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
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
(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 | 2A | 2B | 2C | 2D | 3A | 3B | 3C | 3D | 4A | 4B | 4C | 4D | 4E | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 12I | 12J |
order | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 |
size | 1 | 1 | 18 | 18 | 18 | 2 | 6 | 6 | 12 | 2 | 9 | 9 | 18 | 18 | 2 | 6 | 6 | 12 | 18 | 18 | 36 | 36 | 2 | 2 | 12 | 12 | 12 | 12 | 18 | 18 | 36 | 36 |
32 irreducible representations
dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 |
type | + | + | + | + | + | + | + | + | + | + | + | + | - | + | + | + | + | |||
image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | S3 | D6 | D6 | D6 | C4oD4 | S32 | D4:2S3 | Q8:3S3 | C2xS32 | D12:S3 | C32:D6 | C2xC32:D6 | C12.84S32 |
kernel | C12.84S32 | C6.S32 | He3:3D4 | He3:3Q8 | He3:4D4 | C4xHe3:C2 | C32:4Q8 | C12:S3 | C3:Dic3 | C3xC12 | C2xC3:S3 | He3 | C12 | C32 | C32 | C6 | C3 | C4 | C2 | C1 |
# reps | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 |
Matrix representation of C12.84S32 ►in GL6(F13)
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 | 1 | 0 | 0 | 0 | 0 |
0 | 0 | 1 | 0 | 0 | 0 |
1 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 1 |
0 | 0 | 0 | 1 | 0 | 0 |
0 | 0 | 0 | 0 | 1 | 0 |
0 | 0 | 0 | 1 | 0 | 0 |
0 | 0 | 0 | 0 | 1 | 0 |
0 | 0 | 0 | 0 | 0 | 1 |
12 | 0 | 0 | 0 | 0 | 0 |
0 | 12 | 0 | 0 | 0 | 0 |
0 | 0 | 12 | 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 | 5 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 5 |
0 | 0 | 0 | 0 | 5 | 0 |
8 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 8 | 0 | 0 | 0 |
0 | 8 | 0 | 0 | 0 | 0 |
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