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## G = C2×C3⋊C8order 48 = 24·3

### Direct product of C2 and C3⋊C8

Aliases: C2×C3⋊C8, C6⋊C8, C12.3C4, C4.14D6, C4.3Dic3, C12.14C22, C22.2Dic3, C4(C3⋊C8), C32(C2×C8), (C2×C4).5S3, C6.5(C2×C4), (C2×C6).2C4, (C2×C12).5C2, C2.1(C2×Dic3), SmallGroup(48,9)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — C2×C3⋊C8
 Chief series C1 — C3 — C6 — C12 — C3⋊C8 — C2×C3⋊C8
 Lower central C3 — C2×C3⋊C8
 Upper central C1 — C2×C4

Generators and relations for C2×C3⋊C8
G = < a,b,c | a2=b3=c8=1, ab=ba, ac=ca, cbc-1=b-1 >

Character table of C2×C3⋊C8

 class 1 2A 2B 2C 3 4A 4B 4C 4D 6A 6B 6C 8A 8B 8C 8D 8E 8F 8G 8H 12A 12B 12C 12D size 1 1 1 1 2 1 1 1 1 2 2 2 3 3 3 3 3 3 3 3 2 2 2 2 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 -i i -i -i i i -i i -1 -1 -1 -1 linear of order 4 ρ6 1 1 -1 -1 1 1 -1 -1 1 -1 1 -1 i i -i -i -i -i i i -1 1 -1 1 linear of order 4 ρ7 1 1 1 1 1 -1 -1 -1 -1 1 1 1 i -i i i -i -i i -i -1 -1 -1 -1 linear of order 4 ρ8 1 1 -1 -1 1 1 -1 -1 1 -1 1 -1 -i -i i i i i -i -i -1 1 -1 1 linear of order 4 ρ9 1 -1 1 -1 1 i -i i -i 1 -1 -1 ζ87 ζ85 ζ83 ζ87 ζ8 ζ85 ζ83 ζ8 -i i i -i linear of order 8 ρ10 1 -1 -1 1 1 i i -i -i -1 -1 1 ζ85 ζ83 ζ85 ζ8 ζ83 ζ87 ζ8 ζ87 i i -i -i linear of order 8 ρ11 1 -1 1 -1 1 -i i -i i 1 -1 -1 ζ8 ζ83 ζ85 ζ8 ζ87 ζ83 ζ85 ζ87 i -i -i i linear of order 8 ρ12 1 -1 -1 1 1 -i -i i i -1 -1 1 ζ83 ζ85 ζ83 ζ87 ζ85 ζ8 ζ87 ζ8 -i -i i i linear of order 8 ρ13 1 -1 1 -1 1 i -i i -i 1 -1 -1 ζ83 ζ8 ζ87 ζ83 ζ85 ζ8 ζ87 ζ85 -i i i -i linear of order 8 ρ14 1 -1 -1 1 1 i i -i -i -1 -1 1 ζ8 ζ87 ζ8 ζ85 ζ87 ζ83 ζ85 ζ83 i i -i -i linear of order 8 ρ15 1 -1 -1 1 1 -i -i i i -1 -1 1 ζ87 ζ8 ζ87 ζ83 ζ8 ζ85 ζ83 ζ85 -i -i i i linear of order 8 ρ16 1 -1 1 -1 1 -i i -i i 1 -1 -1 ζ85 ζ87 ζ8 ζ85 ζ83 ζ87 ζ8 ζ83 i -i -i i linear of order 8 ρ17 2 2 -2 -2 -1 -2 2 2 -2 1 -1 1 0 0 0 0 0 0 0 0 -1 1 -1 1 orthogonal lifted from D6 ρ18 2 2 2 2 -1 2 2 2 2 -1 -1 -1 0 0 0 0 0 0 0 0 -1 -1 -1 -1 orthogonal lifted from S3 ρ19 2 2 -2 -2 -1 2 -2 -2 2 1 -1 1 0 0 0 0 0 0 0 0 1 -1 1 -1 symplectic lifted from Dic3, Schur index 2 ρ20 2 2 2 2 -1 -2 -2 -2 -2 -1 -1 -1 0 0 0 0 0 0 0 0 1 1 1 1 symplectic lifted from Dic3, Schur index 2 ρ21 2 -2 2 -2 -1 -2i 2i -2i 2i -1 1 1 0 0 0 0 0 0 0 0 -i i i -i complex lifted from C3⋊C8 ρ22 2 -2 -2 2 -1 2i 2i -2i -2i 1 1 -1 0 0 0 0 0 0 0 0 -i -i i i complex lifted from C3⋊C8 ρ23 2 -2 -2 2 -1 -2i -2i 2i 2i 1 1 -1 0 0 0 0 0 0 0 0 i i -i -i complex lifted from C3⋊C8 ρ24 2 -2 2 -2 -1 2i -2i 2i -2i -1 1 1 0 0 0 0 0 0 0 0 i -i -i i complex lifted from C3⋊C8

