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## G = C2×C32⋊2C8order 144 = 24·32

### Direct product of C2 and C32⋊2C8

Aliases: C2×C322C8, C62.1C4, (C3×C6)⋊2C8, C325(C2×C8), C3⋊Dic3.5C4, C22.2(C32⋊C4), C3⋊Dic3.9C22, (C3×C6).5(C2×C4), C2.3(C2×C32⋊C4), (C2×C3⋊Dic3).6C2, SmallGroup(144,134)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C2×C32⋊2C8
 Chief series C1 — C32 — C3×C6 — C3⋊Dic3 — C32⋊2C8 — C2×C32⋊2C8
 Lower central C32 — C2×C32⋊2C8
 Upper central C1 — C22

Generators and relations for C2×C322C8
G = < a,b,c,d | a2=b3=c3=d8=1, ab=ba, ac=ca, ad=da, dcd-1=bc=cb, dbd-1=b-1c >

Character table of C2×C322C8

 class 1 2A 2B 2C 3A 3B 4A 4B 4C 4D 6A 6B 6C 6D 6E 6F 8A 8B 8C 8D 8E 8F 8G 8H size 1 1 1 1 4 4 9 9 9 9 4 4 4 4 4 4 9 9 9 9 9 9 9 9 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 linear of order 2 ρ3 1 -1 -1 1 1 1 -1 1 1 -1 -1 1 1 -1 -1 -1 1 1 1 1 -1 -1 -1 -1 linear of order 2 ρ4 1 -1 -1 1 1 1 -1 1 1 -1 -1 1 1 -1 -1 -1 -1 -1 -1 -1 1 1 1 1 linear of order 2 ρ5 1 1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 1 1 -i -i i i -i -i i i linear of order 4 ρ6 1 1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 1 1 i i -i -i i i -i -i linear of order 4 ρ7 1 -1 -1 1 1 1 1 -1 -1 1 -1 1 1 -1 -1 -1 -i -i i i i i -i -i linear of order 4 ρ8 1 -1 -1 1 1 1 1 -1 -1 1 -1 1 1 -1 -1 -1 i i -i -i -i -i i i linear of order 4 ρ9 1 -1 1 -1 1 1 i i -i -i -1 -1 -1 -1 1 1 ζ87 ζ83 ζ85 ζ8 ζ87 ζ83 ζ85 ζ8 linear of order 8 ρ10 1 -1 1 -1 1 1 i i -i -i -1 -1 -1 -1 1 1 ζ83 ζ87 ζ8 ζ85 ζ83 ζ87 ζ8 ζ85 linear of order 8 ρ11 1 1 -1 -1 1 1 i -i i -i 1 -1 -1 1 -1 -1 ζ85 ζ8 ζ87 ζ83 ζ8 ζ85 ζ83 ζ87 linear of order 8 ρ12 1 1 -1 -1 1 1 i -i i -i 1 -1 -1 1 -1 -1 ζ8 ζ85 ζ83 ζ87 ζ85 ζ8 ζ87 ζ83 linear of order 8 ρ13 1 -1 1 -1 1 1 -i -i i i -1 -1 -1 -1 1 1 ζ85 ζ8 ζ87 ζ83 ζ85 ζ8 ζ87 ζ83 linear of order 8 ρ14 1 1 -1 -1 1 1 -i i -i i 1 -1 -1 1 -1 -1 ζ83 ζ87 ζ8 ζ85 ζ87 ζ83 ζ85 ζ8 linear of order 8 ρ15 1 1 -1 -1 1 1 -i i -i i 1 -1 -1 1 -1 -1 ζ87 ζ83 ζ85 ζ8 ζ83 ζ87 ζ8 ζ85 linear of order 8 ρ16 1 -1 1 -1 1 1 -i -i i i -1 -1 -1 -1 1 1 ζ8 ζ85 ζ83 ζ87 ζ8 ζ85 ζ83 ζ87 linear of order 8 ρ17 4 4 4 4 -2 1 0 0 0 0 1 -2 1 -2 1 -2 0 0 0 0 0 0 0 0 orthogonal lifted from C32⋊C4 ρ18 4 -4 -4 4 1 -2 0 0 0 0 2 1 -2 -1 2 -1 0 0 0 0 0 0 0 0 orthogonal lifted from C2×C32⋊C4 ρ19 4 4 4 4 1 -2 0 0 0 0 -2 1 -2 1 -2 1 0 0 0 0 0 0 0 0 orthogonal lifted from C32⋊C4 ρ20 4 -4 -4 4 -2 1 0 0 0 0 -1 -2 1 2 -1 2 0 0 0 0 0 0 0 0 orthogonal lifted from C2×C32⋊C4 ρ21 4 -4 4 -4 1 -2 0 0 0 0 2 -1 2 -1 -2 1 0 0 0 0 0 0 0 0 symplectic lifted from C32⋊2C8, Schur index 2 ρ22 4 4 -4 -4 -2 1 0 0 0 0 1 2 -1 -2 -1 2 0 0 0 0 0 0 0 0 symplectic lifted from C32⋊2C8, Schur index 2 ρ23 4 -4 4 -4 -2 1 0 0 0 0 -1 2 -1 2 1 -2 0 0 0 0 0 0 0 0 symplectic lifted from C32⋊2C8, Schur index 2 ρ24 4 4 -4 -4 1 -2 0 0 0 0 -2 -1 2 1 2 -1 0 0 0 0 0 0 0 0 symplectic lifted from C32⋊2C8, Schur index 2

