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## G = C7×GL2(𝔽3)  order 336 = 24·3·7

### Direct product of C7 and GL2(𝔽3)

Aliases: C7×GL2(𝔽3), C14.5S4, SL2(𝔽3)⋊C14, Q8⋊(S3×C7), C2.3(C7×S4), (C7×Q8)⋊2S3, (C7×SL2(𝔽3))⋊4C2, SmallGroup(336,116)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — Q8 — SL2(𝔽3) — C7×GL2(𝔽3)
 Chief series C1 — C2 — Q8 — SL2(𝔽3) — C7×SL2(𝔽3) — C7×GL2(𝔽3)
 Lower central SL2(𝔽3) — C7×GL2(𝔽3)
 Upper central C1 — C14

Generators and relations for C7×GL2(𝔽3)
G = < a,b,c,d,e | a7=b4=d3=e2=1, c2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=ece=b-1, dbd-1=bc, ebe=b2c, dcd-1=b, ede=d-1 >

Smallest permutation representation of C7×GL2(𝔽3)
On 56 points
Generators in S56
(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 49)(50 51 52 53 54 55 56)
(1 50 39 11)(2 51 40 12)(3 52 41 13)(4 53 42 14)(5 54 36 8)(6 55 37 9)(7 56 38 10)(15 49 30 25)(16 43 31 26)(17 44 32 27)(18 45 33 28)(19 46 34 22)(20 47 35 23)(21 48 29 24)
(1 21 39 29)(2 15 40 30)(3 16 41 31)(4 17 42 32)(5 18 36 33)(6 19 37 34)(7 20 38 35)(8 45 54 28)(9 46 55 22)(10 47 56 23)(11 48 50 24)(12 49 51 25)(13 43 52 26)(14 44 53 27)
(8 33 45)(9 34 46)(10 35 47)(11 29 48)(12 30 49)(13 31 43)(14 32 44)(15 25 51)(16 26 52)(17 27 53)(18 28 54)(19 22 55)(20 23 56)(21 24 50)
(1 39)(2 40)(3 41)(4 42)(5 36)(6 37)(7 38)(8 33)(9 34)(10 35)(11 29)(12 30)(13 31)(14 32)(15 51)(16 52)(17 53)(18 54)(19 55)(20 56)(21 50)

G:=sub<Sym(56)| (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)(43,44,45,46,47,48,49)(50,51,52,53,54,55,56), (1,50,39,11)(2,51,40,12)(3,52,41,13)(4,53,42,14)(5,54,36,8)(6,55,37,9)(7,56,38,10)(15,49,30,25)(16,43,31,26)(17,44,32,27)(18,45,33,28)(19,46,34,22)(20,47,35,23)(21,48,29,24), (1,21,39,29)(2,15,40,30)(3,16,41,31)(4,17,42,32)(5,18,36,33)(6,19,37,34)(7,20,38,35)(8,45,54,28)(9,46,55,22)(10,47,56,23)(11,48,50,24)(12,49,51,25)(13,43,52,26)(14,44,53,27), (8,33,45)(9,34,46)(10,35,47)(11,29,48)(12,30,49)(13,31,43)(14,32,44)(15,25,51)(16,26,52)(17,27,53)(18,28,54)(19,22,55)(20,23,56)(21,24,50), (1,39)(2,40)(3,41)(4,42)(5,36)(6,37)(7,38)(8,33)(9,34)(10,35)(11,29)(12,30)(13,31)(14,32)(15,51)(16,52)(17,53)(18,54)(19,55)(20,56)(21,50)>;

G:=Group( (1,2,3,4,5,6,7)(8,9,10,11,12,13,14)(15,16,17,18,19,20,21)(22,23,24,25,26,27,28)(29,30,31,32,33,34,35)(36,37,38,39,40,41,42)(43,44,45,46,47,48,49)(50,51,52,53,54,55,56), (1,50,39,11)(2,51,40,12)(3,52,41,13)(4,53,42,14)(5,54,36,8)(6,55,37,9)(7,56,38,10)(15,49,30,25)(16,43,31,26)(17,44,32,27)(18,45,33,28)(19,46,34,22)(20,47,35,23)(21,48,29,24), (1,21,39,29)(2,15,40,30)(3,16,41,31)(4,17,42,32)(5,18,36,33)(6,19,37,34)(7,20,38,35)(8,45,54,28)(9,46,55,22)(10,47,56,23)(11,48,50,24)(12,49,51,25)(13,43,52,26)(14,44,53,27), (8,33,45)(9,34,46)(10,35,47)(11,29,48)(12,30,49)(13,31,43)(14,32,44)(15,25,51)(16,26,52)(17,27,53)(18,28,54)(19,22,55)(20,23,56)(21,24,50), (1,39)(2,40)(3,41)(4,42)(5,36)(6,37)(7,38)(8,33)(9,34)(10,35)(11,29)(12,30)(13,31)(14,32)(15,51)(16,52)(17,53)(18,54)(19,55)(20,56)(21,50) );