Smallest permutation representation of C2×C3⋊C8
Regular action on 48 points
Generators in S48
(1 19)(2 20)(3 21)(4 22)(5 23)(6 24)(7 17)(8 18)(9 45)(10 46)(11 47)(12 48)(13 41)(14 42)(15 43)(16 44)(25 40)(26 33)(27 34)(28 35)(29 36)(30 37)(31 38)(32 39)
(1 28 45)(2 46 29)(3 30 47)(4 48 31)(5 32 41)(6 42 25)(7 26 43)(8 44 27)(9 19 35)(10 36 20)(11 21 37)(12 38 22)(13 23 39)(14 40 24)(15 17 33)(16 34 18)
(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)| (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,17)(8,18)(9,45)(10,46)(11,47)(12,48)(13,41)(14,42)(15,43)(16,44)(25,40)(26,33)(27,34)(28,35)(29,36)(30,37)(31,38)(32,39), (1,28,45)(2,46,29)(3,30,47)(4,48,31)(5,32,41)(6,42,25)(7,26,43)(8,44,27)(9,19,35)(10,36,20)(11,21,37)(12,38,22)(13,23,39)(14,40,24)(15,17,33)(16,34,18), (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( (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,17)(8,18)(9,45)(10,46)(11,47)(12,48)(13,41)(14,42)(15,43)(16,44)(25,40)(26,33)(27,34)(28,35)(29,36)(30,37)(31,38)(32,39), (1,28,45)(2,46,29)(3,30,47)(4,48,31)(5,32,41)(6,42,25)(7,26,43)(8,44,27)(9,19,35)(10,36,20)(11,21,37)(12,38,22)(13,23,39)(14,40,24)(15,17,33)(16,34,18), (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([[(1,19),(2,20),(3,21),(4,22),(5,23),(6,24),(7,17),(8,18),(9,45),(10,46),(11,47),(12,48),(13,41),(14,42),(15,43),(16,44),(25,40),(26,33),(27,34),(28,35),(29,36),(30,37),(31,38),(32,39)], [(1,28,45),(2,46,29),(3,30,47),(4,48,31),(5,32,41),(6,42,25),(7,26,43),(8,44,27),(9,19,35),(10,36,20),(11,21,37),(12,38,22),(13,23,39),(14,40,24),(15,17,33),(16,34,18)], [(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×C3⋊C8 is a maximal subgroup of
C42.S3  C12⋊C8  C6.Q16  C12.Q8  C6.D8  C6.SD16  C8×Dic3  Dic3⋊C8  C24⋊C4  D6⋊C8  C12.53D4  C12.55D4  D4⋊Dic3  Q82Dic3  S3×C2×C8  D12.C4  D4.Dic3  Q8.13D6  C2.U2(𝔽3)  C60.C4  C337(C2×C8)
C2×C3⋊C8 is a maximal quotient of
C12⋊C8  C12.C8  C12.55D4  C60.C4  C337(C2×C8)

Matrix representation of C2×C3⋊C8 in GL3(𝔽73) generated by

 72 0 0 0 1 0 0 0 1
,
 1 0 0 0 72 72 0 1 0
,
 63 0 0 0 66 41 0 48 7
G:=sub<GL(3,GF(73))| [72,0,0,0,1,0,0,0,1],[1,0,0,0,72,1,0,72,0],[63,0,0,0,66,48,0,41,7] >;

C2×C3⋊C8 in GAP, Magma, Sage, TeX

C_2\times C_3\rtimes C_8
% in TeX

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

G:=SmallGroup(48,9);
// by ID

G=gap.SmallGroup(48,9);
# by ID

G:=PCGroup([5,-2,-2,-2,-2,-3,20,42,804]);
// Polycyclic

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

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