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G = C82⋊C3  order 192 = 26·3

The semidirect product of C82 and C3 acting faithfully

metabelian, soluble, monomial, A-group

Aliases: C82⋊C3, C42.2A4, C22.(C42⋊C3), SmallGroup(192,3)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C82 — C82⋊C3
Chief series
C1 — C22 — C42 — C82 — C82⋊C3
Lower central
C82 — C82⋊C3
Upper central
C1

Generators and relations for C82⋊C3
 G = < a,b,c | a8=b8=c3=1, ab=ba, cac-1=ab-1, cbc-1=a3b6 >

C1
3C2
64C3
C22
3C4
3C4
3C2×C4
3C8
3C8
3C8
3C8
16A4
C42
3C2×C8
3C2×C8
3C4×C8
4C42⋊C3
C82
C82⋊C3

Character table of C82⋊C3

 class 123A3B4A4B4C4D8A8B8C8D8E8F8G8H8I8J8K8L8M8N8O8P
 size 13646433333333333333333333
ρ1111111111111111111111111    trivial
ρ211ζ3ζ3211111111111111111111    linear of order 3
ρ311ζ32ζ311111111111111111111    linear of order 3
ρ433003333-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1    orthogonal lifted from A4
ρ53300-1-1-1-1-1+2i-1+2i11111111-1-2i-1-2i-1-2i-1-2i-1+2i-1+2i    complex lifted from C42⋊C3
ρ63300-1-1-1-1-1-2i-1-2i11111111-1+2i-1+2i-1+2i-1+2i-1-2i-1-2i    complex lifted from C42⋊C3
ρ73300-1-1-1-111-1+2i-1+2i-1+2i-1+2i-1-2i-1-2i-1-2i-1-2i111111    complex lifted from C42⋊C3
ρ83300-1-1-1-111-1-2i-1-2i-1-2i-1-2i-1+2i-1+2i-1+2i-1+2i111111    complex lifted from C42⋊C3
ρ93-1001-1+2i-1-2i11+√2-1-√-2ζ86+2ζ8iζ86+2ζ85i2ζ83+ζ82-i2ζ87+ζ82-i1+√2-1+√-21-√2-1-√-21-√2-1+√-2    complex faithful
ρ103-100-1-2i11-1+2i2ζ83+ζ82-i1+√2-1+√-21-√2-1-√-21-√2-1+√-21+√2-1-√-2ζ86+2ζ85iζ86+2ζ8i2ζ87+ζ82-i    complex faithful
ρ113-100-1+2i11-1-2iiζ86+2ζ8-1+√-21-√2-1-√-21+√2-1+√-21+√2-1-√-21-√2-i2ζ87+ζ82-i2ζ83+ζ82iζ86+2ζ85    complex faithful
ρ123-100-1-2i11-1+2i-i2ζ83+ζ82-1+√-21+√2-1-√-21-√2-1+√-21-√2-1-√-21+√2iζ86+2ζ85iζ86+2ζ8-i2ζ87+ζ82    complex faithful
ρ133-1001-1-2i-1+2i11-√2-1-√-22ζ83+ζ82-i2ζ87+ζ82-iζ86+2ζ8iζ86+2ζ85i1-√2-1+√-21+√2-1-√-21+√2-1+√-2    complex faithful
ρ143-100-1+2i11-1-2iζ86+2ζ8i1-√2-1+√-21+√2-1-√-21+√2-1+√-21-√2-1-√-22ζ87+ζ82-i2ζ83+ζ82-iζ86+2ζ85i    complex faithful
ρ153-100-1+2i11-1-2iζ86+2ζ85i1+√2-1-√-21-√2-1+√-21-√2-1-√-21+√2-1+√-22ζ83+ζ82-i2ζ87+ζ82-iζ86+2ζ8i    complex faithful
ρ163-100-1-2i11-1+2i-i2ζ87+ζ82-1-√-21-√2-1+√-21+√2-1-√-21+√2-1+√-21-√2iζ86+2ζ8iζ86+2ζ85-i2ζ83+ζ82    complex faithful
ρ173-1001-1+2i-1-2i11-√2-1+√-2ζ86+2ζ85iζ86+2ζ8i2ζ87+ζ82-i2ζ83+ζ82-i1-√2-1-√-21+√2-1+√-21+√2-1-√-2    complex faithful
ρ183-1001-1-2i-1+2i1-1-√-21-√2-i2ζ83+ζ82-i2ζ87+ζ82iζ86+2ζ8iζ86+2ζ85-1+√-21-√2-1-√-21+√2-1+√-21+√2    complex faithful
ρ193-1001-1-2i-1+2i1-1+√-21+√2-i2ζ87+ζ82-i2ζ83+ζ82iζ86+2ζ85iζ86+2ζ8-1-√-21+√2-1+√-21-√2-1-√-21-√2    complex faithful
ρ203-1001-1-2i-1+2i11+√2-1+√-22ζ87+ζ82-i2ζ83+ζ82-iζ86+2ζ85iζ86+2ζ8i1+√2-1-√-21-√2-1+√-21-√2-1-√-2    complex faithful
ρ213-100-1-2i11-1+2i2ζ87+ζ82-i1-√2-1-√-21+√2-1+√-21+√2-1-√-21-√2-1+√-2ζ86+2ζ8iζ86+2ζ85i2ζ83+ζ82-i    complex faithful
ρ223-1001-1+2i-1-2i1-1-√-21+√2iζ86+2ζ8iζ86+2ζ85-i2ζ83+ζ82-i2ζ87+ζ82-1+√-21+√2-1-√-21-√2-1+√-21-√2    complex faithful
ρ233-1001-1+2i-1-2i1-1+√-21-√2iζ86+2ζ85iζ86+2ζ8-i2ζ87+ζ82-i2ζ83+ζ82-1-√-21-√2-1+√-21+√2-1-√-21+√2    complex faithful
ρ243-100-1+2i11-1-2iiζ86+2ζ85-1-√-21+√2-1+√-21-√2-1-√-21-√2-1+√-21+√2-i2ζ83+ζ82-i2ζ87+ζ82iζ86+2ζ8    complex faithful

