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G = C7×SL2(𝔽3)  order 168 = 23·3·7

Direct product of C7 and SL2(𝔽3)

direct product, non-abelian, soluble

Aliases: C7×SL2(𝔽3), Q8⋊C21, C14.2A4, C2.(C7×A4), (C7×Q8)⋊1C3, SmallGroup(168,22)

Series: Derived Chief Lower central Upper central

C1C2Q8 — C7×SL2(𝔽3)
C1C2Q8C7×Q8 — C7×SL2(𝔽3)
Q8 — C7×SL2(𝔽3)
C1C14

Generators and relations for C7×SL2(𝔽3)
 G = < a,b,c,d | a7=b4=d3=1, c2=b2, ab=ba, ac=ca, ad=da, cbc-1=b-1, dbd-1=c, dcd-1=bc >

4C3
3C4
4C6
4C21
3C28
4C42

Smallest permutation representation of C7×SL2(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 40 31 48)(2 41 32 49)(3 42 33 43)(4 36 34 44)(5 37 35 45)(6 38 29 46)(7 39 30 47)(8 26 54 17)(9 27 55 18)(10 28 56 19)(11 22 50 20)(12 23 51 21)(13 24 52 15)(14 25 53 16)
(1 22 31 20)(2 23 32 21)(3 24 33 15)(4 25 34 16)(5 26 35 17)(6 27 29 18)(7 28 30 19)(8 45 54 37)(9 46 55 38)(10 47 56 39)(11 48 50 40)(12 49 51 41)(13 43 52 42)(14 44 53 36)
(8 26 37)(9 27 38)(10 28 39)(11 22 40)(12 23 41)(13 24 42)(14 25 36)(15 43 52)(16 44 53)(17 45 54)(18 46 55)(19 47 56)(20 48 50)(21 49 51)

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

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

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

C7×SL2(𝔽3) is a maximal subgroup of   Q8.D21  Q8⋊D21  Dic7.2A4

49 conjugacy classes

class 1  2 3A3B 4 6A6B7A···7F14A···14F21A···21L28A···28F42A···42L
order12334667···714···1421···2128···2842···42
size11446441···11···14···46···64···4

49 irreducible representations

dim111122233
type+-+
imageC1C3C7C21SL2(𝔽3)SL2(𝔽3)C7×SL2(𝔽3)A4C7×A4
kernelC7×SL2(𝔽3)C7×Q8SL2(𝔽3)Q8C7C7C1C14C2
# reps12612121816

Matrix representation of C7×SL2(𝔽3) in GL2(𝔽29) generated by

250
025
,
312
426
,
523
1424
,
016
928
G:=sub<GL(2,GF(29))| [25,0,0,25],[3,4,12,26],[5,14,23,24],[0,9,16,28] >;

C7×SL2(𝔽3) in GAP, Magma, Sage, TeX

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

G:=Group("C7xSL(2,3)");
// GroupNames label

G:=SmallGroup(168,22);
// by ID

G=gap.SmallGroup(168,22);
# by ID

G:=PCGroup([5,-3,-7,-2,2,-2,632,72,1263,133,58]);
// Polycyclic

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

Export

Subgroup lattice of C7×SL2(𝔽3) in TeX

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