Smallest permutation representation of C2×C322C8
On 48 points
Generators in S48
(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 9)(8 10)(17 33)(18 34)(19 35)(20 36)(21 37)(22 38)(23 39)(24 40)(25 45)(26 46)(27 47)(28 48)(29 41)(30 42)(31 43)(32 44)
(2 32 22)(4 24 26)(6 28 18)(8 20 30)(10 36 42)(12 44 38)(14 40 46)(16 48 34)
(1 31 21)(2 32 22)(3 23 25)(4 24 26)(5 27 17)(6 28 18)(7 19 29)(8 20 30)(9 35 41)(10 36 42)(11 43 37)(12 44 38)(13 39 45)(14 40 46)(15 47 33)(16 48 34)
(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)| (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,9)(8,10)(17,33)(18,34)(19,35)(20,36)(21,37)(22,38)(23,39)(24,40)(25,45)(26,46)(27,47)(28,48)(29,41)(30,42)(31,43)(32,44), (2,32,22)(4,24,26)(6,28,18)(8,20,30)(10,36,42)(12,44,38)(14,40,46)(16,48,34), (1,31,21)(2,32,22)(3,23,25)(4,24,26)(5,27,17)(6,28,18)(7,19,29)(8,20,30)(9,35,41)(10,36,42)(11,43,37)(12,44,38)(13,39,45)(14,40,46)(15,47,33)(16,48,34), (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( (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,9)(8,10)(17,33)(18,34)(19,35)(20,36)(21,37)(22,38)(23,39)(24,40)(25,45)(26,46)(27,47)(28,48)(29,41)(30,42)(31,43)(32,44), (2,32,22)(4,24,26)(6,28,18)(8,20,30)(10,36,42)(12,44,38)(14,40,46)(16,48,34), (1,31,21)(2,32,22)(3,23,25)(4,24,26)(5,27,17)(6,28,18)(7,19,29)(8,20,30)(9,35,41)(10,36,42)(11,43,37)(12,44,38)(13,39,45)(14,40,46)(15,47,33)(16,48,34), (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([(1,11),(2,12),(3,13),(4,14),(5,15),(6,16),(7,9),(8,10),(17,33),(18,34),(19,35),(20,36),(21,37),(22,38),(23,39),(24,40),(25,45),(26,46),(27,47),(28,48),(29,41),(30,42),(31,43),(32,44)], [(2,32,22),(4,24,26),(6,28,18),(8,20,30),(10,36,42),(12,44,38),(14,40,46),(16,48,34)], [(1,31,21),(2,32,22),(3,23,25),(4,24,26),(5,27,17),(6,28,18),(7,19,29),(8,20,30),(9,35,41),(10,36,42),(11,43,37),(12,44,38),(13,39,45),(14,40,46),(15,47,33),(16,48,34)], [(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)])

C2×C322C8 is a maximal subgroup of
C62.3D4  C62.4D4  C62.6D4  C62.7D4  C62.2Q8  (C3×C12)⋊4C8  C322C8⋊C4  C62.6(C2×C4)  C325(C4⋊C8)  C623C8  C22.F9  C62.13D4  C62.(C2×C4)
C2×C322C8 is a maximal quotient of
C62.4C8  (C3×C12)⋊4C8  C623C8

Matrix representation of C2×C322C8 in GL6(𝔽73)

 1 0 0 0 0 0 0 72 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 1
,
 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 72 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 72 72
,
 51 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 41 51 0 0 0 0 10 32 0 0

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,1,72],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,1,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,1,72],[51,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,41,10,0,0,0,0,51,32,0,0,1,0,0,0,0,0,0,1,0,0] >;

C2×C322C8 in GAP, Magma, Sage, TeX

C_2\times C_3^2\rtimes_2C_8
% in TeX

G:=Group("C2xC3^2:2C8");
// GroupNames label

G:=SmallGroup(144,134);
// by ID

G=gap.SmallGroup(144,134);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,3,24,50,3364,256,4613,881]);
// Polycyclic

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

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