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G = C12.84S32order 432 = 24·33

4th non-split extension by C12 of S32 acting via S32/C3:S3=C2

non-abelian, supersoluble, monomial

Aliases: C12.84S32, C12:S3:4S3, He3:2(C4oD4), He3:3D4:3C2, He3:4D4:5C2, He3:3Q8:5C2, (C3xC12).22D6, C3:Dic3.1D6, C32:4Q8:4S3, C4.11(C32:D6), (C2xHe3).3C23, C3.3(D12:S3), C32:2(D4:2S3), C32:1(Q8:3S3), C32:C12.1C22, (C4xHe3).18C22, He3:3C4.14C22, C6.77(C2xS32), (C2xC3:S3).2D6, C6.S32:2C2, C2.6(C2xC32:D6), (C4xHe3:C2):2C2, (C3xC6).3(C22xS3), (C2xC32:C6).1C22, (C2xHe3:C2).11C22, SmallGroup(432,296)

Series: Derived Chief Lower central Upper central

C1C3C2xHe3 — C12.84S32
C1C3C32He3C2xHe3C2xC32:C6C6.S32 — C12.84S32
He3C2xHe3 — C12.84S32
C1C2C4

Generators and relations for C12.84S32
 G = < a,b,c,d,e | a12=b3=d3=e2=1, c2=a6, ab=ba, cac-1=eae=a-1, ad=da, cbc-1=b-1, dbd-1=a8b, be=eb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 927 in 156 conjugacy classes, 35 normal (23 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C2xC4, D4, Q8, C32, C32, Dic3, C12, C12, D6, C2xC6, C4oD4, C3xS3, C3:S3, C3xC6, C3xC6, Dic6, C4xS3, D12, C2xDic3, C3:D4, C2xC12, C3xD4, C3xQ8, He3, C3xDic3, C3:Dic3, C3xC12, C3xC12, S3xC6, C2xC3:S3, C4oD12, D4:2S3, Q8:3S3, C32:C6, He3:C2, C2xHe3, S3xDic3, C6.D6, D6:S3, C3:D12, C3xDic6, S3xC12, C3xD12, C32:4Q8, C12:S3, C32:C12, He3:3C4, C4xHe3, C2xC32:C6, C2xHe3:C2, D12:5S3, D6.6D6, C6.S32, He3:3D4, He3:3Q8, He3:4D4, C4xHe3:C2, C12.84S32
Quotients: C1, C2, C22, S3, C23, D6, C4oD4, C22xS3, S32, D4:2S3, Q8:3S3, C2xS32, C32:D6, D12:S3, C2xC32:D6, C12.84S32

Smallest permutation representation of C12.84S32
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 26)(2 22 27)(3 23 28)(4 24 29)(5 13 30)(6 14 31)(7 15 32)(8 16 33)(9 17 34)(10 18 35)(11 19 36)(12 20 25)(37 56 62)(38 57 63)(39 58 64)(40 59 65)(41 60 66)(42 49 67)(43 50 68)(44 51 69)(45 52 70)(46 53 71)(47 54 72)(48 55 61)
(1 50 7 56)(2 49 8 55)(3 60 9 54)(4 59 10 53)(5 58 11 52)(6 57 12 51)(13 39 19 45)(14 38 20 44)(15 37 21 43)(16 48 22 42)(17 47 23 41)(18 46 24 40)(25 69 31 63)(26 68 32 62)(27 67 33 61)(28 66 34 72)(29 65 35 71)(30 64 36 70)
(13 17 21)(14 18 22)(15 19 23)(16 20 24)(25 33 29)(26 34 30)(27 35 31)(28 36 32)(37 45 41)(38 46 42)(39 47 43)(40 48 44)(61 65 69)(62 66 70)(63 67 71)(64 68 72)
(1 53)(2 52)(3 51)(4 50)(5 49)(6 60)(7 59)(8 58)(9 57)(10 56)(11 55)(12 54)(13 67)(14 66)(15 65)(16 64)(17 63)(18 62)(19 61)(20 72)(21 71)(22 70)(23 69)(24 68)(25 47)(26 46)(27 45)(28 44)(29 43)(30 42)(31 41)(32 40)(33 39)(34 38)(35 37)(36 48)

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

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

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

32 conjugacy classes

class 1 2A2B2C2D3A3B3C3D4A4B4C4D4E6A6B6C6D6E6F6G6H12A12B12C12D12E12F12G12H12I12J
order122223333444446666666612121212121212121212
size111818182661229918182661218183636221212121218183636

32 irreducible representations

dim11111122222244444666
type++++++++++++-++++
imageC1C2C2C2C2C2S3S3D6D6D6C4oD4S32D4:2S3Q8:3S3C2xS32D12:S3C32:D6C2xC32:D6C12.84S32
kernelC12.84S32C6.S32He3:3D4He3:3Q8He3:4D4C4xHe3:C2C32:4Q8C12:S3C3:Dic3C3xC12C2xC3:S3He3C12C32C32C6C3C4C2C1
# reps12211111222211112224

Matrix representation of C12.84S32 in GL6(F13)

600000
060000
006000
0001100
0000110
0000011
,
010000
001000
100000
000001
000100
000010
,
000100
000010
000001
1200000
0120000
0012000
,
100000
030000
009000
000100
000030
000009
,
000500
000005
000050
800000
008000
080000

G:=sub<GL(6,GF(13))| [6,0,0,0,0,0,0,6,0,0,0,0,0,0,6,0,0,0,0,0,0,11,0,0,0,0,0,0,11,0,0,0,0,0,0,11],[0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0,0],[0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0],[1,0,0,0,0,0,0,3,0,0,0,0,0,0,9,0,0,0,0,0,0,1,0,0,0,0,0,0,3,0,0,0,0,0,0,9],[0,0,0,8,0,0,0,0,0,0,0,8,0,0,0,0,8,0,5,0,0,0,0,0,0,0,5,0,0,0,0,5,0,0,0,0] >;

C12.84S32 in GAP, Magma, Sage, TeX

C_{12}._{84}S_3^2
% in TeX

G:=Group("C12.84S3^2");
// GroupNames label

G:=SmallGroup(432,296);
// by ID

G=gap.SmallGroup(432,296);
# by ID

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

G:=Group<a,b,c,d,e|a^12=b^3=d^3=e^2=1,c^2=a^6,a*b=b*a,c*a*c^-1=e*a*e=a^-1,a*d=d*a,c*b*c^-1=b^-1,d*b*d^-1=a^8*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations

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