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G = C322D8order 144 = 24·32

1st semidirect product of C32 and D8 acting via D8/C4=C22

metabelian, supersoluble, monomial

Aliases: C322D8, D122S3, C12.9D6, C4.8S32, (C3×C6).6D4, C32(D4⋊S3), (C3×D12)⋊1C2, C324C81C2, C6.7(C3⋊D4), (C3×C12).1C22, C2.3(D6⋊S3), SmallGroup(144,56)

Series: Derived Chief Lower central Upper central

C1C3×C12 — C322D8
C1C3C32C3×C6C3×C12C3×D12 — C322D8
C32C3×C6C3×C12 — C322D8
C1C2C4

Generators and relations for C322D8
 G = < a,b,c,d | a3=b3=c8=d2=1, ab=ba, cac-1=a-1, ad=da, cbc-1=dbd=b-1, dcd=c-1 >

12C2
12C2
2C3
6C22
6C22
2C6
4S3
4S3
12C6
12C6
3D4
3D4
9C8
2D6
2C12
2D6
6C2×C6
6C2×C6
4C3×S3
4C3×S3
9D8
3C3×D4
3C3⋊C8
3C3⋊C8
3C3×D4
6C3⋊C8
2S3×C6
2S3×C6
3D4⋊S3
3D4⋊S3

Character table of C322D8

 class 12A2B2C3A3B3C46A6B6C6D6E6F6G8A8B12A12B12C12D
 size 11121222422241212121218184444
ρ1111111111111111111111    trivial
ρ211-1-11111111-1-1-1-1111111    linear of order 2
ρ3111-11111111-1-111-1-11111    linear of order 2
ρ411-11111111111-1-1-1-11111    linear of order 2
ρ52200222-2222000000-2-2-2-2    orthogonal lifted from D4
ρ622-202-1-12-12-10011002-1-1-1    orthogonal lifted from D6
ρ722202-1-12-12-100-1-1002-1-1-1    orthogonal lifted from S3
ρ82202-12-122-1-1-1-10000-12-1-1    orthogonal lifted from S3
ρ9220-2-12-122-1-1110000-12-1-1    orthogonal lifted from D6
ρ102-2002220-2-2-200002-20000    orthogonal lifted from D8
ρ112-2002220-2-2-20000-220000    orthogonal lifted from D8
ρ1222002-1-1-2-12-100--3-300-2111    complex lifted from C3⋊D4
ρ1322002-1-1-2-12-100-3--300-2111    complex lifted from C3⋊D4
ρ142200-12-1-22-1-1-3--300001-211    complex lifted from C3⋊D4
ρ152200-12-1-22-1-1--3-300001-211    complex lifted from C3⋊D4
ρ164-4004-2-202-420000000000    orthogonal lifted from D4⋊S3, Schur index 2
ρ174-400-24-20-4220000000000    orthogonal lifted from D4⋊S3, Schur index 2
ρ184400-2-214-2-21000000-2-211    orthogonal lifted from S32
ρ194400-2-21-4-2-2100000022-1-1    symplectic lifted from D6⋊S3, Schur index 2
ρ204-400-2-21022-1000000003i-3i    complex faithful
ρ214-400-2-21022-100000000-3i3i    complex faithful

Smallest permutation representation of C322D8
On 48 points
Generators in S48
(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 20 26)(10 27 21)(11 22 28)(12 29 23)(13 24 30)(14 31 17)(15 18 32)(16 25 19)
(1 46 35)(2 36 47)(3 48 37)(4 38 41)(5 42 39)(6 40 43)(7 44 33)(8 34 45)(9 20 26)(10 27 21)(11 22 28)(12 29 23)(13 24 30)(14 31 17)(15 18 32)(16 25 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 23)(2 22)(3 21)(4 20)(5 19)(6 18)(7 17)(8 24)(9 38)(10 37)(11 36)(12 35)(13 34)(14 33)(15 40)(16 39)(25 42)(26 41)(27 48)(28 47)(29 46)(30 45)(31 44)(32 43)

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

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

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

C322D8 is a maximal subgroup of
C32⋊D16  C32⋊SD32  C244D6  C246D6  D12.2D6  D12.30D6  D1220D6  S3×D4⋊S3  D129D6  D126D6  D12.12D6  D36⋊S3  He33D8  C336D8  C339D8
C322D8 is a maximal quotient of
C322D16  D24.S3  C322Q32  D123Dic3  C12.8Dic6  D36⋊S3  He32D8  C336D8  C339D8

Matrix representation of C322D8 in GL4(𝔽5) generated by

4011
1421
2444
2111
,
4340
0401
1203
0400
,
0023
1114
0433
0011
,
2112
4231
1122
1044
G:=sub<GL(4,GF(5))| [4,1,2,2,0,4,4,1,1,2,4,1,1,1,4,1],[4,0,1,0,3,4,2,4,4,0,0,0,0,1,3,0],[0,1,0,0,0,1,4,0,2,1,3,1,3,4,3,1],[2,4,1,1,1,2,1,0,1,3,2,4,2,1,2,4] >;

C322D8 in GAP, Magma, Sage, TeX

C_3^2\rtimes_2D_8
% in TeX

G:=Group("C3^2:2D8");
// GroupNames label

G:=SmallGroup(144,56);
// by ID

G=gap.SmallGroup(144,56);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-3,73,218,116,50,490,3461]);
// Polycyclic

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

Export

Subgroup lattice of C322D8 in TeX
Character table of C322D8 in TeX

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