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G = C82⋊C3order 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

C1C82 — C82⋊C3
C1C22C42C82 — C82⋊C3
C82 — C82⋊C3
C1

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

3C2
64C3
3C4
3C4
3C2×C4
3C8
3C8
3C8
3C8
16A4
3C2×C8
3C2×C8
3C4×C8
4C42⋊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+2ζ85i8382-i8782-i1+2-1+-21-2-1--21-2-1+-2    complex faithful
ρ103-100-1-2i11-1+2i8382-i1+2-1+-21-2-1--21-2-1+-21+2-1--2ζ86+2ζ85iζ86+2ζ8i8782-i    complex faithful
ρ113-100-1+2i11-1-2iiζ86+2ζ8-1+-21-2-1--21+2-1+-21+2-1--21-2-i8782-i8382iζ86+2ζ85    complex faithful
ρ123-100-1-2i11-1+2i-i8382-1+-21+2-1--21-2-1+-21-2-1--21+2iζ86+2ζ85iζ86+2ζ8-i8782    complex faithful
ρ133-1001-1-2i-1+2i11-2-1--28382-i8782-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--28782-i8382-iζ86+2ζ85i    complex faithful
ρ153-100-1+2i11-1-2iζ86+2ζ85i1+2-1--21-2-1+-21-2-1--21+2-1+-28382-i8782-iζ86+2ζ8i    complex faithful
ρ163-100-1-2i11-1+2i-i8782-1--21-2-1+-21+2-1--21+2-1+-21-2iζ86+2ζ8iζ86+2ζ85-i8382    complex faithful
ρ173-1001-1+2i-1-2i11-2-1+-2ζ86+2ζ85iζ86+2ζ8i8782-i8382-i1-2-1--21+2-1+-21+2-1--2    complex faithful
ρ183-1001-1-2i-1+2i1-1--21-2-i8382-i8782iζ86+2ζ8iζ86+2ζ85-1+-21-2-1--21+2-1+-21+2    complex faithful
ρ193-1001-1-2i-1+2i1-1+-21+2-i8782-i8382iζ86+2ζ85iζ86+2ζ8-1--21+2-1+-21-2-1--21-2    complex faithful
ρ203-1001-1-2i-1+2i11+2-1+-28782-i8382-iζ86+2ζ85iζ86+2ζ8i1+2-1--21-2-1+-21-2-1--2    complex faithful
ρ213-100-1-2i11-1+2i8782-i1-2-1--21+2-1+-21+2-1--21-2-1+-2ζ86+2ζ8iζ86+2ζ85i8382-i    complex faithful
ρ223-1001-1+2i-1-2i1-1--21+2iζ86+2ζ8iζ86+2ζ85-i8382-i8782-1+-21+2-1--21-2-1+-21-2    complex faithful
ρ233-1001-1+2i-1-2i1-1+-21-2iζ86+2ζ85iζ86+2ζ8-i8782-i8382-1--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-i8382-i8782iζ86+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 23 9)(2 17 11)(3 19 13)(4 21 15)(5 18 16)(6 20 10)(7 22 12)(8 24 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), (1,23,9)(2,17,11)(3,19,13)(4,21,15)(5,18,16)(6,20,10)(7,22,12)(8,24,14)>;

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

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

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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