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G = C12.85S32order 432 = 24·33

5th non-split extension by C12 of S32 acting via S32/C3:S3=C2

non-abelian, supersoluble, monomial

Aliases: C12.85S32, He3:3(C2xQ8), C32:2(S3xQ8), He3:2Q8:3C2, He3:3Q8:6C2, He3:C2:2Q8, (C3xC12).23D6, C3:Dic3.2D6, C32:4Q8:5S3, C4.12(C32:D6), (C2xHe3).5C23, C32:C12.2C22, (C4xHe3).19C22, C3.2(Dic3.D6), He3:3C4.12C22, C6.79(C2xS32), C2.8(C2xC32:D6), He3:(C2xC4).1C2, (C3xC6).5(C22xS3), (C4xHe3:C2).2C2, (C2xHe3:C2).13C22, SmallGroup(432,298)

Series: Derived Chief Lower central Upper central

C1C3C2xHe3 — C12.85S32
C1C3C32He3C2xHe3C32:C12He3:(C2xC4) — C12.85S32
He3C2xHe3 — C12.85S32
C1C2C4

Generators and relations for C12.85S32
 G = < a,b,c,d,e,f | a3=b3=c3=d2=e4=1, f2=e2, ab=ba, cac-1=ab-1, dad=eae-1=faf-1=a-1, bc=cb, bd=db, be=eb, fbf-1=b-1, dcd=ece-1=c-1, cf=fc, de=ed, df=fd, fef-1=e-1 >

Subgroups: 787 in 149 conjugacy classes, 37 normal (13 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C2xC4, Q8, C32, C32, Dic3, C12, C12, D6, C2xC6, C2xQ8, C3xS3, C3xC6, C3xC6, Dic6, C4xS3, C2xDic3, C2xC12, C3xQ8, He3, C3xDic3, C3:Dic3, C3xC12, C3xC12, S3xC6, C2xDic6, S3xQ8, He3:C2, C2xHe3, S3xDic3, C32:2Q8, C3xDic6, S3xC12, C32:4Q8, C32:C12, He3:3C4, C4xHe3, C2xHe3:C2, S3xDic6, He3:2Q8, He3:(C2xC4), He3:3Q8, C4xHe3:C2, C12.85S32
Quotients: C1, C2, C22, S3, Q8, C23, D6, C2xQ8, C22xS3, S32, S3xQ8, C2xS32, C32:D6, Dic3.D6, C2xC32:D6, C12.85S32

Smallest permutation representation of C12.85S32
On 72 points
Generators in S72
(1 8 45)(2 46 5)(3 6 47)(4 48 7)(9 66 43)(10 44 67)(11 68 41)(12 42 65)(13 21 37)(14 38 22)(15 23 39)(16 40 24)(17 60 62)(18 63 57)(19 58 64)(20 61 59)(25 49 71)(26 72 50)(27 51 69)(28 70 52)(29 55 36)(30 33 56)(31 53 34)(32 35 54)
(1 30 16)(2 31 13)(3 32 14)(4 29 15)(5 34 37)(6 35 38)(7 36 39)(8 33 40)(9 69 63)(10 70 64)(11 71 61)(12 72 62)(17 42 50)(18 43 51)(19 44 52)(20 41 49)(21 46 53)(22 47 54)(23 48 55)(24 45 56)(25 59 68)(26 60 65)(27 57 66)(28 58 67)
(5 37 34)(6 35 38)(7 39 36)(8 33 40)(9 69 63)(10 64 70)(11 71 61)(12 62 72)(21 46 53)(22 54 47)(23 48 55)(24 56 45)(25 68 59)(26 60 65)(27 66 57)(28 58 67)
(1 3)(2 4)(5 48)(6 45)(7 46)(8 47)(9 68)(10 65)(11 66)(12 67)(13 15)(14 16)(17 19)(18 20)(21 39)(22 40)(23 37)(24 38)(25 69)(26 70)(27 71)(28 72)(29 31)(30 32)(33 54)(34 55)(35 56)(36 53)(41 43)(42 44)(49 51)(50 52)(57 61)(58 62)(59 63)(60 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 18 3 20)(2 17 4 19)(5 60 7 58)(6 59 8 57)(9 22 11 24)(10 21 12 23)(13 42 15 44)(14 41 16 43)(25 33 27 35)(26 36 28 34)(29 52 31 50)(30 51 32 49)(37 65 39 67)(38 68 40 66)(45 63 47 61)(46 62 48 64)(53 72 55 70)(54 71 56 69)

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

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

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

32 conjugacy classes

class 1 2A2B2C3A3B3C3D4A4B···4F6A6B6C6D6E6F12A12B12C12D12E12F12G12H12I12J12K12L
order1222333344···4666666121212121212121212121212
size119926612218···182661218182212121212181836363636

32 irreducible representations

dim1111122224444666
type++++++-+++-+++-
imageC1C2C2C2C2S3Q8D6D6S32S3xQ8C2xS32Dic3.D6C32:D6C2xC32:D6C12.85S32
kernelC12.85S32He3:2Q8He3:(C2xC4)He3:3Q8C4xHe3:C2C32:4Q8He3:C2C3:Dic3C3xC12C12C32C6C3C4C2C1
# reps1222122421212224

Matrix representation of C12.85S32 in GL6(F13)

0012100
11111200
0012010
0012001
0012000
1012000
,
1210000
1200000
1200100
01121200
1200001
01001212
,
100000
010000
11121200
001000
000001
11001212
,
1200000
0120000
0000120
0000012
0012000
0001200
,
370000
6100000
6000107
070063
6010700
076300
,
1140000
220000
1100042
2400119
1104200
2411900

G:=sub<GL(6,GF(13))| [0,1,0,0,0,1,0,1,0,0,0,0,12,11,12,12,12,12,1,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0],[12,12,12,0,12,0,1,0,0,1,0,1,0,0,0,12,0,0,0,0,1,12,0,0,0,0,0,0,0,12,0,0,0,0,1,12],[1,0,1,0,0,1,0,1,1,0,0,1,0,0,12,1,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,1,12],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,12,0,0,0,0,0,0,12,0,0],[3,6,6,0,6,0,7,10,0,7,0,7,0,0,0,0,10,6,0,0,0,0,7,3,0,0,10,6,0,0,0,0,7,3,0,0],[11,2,11,2,11,2,4,2,0,4,0,4,0,0,0,0,4,11,0,0,0,0,2,9,0,0,4,11,0,0,0,0,2,9,0,0] >;

C12.85S32 in GAP, Magma, Sage, TeX

C_{12}._{85}S_3^2
% in TeX

G:=Group("C12.85S3^2");
// GroupNames label

G:=SmallGroup(432,298);
// by ID

G=gap.SmallGroup(432,298);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,56,64,254,135,58,571,4037,537,14118,7069]);
// Polycyclic

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

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