direct product, metabelian, supersoluble, monomial
Aliases: C32×D4.S3, C12.2C62, C33⋊22SD16, D4.(S3×C32), C12.58(S3×C6), Dic6⋊2(C3×C6), (C3×Dic6)⋊3C6, C6.8(D4×C32), (C3×C12).185D6, (D4×C33).1C2, C3⋊2(C32×SD16), (C32×C6).67D4, (C32×Dic6)⋊5C2, (D4×C32).11C6, (D4×C32).17S3, C32⋊12(C3×SD16), (C32×C12).24C22, (C3×C3⋊C8)⋊9C6, C3⋊C8⋊2(C3×C6), C4.2(S3×C3×C6), (C32×C3⋊C8)⋊16C2, (C3×D4).1(C3×C6), (C3×C6).63(C3×D4), C6.54(C3×C3⋊D4), (C3×C12).43(C2×C6), (C3×D4).16(C3×S3), C2.5(C32×C3⋊D4), (C3×C6).123(C3⋊D4), SmallGroup(432,476)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C32×D4.S3
G = < a,b,c,d,e,f | a3=b3=c4=d2=e3=1, f2=c2, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, dcd=fcf-1=c-1, ce=ec, de=ed, fdf-1=cd, fef-1=e-1 >
Subgroups: 400 in 184 conjugacy classes, 66 normal (22 characteristic)
C1, C2, C2, C3, C3, C3, C4, C4, C22, C6, C6, C6, C8, D4, Q8, C32, C32, C32, Dic3, C12, C12, C12, C2×C6, SD16, C3×C6, C3×C6, C3×C6, C3⋊C8, C24, Dic6, C3×D4, C3×D4, C3×D4, C3×Q8, C33, C3×Dic3, C3×C12, C3×C12, C3×C12, C62, D4.S3, C3×SD16, C32×C6, C32×C6, C3×C3⋊C8, C3×C24, C3×Dic6, D4×C32, D4×C32, D4×C32, Q8×C32, C32×Dic3, C32×C12, C3×C62, C3×D4.S3, C32×SD16, C32×C3⋊C8, C32×Dic6, D4×C33, C32×D4.S3
Quotients: C1, C2, C3, C22, S3, C6, D4, C32, D6, C2×C6, SD16, C3×S3, C3×C6, C3⋊D4, C3×D4, S3×C6, C62, D4.S3, C3×SD16, S3×C32, C3×C3⋊D4, D4×C32, S3×C3×C6, C3×D4.S3, C32×SD16, C32×C3⋊D4, C32×D4.S3
►On 72 points
Generators in S72
(1 43 29)(2 44 30)(3 41 31)(4 42 32)(5 63 45)(6 64 46)(7 61 47)(8 62 48)(9 60 55)(10 57 56)(11 58 53)(12 59 54)(13 39 23)(14 40 24)(15 37 21)(16 38 22)(17 34 28)(18 35 25)(19 36 26)(20 33 27)(49 71 66)(50 72 67)(51 69 68)(52 70 65)
(1 17 16)(2 18 13)(3 19 14)(4 20 15)(5 69 10)(6 70 11)(7 71 12)(8 72 9)(21 32 27)(22 29 28)(23 30 25)(24 31 26)(33 37 42)(34 38 43)(35 39 44)(36 40 41)(45 51 56)(46 52 53)(47 49 54)(48 50 55)(57 63 68)(58 64 65)(59 61 66)(60 62 67)
(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 4)(2 3)(6 8)(9 11)(13 14)(15 16)(17 20)(18 19)(21 22)(23 24)(25 26)(27 28)(29 32)(30 31)(33 34)(35 36)(37 38)(39 40)(41 44)(42 43)(46 48)(50 52)(53 55)(58 60)(62 64)(65 67)(70 72)
(1 17 16)(2 18 13)(3 19 14)(4 20 15)(5 10 69)(6 11 70)(7 12 71)(8 9 72)(21 32 27)(22 29 28)(23 30 25)(24 31 26)(33 37 42)(34 38 43)(35 39 44)(36 40 41)(45 56 51)(46 53 52)(47 54 49)(48 55 50)(57 68 63)(58 65 64)(59 66 61)(60 67 62)
(1 48 3 46)(2 47 4 45)(5 44 7 42)(6 43 8 41)(9 40 11 38)(10 39 12 37)(13 54 15 56)(14 53 16 55)(17 50 19 52)(18 49 20 51)(21 57 23 59)(22 60 24 58)(25 66 27 68)(26 65 28 67)(29 62 31 64)(30 61 32 63)(33 69 35 71)(34 72 36 70)
G:=sub<Sym(72)| (1,43,29)(2,44,30)(3,41,31)(4,42,32)(5,63,45)(6,64,46)(7,61,47)(8,62,48)(9,60,55)(10,57,56)(11,58,53)(12,59,54)(13,39,23)(14,40,24)(15,37,21)(16,38,22)(17,34,28)(18,35,25)(19,36,26)(20,33,27)(49,71,66)(50,72,67)(51,69,68)(52,70,65), (1,17,16)(2,18,13)(3,19,14)(4,20,15)(5,69,10)(6,70,11)(7,71,12)(8,72,9)(21,32,27)(22,29,28)(23,30,25)(24,31,26)(33,37,42)(34,38,43)(35,39,44)(36,40,41)(45,51,56)(46,52,53)(47,49,54)(48,50,55)(57,63,68)(58,64,65)(59,61,66)(60,62,67), (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), (1,4)(2,3)(6,8)(9,11)(13,14)(15,16)(17,20)(18,19)(21,22)(23,24)(25,26)(27,28)(29,32)(30,31)(33,34)(35,36)(37,38)(39,40)(41,44)(42,43)(46,48)(50,52)(53,55)(58,60)(62,64)(65,67)(70,72), (1,17,16)(2,18,13)(3,19,14)(4,20,15)(5,10,69)(6,11,70)(7,12,71)(8,9,72)(21,32,27)(22,29,28)(23,30,25)(24,31,26)(33,37,42)(34,38,43)(35,39,44)(36,40,41)(45,56,51)(46,53,52)(47,54,49)(48,55,50)(57,68,63)(58,65,64)(59,66,61)(60,67,62), (1,48,3,46)(2,47,4,45)(5,44,7,42)(6,43,8,41)(9,40,11,38)(10,39,12,37)(13,54,15,56)(14,53,16,55)(17,50,19,52)(18,49,20,51)(21,57,23,59)(22,60,24,58)(25,66,27,68)(26,65,28,67)(29,62,31,64)(30,61,32,63)(33,69,35,71)(34,72,36,70)>;