Permutation representations of C82⋊C3
►On 24 points - transitive group 24T389
Generators in S24
(1 2 3 4)(5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)

(1 8 2 5 3 6 4 7)(9 16 15 14 13 12 11 10)(17 21)(18 22)(19 23)(20 24)

(1 18 9)(2 20 11)(3 22 13)(4 24 15)(5 21 16)(6 23 10)(7 17 12)(8 19 14)
 
G:=sub<Sym(24)| (1,2,3,4)(5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24), (1,8,2,5,3,6,4,7)(9,16,15,14,13,12,11,10)(17,21)(18,22)(19,23)(20,24), (1,18,9)(2,20,11)(3,22,13)(4,24,15)(5,21,16)(6,23,10)(7,17,12)(8,19,14)>;
 
G:=Group( (1,2,3,4)(5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24), (1,8,2,5,3,6,4,7)(9,16,15,14,13,12,11,10)(17,21)(18,22)(19,23)(20,24), (1,18,9)(2,20,11)(3,22,13)(4,24,15)(5,21,16)(6,23,10)(7,17,12)(8,19,14) );
 
G=PermutationGroup([[(1,2,3,4),(5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24)], [(1,8,2,5,3,6,4,7),(9,16,15,14,13,12,11,10),(17,21),(18,22),(19,23),(20,24)], [(1,18,9),(2,20,11),(3,22,13),(4,24,15),(5,21,16),(6,23,10),(7,17,12),(8,19,14)]])
 
G:=TransitiveGroup(24,389);
 

Matrix representation of C82⋊C3 ►in GL3(𝔽73) generated by

4600
0630
0063
,
1000
0220
001
,
010
001
100
G:=sub<GL(3,GF(73))| [46,0,0,0,63,0,0,0,63],[10,0,0,0,22,0,0,0,1],[0,0,1,1,0,0,0,1,0] >;
 

C82⋊C3 in GAP, Magma, Sage, TeX 

C_8^2\rtimes C_3
 
% in TeX
 
G:=Group("C8^2:C3");
 
// GroupNames label
 
G:=SmallGroup(192,3);
 
// by ID
 
G=gap.SmallGroup(192,3);
 
# by ID
 
G:=PCGroup([7,-3,-2,2,-2,2,-2,2,85,176,695,394,4707,360,1264,102,4037,7062]);
 
// Polycyclic
 
G:=Group<a,b,c|a^8=b^8=c^3=1,a*b=b*a,c*a*c^-1=a*b^-1,c*b*c^-1=a^3*b^6>;
 
// generators/relations
 

Export

Subgroup lattice of C82⋊C3 in TeX
Character table of C82⋊C3 in TeX

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