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G = C2×C17⋊C4order 136 = 23·17

Direct product of C2 and C17⋊C4

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

Aliases: C2×C17⋊C4, C34⋊C4, D17⋊C4, D34.C2, D17.C22, C17⋊(C2×C4), SmallGroup(136,13)

Series: Derived Chief Lower central Upper central

C1C17 — C2×C17⋊C4
C1C17D17C17⋊C4 — C2×C17⋊C4
C17 — C2×C17⋊C4
C1C2

Generators and relations for C2×C17⋊C4
 G = < a,b,c | a2=b17=c4=1, ab=ba, ac=ca, cbc-1=b4 >

17C2
17C2
17C4
17C22
17C4
17C2×C4

Character table of C2×C17⋊C4

 class 12A2B2C4A4B4C4D17A17B17C17D34A34B34C34D
 size 1117171717171744444444
ρ11111111111111111    trivial
ρ21-11-1-11-111111-1-1-1-1    linear of order 2
ρ31-11-11-11-11111-1-1-1-1    linear of order 2
ρ41111-1-1-1-111111111    linear of order 2
ρ511-1-1ii-i-i11111111    linear of order 4
ρ61-1-11-iii-i1111-1-1-1-1    linear of order 4
ρ711-1-1-i-iii11111111    linear of order 4
ρ81-1-11i-i-ii1111-1-1-1-1    linear of order 4
ρ94-4000000ζ1716171317417ζ17111710177176ζ17141712175173ζ1715179178172171617131741717111710177176171417121751731715179178172    orthogonal faithful
ρ104-4000000ζ17111710177176ζ1715179178172ζ1716171317417ζ17141712175173171117101771761715179178172171617131741717141712175173    orthogonal faithful
ρ1144000000ζ17141712175173ζ1716171317417ζ1715179178172ζ17111710177176ζ17141712175173ζ1716171317417ζ1715179178172ζ17111710177176    orthogonal lifted from C17⋊C4
ρ124-4000000ζ17141712175173ζ1716171317417ζ1715179178172ζ17111710177176171417121751731716171317417171517917817217111710177176    orthogonal faithful
ρ1344000000ζ1716171317417ζ17111710177176ζ17141712175173ζ1715179178172ζ1716171317417ζ17111710177176ζ17141712175173ζ1715179178172    orthogonal lifted from C17⋊C4
ρ1444000000ζ17111710177176ζ1715179178172ζ1716171317417ζ17141712175173ζ17111710177176ζ1715179178172ζ1716171317417ζ17141712175173    orthogonal lifted from C17⋊C4
ρ1544000000ζ1715179178172ζ17141712175173ζ17111710177176ζ1716171317417ζ1715179178172ζ17141712175173ζ17111710177176ζ1716171317417    orthogonal lifted from C17⋊C4
ρ164-4000000ζ1715179178172ζ17141712175173ζ17111710177176ζ1716171317417171517917817217141712175173171117101771761716171317417    orthogonal faithful

Smallest permutation representation of C2×C17⋊C4
On 34 points
Generators in S34
(1 18)(2 19)(3 20)(4 21)(5 22)(6 23)(7 24)(8 25)(9 26)(10 27)(11 28)(12 29)(13 30)(14 31)(15 32)(16 33)(17 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)
(2 14 17 5)(3 10 16 9)(4 6 15 13)(7 11 12 8)(19 31 34 22)(20 27 33 26)(21 23 32 30)(24 28 29 25)

G:=sub<Sym(34)| (1,18)(2,19)(3,20)(4,21)(5,22)(6,23)(7,24)(8,25)(9,26)(10,27)(11,28)(12,29)(13,30)(14,31)(15,32)(16,33)(17,34), (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), (2,14,17,5)(3,10,16,9)(4,6,15,13)(7,11,12,8)(19,31,34,22)(20,27,33,26)(21,23,32,30)(24,28,29,25)>;

G:=Group( (1,18)(2,19)(3,20)(4,21)(5,22)(6,23)(7,24)(8,25)(9,26)(10,27)(11,28)(12,29)(13,30)(14,31)(15,32)(16,33)(17,34), (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), (2,14,17,5)(3,10,16,9)(4,6,15,13)(7,11,12,8)(19,31,34,22)(20,27,33,26)(21,23,32,30)(24,28,29,25) );

G=PermutationGroup([[(1,18),(2,19),(3,20),(4,21),(5,22),(6,23),(7,24),(8,25),(9,26),(10,27),(11,28),(12,29),(13,30),(14,31),(15,32),(16,33),(17,34)], [(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)], [(2,14,17,5),(3,10,16,9),(4,6,15,13),(7,11,12,8),(19,31,34,22),(20,27,33,26),(21,23,32,30),(24,28,29,25)]])

C2×C17⋊C4 is a maximal subgroup of   C68⋊C4  D17.D4
C2×C17⋊C4 is a maximal quotient of   C68.C4  D34.4C4  C68⋊C4  C17⋊M4(2)  D17.D4

Matrix representation of C2×C17⋊C4 in GL4(𝔽137) generated by

136000
013600
001360
000136
,
7511474136
742121115
32886111
119114966
,
11477107132
54491518
201129372
105116718
G:=sub<GL(4,GF(137))| [136,0,0,0,0,136,0,0,0,0,136,0,0,0,0,136],[75,7,32,119,114,42,88,11,74,121,6,49,136,115,111,66],[114,54,20,105,77,49,112,11,107,15,93,67,132,18,72,18] >;

C2×C17⋊C4 in GAP, Magma, Sage, TeX

C_2\times C_{17}\rtimes C_4
% in TeX

G:=Group("C2xC17:C4");
// GroupNames label

G:=SmallGroup(136,13);
// by ID

G=gap.SmallGroup(136,13);
# by ID

G:=PCGroup([4,-2,-2,-2,-17,16,1667,523]);
// Polycyclic

G:=Group<a,b,c|a^2=b^17=c^4=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^4>;
// generators/relations

Export

Subgroup lattice of C2×C17⋊C4 in TeX
Character table of C2×C17⋊C4 in TeX

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