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G = C9×SL2(𝔽3)  order 216 = 23·33

Direct product of C9 and SL2(𝔽3)

direct product, non-abelian, soluble

Aliases: C9×SL2(𝔽3), C18.4A4, C2.(C9×A4), Q8⋊C93C3, Q81(C3×C9), C6.1(C3×A4), (Q8×C9)⋊1C3, (C3×Q8).1C32, C3.1(C3×SL2(𝔽3)), (C3×SL2(𝔽3)).2C3, SmallGroup(216,38)

Series: Derived Chief Lower central Upper central

C1C2Q8 — C9×SL2(𝔽3)
C1C2Q8C3×Q8C3×SL2(𝔽3) — C9×SL2(𝔽3)
Q8 — C9×SL2(𝔽3)
C1C18

Generators and relations for C9×SL2(𝔽3)
 G = < a,b,c,d | a9=b4=d3=1, c2=b2, ab=ba, ac=ca, ad=da, cbc-1=b-1, dbd-1=c, dcd-1=bc >

4C3
4C3
4C3
3C4
4C6
4C6
4C6
4C9
4C32
4C9
3C12
4C3×C6
4C18
4C18
4C3×C9
3C36
4C3×C18

Smallest permutation representation of C9×SL2(𝔽3)
On 72 points
Generators in S72
(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 57 58 59 60 61 62 63)(64 65 66 67 68 69 70 71 72)
(1 21 14 64)(2 22 15 65)(3 23 16 66)(4 24 17 67)(5 25 18 68)(6 26 10 69)(7 27 11 70)(8 19 12 71)(9 20 13 72)(28 39 50 59)(29 40 51 60)(30 41 52 61)(31 42 53 62)(32 43 54 63)(33 44 46 55)(34 45 47 56)(35 37 48 57)(36 38 49 58)
(1 50 14 28)(2 51 15 29)(3 52 16 30)(4 53 17 31)(5 54 18 32)(6 46 10 33)(7 47 11 34)(8 48 12 35)(9 49 13 36)(19 37 71 57)(20 38 72 58)(21 39 64 59)(22 40 65 60)(23 41 66 61)(24 42 67 62)(25 43 68 63)(26 44 69 55)(27 45 70 56)
(1 7 4)(2 8 5)(3 9 6)(10 16 13)(11 17 14)(12 18 15)(19 43 51)(20 44 52)(21 45 53)(22 37 54)(23 38 46)(24 39 47)(25 40 48)(26 41 49)(27 42 50)(28 70 62)(29 71 63)(30 72 55)(31 64 56)(32 65 57)(33 66 58)(34 67 59)(35 68 60)(36 69 61)

G:=sub<Sym(72)| (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)(55,56,57,58,59,60,61,62,63)(64,65,66,67,68,69,70,71,72), (1,21,14,64)(2,22,15,65)(3,23,16,66)(4,24,17,67)(5,25,18,68)(6,26,10,69)(7,27,11,70)(8,19,12,71)(9,20,13,72)(28,39,50,59)(29,40,51,60)(30,41,52,61)(31,42,53,62)(32,43,54,63)(33,44,46,55)(34,45,47,56)(35,37,48,57)(36,38,49,58), (1,50,14,28)(2,51,15,29)(3,52,16,30)(4,53,17,31)(5,54,18,32)(6,46,10,33)(7,47,11,34)(8,48,12,35)(9,49,13,36)(19,37,71,57)(20,38,72,58)(21,39,64,59)(22,40,65,60)(23,41,66,61)(24,42,67,62)(25,43,68,63)(26,44,69,55)(27,45,70,56), (1,7,4)(2,8,5)(3,9,6)(10,16,13)(11,17,14)(12,18,15)(19,43,51)(20,44,52)(21,45,53)(22,37,54)(23,38,46)(24,39,47)(25,40,48)(26,41,49)(27,42,50)(28,70,62)(29,71,63)(30,72,55)(31,64,56)(32,65,57)(33,66,58)(34,67,59)(35,68,60)(36,69,61)>;

