123A3B4A4B6A6B8A8B8C8D12A12B12C12D16A16B16C16D16E16F16G16H
 size 114411449999444499999999
ρ1111111111111111111111111    trivial
ρ21111111111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ311111111-1-1-1-11111i-i-i-iiii-i    linear of order 4
ρ411111111-1-1-1-11111-iiii-i-i-ii    linear of order 4
ρ51111-1-111ii-i-i-1-1-1-1ζ83ζ85ζ8ζ8ζ87ζ87ζ83ζ85    linear of order 8
ρ61111-1-111ii-i-i-1-1-1-1ζ87ζ8ζ85ζ85ζ83ζ83ζ87ζ8    linear of order 8
ρ71111-1-111-i-iii-1-1-1-1ζ85ζ83ζ87ζ87ζ8ζ8ζ85ζ83    linear of order 8
ρ81111-1-111-i-iii-1-1-1-1ζ8ζ87ζ83ζ83ζ85ζ85ζ8ζ87    linear of order 8
ρ91-111i-i-1-1ζ1614ζ166ζ1610ζ162-i-iiiζ169ζ1615ζ163ζ1611ζ165ζ1613ζ16ζ167    linear of order 16
ρ101-111-ii-1-1ζ1610ζ162ζ1614ζ166ii-i-iζ163ζ165ζ16ζ169ζ167ζ1615ζ1611ζ1613    linear of order 16
ρ111-111-ii-1-1ζ1610ζ162ζ1614ζ166ii-i-iζ1611ζ1613ζ169ζ16ζ1615ζ167ζ163ζ165    linear of order 16
ρ121-111i-i-1-1ζ1614ζ166ζ1610ζ162-i-iiiζ16ζ167ζ1611ζ163ζ1613ζ165ζ169ζ1615    linear of order 16
ρ131-111-ii-1-1ζ162ζ1610ζ166ζ1614ii-i-iζ167ζ16ζ1613ζ165ζ1611ζ163ζ1615ζ169    linear of order 16
ρ141-111i-i-1-1ζ166ζ1614ζ162ζ1610-i-iiiζ165ζ163ζ167ζ1615ζ16ζ169ζ1613ζ1611    linear of order 16
ρ151-111-ii-1-1ζ162ζ1610ζ166ζ1614ii-i-iζ1615ζ169ζ165ζ1613ζ163ζ1611ζ167ζ16    linear of order 16
ρ161-111i-i-1-1ζ166ζ1614ζ162ζ1610-i-iiiζ1613ζ1611ζ1615ζ167ζ169ζ16ζ165ζ163    linear of order 16
ρ1744-21441-20000-211-200000000    orthogonal lifted from C32⋊C4
ρ18441-244-2100001-2-2100000000    orthogonal lifted from C32⋊C4
ρ19441-2-4-4-210000-122-100000000    symplectic lifted from C322C8, Schur index 2
ρ2044-21-4-41-200002-1-1200000000    symplectic lifted from C322C8, Schur index 2
ρ214-4-214i-4i-1200002i-ii-2i00000000    complex faithful
ρ224-4-21-4i4i-120000-2ii-i2i00000000    complex faithful
ρ234-41-2-4i4i2-10000i-2i2i-i00000000    complex faithful
ρ244-41-24i-4i2-10000-i2i-2ii00000000    complex faithful

Smallest permutation representation of C322C16
On 48 points
Generators in S48
(2 38 21)(4 23 40)(6 42 25)(8 27 44)(10 46 29)(12 31 48)(14 34 17)(16 19 36)
(1 37 20)(2 38 21)(3 22 39)(4 23 40)(5 41 24)(6 42 25)(7 26 43)(8 27 44)(9 45 28)(10 46 29)(11 30 47)(12 31 48)(13 33 32)(14 34 17)(15 18 35)(16 19 36)
(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)

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

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

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

C322C16 is a maximal subgroup of
C32⋊C32  C32⋊D16  C32⋊SD32  C32⋊Q32  C3⋊S33C16  C323M5(2)  C62.4C8  C334C16
C322C16 is a maximal quotient of
C322C32  He32C16  C334C16

Matrix representation of C322C16 in GL4(𝔽5) generated by

2020
0104
4020
0303
,
1000
0301
0010
0201
,
0303
1040
0300
0030
G:=sub<GL(4,GF(5))| [2,0,4,0,0,1,0,3,2,0,2,0,0,4,0,3],[1,0,0,0,0,3,0,2,0,0,1,0,0,1,0,1],[0,1,0,0,3,0,3,0,0,4,0,3,3,0,0,0] >;

C322C16 in GAP, Magma, Sage, TeX

C_3^2\rtimes_2C_{16}
% in TeX

G:=Group("C3^2:2C16");
// GroupNames label

G:=SmallGroup(144,51);
// by ID

G=gap.SmallGroup(144,51);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,3,12,31,50,3364,490,4613,1739]);
// Polycyclic

G:=Group<a,b,c|a^3=b^3=c^16=1,c*b*c^-1=a*b=b*a,c*a*c^-1=a^-1*b>;
// generators/relations

Export

Subgroup lattice of C322C16 in TeX
Character table of C322C16 in TeX

׿
×
𝔽