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G = C2×C3⋊C8  order 48 = 24·3

Direct product of C2 and C3⋊C8

direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary

Aliases: C2×C3⋊C8, C6⋊C8, C12.3C4, C4.14D6, C4.3Dic3, C12.14C22, C22.2Dic3, C4○(C3⋊C8), C3⋊2(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 >

C1
C2
C2
C2
C3
C4
C22
C4
C6
C6
C6
C2×C4
3C8
3C8
C2×C6
C12
C12
3C2×C8
C3⋊C8
C2×C12
C3⋊C8
C2×C3⋊C8

Character table of C2×C3⋊C8

 class 12A2B2C34A4B4C4D6A6B6C8A8B8C8D8E8F8G8H12A12B12C12D
 size 111121111222333333332222
ρ1111111111111111111111111    trivial
ρ2111111111111-1-1-1-1-1-1-1-11111    linear of order 2
ρ311-1-11-111-1-11-1-1111-1-1-111-11-1    linear of order 2
ρ411-1-11-111-1-11-11-1-1-1111-11-11-1    linear of order 2
ρ511111-1-1-1-1111-ii-i-iii-ii-1-1-1-1    linear of order 4
ρ611-1-111-1-11-11-1ii-i-i-i-iii-11-11    linear of order 4
ρ711111-1-1-1-1111i-iii-i-ii-i-1-1-1-1    linear of order 4
ρ811-1-111-1-11-11-1-i-iiiii-i-i-11-11    linear of order 4
ρ91-11-11i-ii-i1-1-1ζ87ζ85ζ83ζ87ζ8ζ85ζ83ζ8-iii-i    linear of order 8
ρ101-1-111ii-i-i-1-11ζ85ζ83ζ85ζ8ζ83ζ87ζ8ζ87ii-i-i    linear of order 8
ρ111-11-11-ii-ii1-1-1ζ8ζ83ζ85ζ8ζ87ζ83ζ85ζ87i-i-ii    linear of order 8
ρ121-1-111-i-iii-1-11ζ83ζ85ζ83ζ87ζ85ζ8ζ87ζ8-i-iii    linear of order 8
ρ131-11-11i-ii-i1-1-1ζ83ζ8ζ87ζ83ζ85ζ8ζ87ζ85-iii-i    linear of order 8
ρ141-1-111ii-i-i-1-11ζ8ζ87ζ8ζ85ζ87ζ83ζ85ζ83ii-i-i    linear of order 8
ρ151-1-111-i-iii-1-11ζ87ζ8ζ87ζ83ζ8ζ85ζ83ζ85-i-iii    linear of order 8
ρ161-11-11-ii-ii1-1-1ζ85ζ87ζ8ζ85ζ83ζ87ζ8ζ83i-i-ii    linear of order 8
ρ1722-2-2-1-222-21-1100000000-11-11    orthogonal lifted from D6
ρ182222-12222-1-1-100000000-1-1-1-1    orthogonal lifted from S3
ρ1922-2-2-12-2-221-11000000001-11-1    symplectic lifted from Dic3, Schur index 2
ρ202222-1-2-2-2-2-1-1-1000000001111    symplectic lifted from Dic3, Schur index 2
ρ212-22-2-1-2i2i-2i2i-11100000000-iii-i    complex lifted from C3⋊C8
ρ222-2-22-12i2i-2i-2i11-100000000-i-iii    complex lifted from C3⋊C8
ρ232-2-22-1-2i-2i2i2i11-100000000ii-i-i    complex lifted from C3⋊C8
ρ242-22-2-12i-2i2i-2i-11100000000i-i-ii    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  Q8⋊2Dic3  S3×C2×C8  D12.C4  D4.Dic3  Q8.13D6  C2.U2(𝔽3)  C60.C4  C33⋊7(C2×C8)
C2×C3⋊C8 is a maximal quotient of
 C12⋊C8  C12.C8  C12.55D4  C60.C4  C33⋊7(C2×C8)

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

7200
010
001
,
100
07272
010
,
6300
06641
0487
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
 

Export

Subgroup lattice of C2×C3⋊C8 in TeX
Character table of C2×C3⋊C8 in TeX

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