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## G = C32⋊5GL2(𝔽3)  order 432 = 24·33

### 2nd semidirect product of C32 and GL2(𝔽3) acting via GL2(𝔽3)/SL2(𝔽3)=C2

Aliases: C325GL2(𝔽3), C3⋊(C6.6S4), (C3×C6).23S4, C6.10(C3⋊S4), Q8⋊(C33⋊C2), (Q8×C32)⋊10S3, SL2(𝔽3)⋊(C3⋊S3), (C3×SL2(𝔽3))⋊4S3, C2.3(C324S4), (C32×SL2(𝔽3))⋊2C2, (C3×Q8)⋊1(C3⋊S3), SmallGroup(432,620)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — Q8 — C32×SL2(𝔽3) — C32⋊5GL2(𝔽3)
 Chief series C1 — C2 — Q8 — C3×Q8 — Q8×C32 — C32×SL2(𝔽3) — C32⋊5GL2(𝔽3)
 Lower central C32×SL2(𝔽3) — C32⋊5GL2(𝔽3)
 Upper central C1 — C2

Generators and relations for C325GL2(𝔽3)
G = < a,b,c,d,e,f | a3=b3=c4=e3=f2=1, d2=c2, ab=ba, ac=ca, ad=da, ae=ea, faf=a-1, bc=cb, bd=db, be=eb, fbf=b-1, dcd-1=fdf=c-1, ece-1=cd, fcf=c2d, ede-1=c, fef=e-1 >

Subgroups: 2362 in 208 conjugacy classes, 41 normal (8 characteristic)
C1, C2, C2, C3, C3, C4, C22, S3, C6, C6, C8, D4, Q8, C32, C32, C12, D6, SD16, C3⋊S3, C3×C6, C3×C6, C3⋊C8, SL2(𝔽3), D12, C3×Q8, C33, C3×C12, C2×C3⋊S3, Q82S3, GL2(𝔽3), C33⋊C2, C32×C6, C324C8, C3×SL2(𝔽3), C12⋊S3, Q8×C32, C2×C33⋊C2, C3211SD16, C6.6S4, C32×SL2(𝔽3), C325GL2(𝔽3)
Quotients: C1, C2, S3, C3⋊S3, S4, GL2(𝔽3), C33⋊C2, C3⋊S4, C6.6S4, C324S4, C325GL2(𝔽3)

Smallest permutation representation of C325GL2(𝔽3)
On 72 points
Generators in S72
(1 54 30)(2 55 31)(3 56 32)(4 53 29)(5 51 27)(6 52 28)(7 49 25)(8 50 26)(9 57 33)(10 58 34)(11 59 35)(12 60 36)(13 61 37)(14 62 38)(15 63 39)(16 64 40)(17 65 41)(18 66 42)(19 67 43)(20 68 44)(21 69 45)(22 70 46)(23 71 47)(24 72 48)
(1 22 14)(2 23 15)(3 24 16)(4 21 13)(5 67 59)(6 68 60)(7 65 57)(8 66 58)(9 25 17)(10 26 18)(11 27 19)(12 28 20)(29 45 37)(30 46 38)(31 47 39)(32 48 40)(33 49 41)(34 50 42)(35 51 43)(36 52 44)(53 69 61)(54 70 62)(55 71 63)(56 72 64)
(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 11 3 9)(2 10 4 12)(5 72 7 70)(6 71 8 69)(13 20 15 18)(14 19 16 17)(21 28 23 26)(22 27 24 25)(29 36 31 34)(30 35 32 33)(37 44 39 42)(38 43 40 41)(45 52 47 50)(46 51 48 49)(53 60 55 58)(54 59 56 57)(61 68 63 66)(62 67 64 65)
(1 70 38)(2 5 42)(3 72 40)(4 7 44)(6 37 9)(8 39 11)(10 71 43)(12 69 41)(13 57 52)(14 54 46)(15 59 50)(16 56 48)(17 60 45)(18 55 51)(19 58 47)(20 53 49)(21 65 36)(22 62 30)(23 67 34)(24 64 32)(25 68 29)(26 63 35)(27 66 31)(28 61 33)
(2 9)(4 11)(5 37)(6 42)(7 39)(8 44)(10 12)(13 27)(14 22)(15 25)(16 24)(17 23)(18 28)(19 21)(20 26)(29 59)(30 54)(31 57)(32 56)(33 55)(34 60)(35 53)(36 58)(38 70)(40 72)(41 71)(43 69)(45 67)(46 62)(47 65)(48 64)(49 63)(50 68)(51 61)(52 66)

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

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

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

36 conjugacy classes

 class 1 2A 2B 3A 3B 3C 3D 3E ··· 3M 4 6A 6B 6C 6D 6E ··· 6M 8A 8B 12A 12B 12C 12D order 1 2 2 3 3 3 3 3 ··· 3 4 6 6 6 6 6 ··· 6 8 8 12 12 12 12 size 1 1 108 2 2 2 2 8 ··· 8 6 2 2 2 2 8 ··· 8 54 54 12 12 12 12

36 irreducible representations

 dim 1 1 2 2 2 3 4 4 6 type + + + + + + + + image C1 C2 S3 S3 GL2(𝔽3) S4 GL2(𝔽3) C6.6S4 C3⋊S4 kernel C32⋊5GL2(𝔽3) C32×SL2(𝔽3) C3×SL2(𝔽3) Q8×C32 C32 C3×C6 C32 C3 C6 # reps 1 1 12 1 2 2 1 12 4

Matrix representation of C325GL2(𝔽3) in GL6(𝔽73)

 72 1 0 0 0 0 72 0 0 0 0 0 0 0 0 72 0 0 0 0 1 72 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 72 0 0 0 0 1 72 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 32 21 0 0 0 0 52 41
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 20 21 0 0 0 0 40 53
,
 0 72 0 0 0 0 1 72 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 52 41 0 0 0 0 20 20
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 1 72

G:=sub<GL(6,GF(73))| [72,72,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,72,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,72,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,32,52,0,0,0,0,21,41],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,20,40,0,0,0,0,21,53],[0,1,0,0,0,0,72,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,52,20,0,0,0,0,41,20],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,1,0,0,0,0,0,72] >;

C325GL2(𝔽3) in GAP, Magma, Sage, TeX

C_3^2\rtimes_5{\rm GL}_2({\mathbb F}_3)
% in TeX

G:=Group("C3^2:5GL(2,3)");
// GroupNames label

G:=SmallGroup(432,620);
// by ID

G=gap.SmallGroup(432,620);
# by ID

G:=PCGroup([7,-2,-3,-3,-3,-2,2,-2,57,254,1011,3784,5681,172,2273,3414,285,124]);
// Polycyclic

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

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