G=PermutationGroup([(1,2,3,4,5,6,7),(8,9,10,11,12,13,14),(15,16,17,18,19,20,21),(22,23,24,25,26,27,28),(29,30,31,32,33,34,35),(36,37,38,39,40,41,42),(43,44,45,46,47,48,49),(50,51,52,53,54,55,56)], [(1,50,39,11),(2,51,40,12),(3,52,41,13),(4,53,42,14),(5,54,36,8),(6,55,37,9),(7,56,38,10),(15,49,30,25),(16,43,31,26),(17,44,32,27),(18,45,33,28),(19,46,34,22),(20,47,35,23),(21,48,29,24)], [(1,21,39,29),(2,15,40,30),(3,16,41,31),(4,17,42,32),(5,18,36,33),(6,19,37,34),(7,20,38,35),(8,45,54,28),(9,46,55,22),(10,47,56,23),(11,48,50,24),(12,49,51,25),(13,43,52,26),(14,44,53,27)], [(8,33,45),(9,34,46),(10,35,47),(11,29,48),(12,30,49),(13,31,43),(14,32,44),(15,25,51),(16,26,52),(17,27,53),(18,28,54),(19,22,55),(20,23,56),(21,24,50)], [(1,39),(2,40),(3,41),(4,42),(5,36),(6,37),(7,38),(8,33),(9,34),(10,35),(11,29),(12,30),(13,31),(14,32),(15,51),(16,52),(17,53),(18,54),(19,55),(20,56),(21,50)])

56 conjugacy classes

 class 1 2A 2B 3 4 6 7A ··· 7F 8A 8B 14A ··· 14F 14G ··· 14L 21A ··· 21F 28A ··· 28F 42A ··· 42F 56A ··· 56L order 1 2 2 3 4 6 7 ··· 7 8 8 14 ··· 14 14 ··· 14 21 ··· 21 28 ··· 28 42 ··· 42 56 ··· 56 size 1 1 12 8 6 8 1 ··· 1 6 6 1 ··· 1 12 ··· 12 8 ··· 8 6 ··· 6 8 ··· 8 6 ··· 6

56 irreducible representations

 dim 1 1 1 1 2 2 2 2 3 3 4 4 type + + + + + image C1 C2 C7 C14 S3 S3×C7 GL2(𝔽3) C7×GL2(𝔽3) S4 C7×S4 GL2(𝔽3) C7×GL2(𝔽3) kernel C7×GL2(𝔽3) C7×SL2(𝔽3) GL2(𝔽3) SL2(𝔽3) C7×Q8 Q8 C7 C1 C14 C2 C7 C1 # reps 1 1 6 6 1 6 2 12 2 12 1 6

Matrix representation of C7×GL2(𝔽3) in GL2(𝔽43) generated by

 11 0 0 11
,
 15 39 35 28
,
 42 11 35 1
,
 41 9 14 1
,
 42 9 0 1
G:=sub<GL(2,GF(43))| [11,0,0,11],[15,35,39,28],[42,35,11,1],[41,14,9,1],[42,0,9,1] >;

C7×GL2(𝔽3) in GAP, Magma, Sage, TeX

C_7\times {\rm GL}_2({\mathbb F}_3)
% in TeX

G:=Group("C7xGL(2,3)");
// GroupNames label

G:=SmallGroup(336,116);
// by ID

G=gap.SmallGroup(336,116);
# by ID

G:=PCGroup([6,-2,-7,-3,-2,2,-2,506,2019,447,117,1264,286,202,88]);
// Polycyclic

G:=Group<a,b,c,d,e|a^7=b^4=d^3=e^2=1,c^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=e*c*e=b^-1,d*b*d^-1=b*c,e*b*e=b^2*c,d*c*d^-1=b,e*d*e=d^-1>;
// generators/relations

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