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)(55,56,57,58,59,60,61,62,63)(64,65,66,67,68,69,70,71,72), (1,21,14,64)(2,22,15,65)(3,23,16,66)(4,24,17,67)(5,25,18,68)(6,26,10,69)(7,27,11,70)(8,19,12,71)(9,20,13,72)(28,39,50,59)(29,40,51,60)(30,41,52,61)(31,42,53,62)(32,43,54,63)(33,44,46,55)(34,45,47,56)(35,37,48,57)(36,38,49,58), (1,50,14,28)(2,51,15,29)(3,52,16,30)(4,53,17,31)(5,54,18,32)(6,46,10,33)(7,47,11,34)(8,48,12,35)(9,49,13,36)(19,37,71,57)(20,38,72,58)(21,39,64,59)(22,40,65,60)(23,41,66,61)(24,42,67,62)(25,43,68,63)(26,44,69,55)(27,45,70,56), (1,7,4)(2,8,5)(3,9,6)(10,16,13)(11,17,14)(12,18,15)(19,43,51)(20,44,52)(21,45,53)(22,37,54)(23,38,46)(24,39,47)(25,40,48)(26,41,49)(27,42,50)(28,70,62)(29,71,63)(30,72,55)(31,64,56)(32,65,57)(33,66,58)(34,67,59)(35,68,60)(36,69,61) );

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),(55,56,57,58,59,60,61,62,63),(64,65,66,67,68,69,70,71,72)], [(1,21,14,64),(2,22,15,65),(3,23,16,66),(4,24,17,67),(5,25,18,68),(6,26,10,69),(7,27,11,70),(8,19,12,71),(9,20,13,72),(28,39,50,59),(29,40,51,60),(30,41,52,61),(31,42,53,62),(32,43,54,63),(33,44,46,55),(34,45,47,56),(35,37,48,57),(36,38,49,58)], [(1,50,14,28),(2,51,15,29),(3,52,16,30),(4,53,17,31),(5,54,18,32),(6,46,10,33),(7,47,11,34),(8,48,12,35),(9,49,13,36),(19,37,71,57),(20,38,72,58),(21,39,64,59),(22,40,65,60),(23,41,66,61),(24,42,67,62),(25,43,68,63),(26,44,69,55),(27,45,70,56)], [(1,7,4),(2,8,5),(3,9,6),(10,16,13),(11,17,14),(12,18,15),(19,43,51),(20,44,52),(21,45,53),(22,37,54),(23,38,46),(24,39,47),(25,40,48),(26,41,49),(27,42,50),(28,70,62),(29,71,63),(30,72,55),(31,64,56),(32,65,57),(33,66,58),(34,67,59),(35,68,60),(36,69,61)])

C9×SL2(𝔽3) is a maximal subgroup of   C18.5S4  C18.6S4  Dic9.2A4

63 conjugacy classes

class 1  2 3A3B3C···3H 4 6A6B6C···6H9A···9F9G···9R12A12B18A···18F18G···18R36A···36F
order12333···34666···69···99···9121218···1818···1836···36
size11114···46114···41···14···4661···14···46···6

63 irreducible representations

dim111112222333
type+-+
imageC1C3C3C3C9SL2(𝔽3)SL2(𝔽3)C3×SL2(𝔽3)C9×SL2(𝔽3)A4C3×A4C9×A4
kernelC9×SL2(𝔽3)Q8⋊C9Q8×C9C3×SL2(𝔽3)SL2(𝔽3)C9C9C3C1C18C6C2
# reps14221812618126

Matrix representation of C9×SL2(𝔽3) in GL2(𝔽19) generated by

160
016
,
010
170
,
124
167
,
79
011
G:=sub<GL(2,GF(19))| [16,0,0,16],[0,17,10,0],[12,16,4,7],[7,0,9,11] >;

C9×SL2(𝔽3) in GAP, Magma, Sage, TeX

C_9\times {\rm SL}_2({\mathbb F}_3)
% in TeX

G:=Group("C9xSL(2,3)");
// GroupNames label

G:=SmallGroup(216,38);
// by ID

G=gap.SmallGroup(216,38);
# by ID

G:=PCGroup([6,-3,-3,-3,-2,2,-2,43,1299,117,2434,202,88]);
// Polycyclic

G:=Group<a,b,c,d|a^9=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 C9×SL2(𝔽3) in TeX

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