G:=Group( (1,43,29)(2,44,30)(3,41,31)(4,42,32)(5,63,45)(6,64,46)(7,61,47)(8,62,48)(9,60,55)(10,57,56)(11,58,53)(12,59,54)(13,39,23)(14,40,24)(15,37,21)(16,38,22)(17,34,28)(18,35,25)(19,36,26)(20,33,27)(49,71,66)(50,72,67)(51,69,68)(52,70,65), (1,17,16)(2,18,13)(3,19,14)(4,20,15)(5,69,10)(6,70,11)(7,71,12)(8,72,9)(21,32,27)(22,29,28)(23,30,25)(24,31,26)(33,37,42)(34,38,43)(35,39,44)(36,40,41)(45,51,56)(46,52,53)(47,49,54)(48,50,55)(57,63,68)(58,64,65)(59,61,66)(60,62,67), (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), (1,4)(2,3)(6,8)(9,11)(13,14)(15,16)(17,20)(18,19)(21,22)(23,24)(25,26)(27,28)(29,32)(30,31)(33,34)(35,36)(37,38)(39,40)(41,44)(42,43)(46,48)(50,52)(53,55)(58,60)(62,64)(65,67)(70,72), (1,17,16)(2,18,13)(3,19,14)(4,20,15)(5,10,69)(6,11,70)(7,12,71)(8,9,72)(21,32,27)(22,29,28)(23,30,25)(24,31,26)(33,37,42)(34,38,43)(35,39,44)(36,40,41)(45,56,51)(46,53,52)(47,54,49)(48,55,50)(57,68,63)(58,65,64)(59,66,61)(60,67,62), (1,48,3,46)(2,47,4,45)(5,44,7,42)(6,43,8,41)(9,40,11,38)(10,39,12,37)(13,54,15,56)(14,53,16,55)(17,50,19,52)(18,49,20,51)(21,57,23,59)(22,60,24,58)(25,66,27,68)(26,65,28,67)(29,62,31,64)(30,61,32,63)(33,69,35,71)(34,72,36,70) );
G=PermutationGroup([[(1,43,29),(2,44,30),(3,41,31),(4,42,32),(5,63,45),(6,64,46),(7,61,47),(8,62,48),(9,60,55),(10,57,56),(11,58,53),(12,59,54),(13,39,23),(14,40,24),(15,37,21),(16,38,22),(17,34,28),(18,35,25),(19,36,26),(20,33,27),(49,71,66),(50,72,67),(51,69,68),(52,70,65)], [(1,17,16),(2,18,13),(3,19,14),(4,20,15),(5,69,10),(6,70,11),(7,71,12),(8,72,9),(21,32,27),(22,29,28),(23,30,25),(24,31,26),(33,37,42),(34,38,43),(35,39,44),(36,40,41),(45,51,56),(46,52,53),(47,49,54),(48,50,55),(57,63,68),(58,64,65),(59,61,66),(60,62,67)], [(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)], [(1,4),(2,3),(6,8),(9,11),(13,14),(15,16),(17,20),(18,19),(21,22),(23,24),(25,26),(27,28),(29,32),(30,31),(33,34),(35,36),(37,38),(39,40),(41,44),(42,43),(46,48),(50,52),(53,55),(58,60),(62,64),(65,67),(70,72)], [(1,17,16),(2,18,13),(3,19,14),(4,20,15),(5,10,69),(6,11,70),(7,12,71),(8,9,72),(21,32,27),(22,29,28),(23,30,25),(24,31,26),(33,37,42),(34,38,43),(35,39,44),(36,40,41),(45,56,51),(46,53,52),(47,54,49),(48,55,50),(57,68,63),(58,65,64),(59,66,61),(60,67,62)], [(1,48,3,46),(2,47,4,45),(5,44,7,42),(6,43,8,41),(9,40,11,38),(10,39,12,37),(13,54,15,56),(14,53,16,55),(17,50,19,52),(18,49,20,51),(21,57,23,59),(22,60,24,58),(25,66,27,68),(26,65,28,67),(29,62,31,64),(30,61,32,63),(33,69,35,71),(34,72,36,70)]])
108 conjugacy classes
| class | 1 | 2A | 2B | 3A | ··· | 3H | 3I | ··· | 3Q | 4A | 4B | 6A | ··· | 6H | 6I | ··· | 6Q | 6R | ··· | 6AQ | 8A | 8B | 12A | ··· | 12H | 12I | ··· | 12Q | 12R | ··· | 12Y | 24A | ··· | 24P |
| order | 1 | 2 | 2 | 3 | ··· | 3 | 3 | ··· | 3 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 6 | ··· | 6 | 8 | 8 | 12 | ··· | 12 | 12 | ··· | 12 | 12 | ··· | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | 12 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 6 | 6 | 2 | ··· | 2 | 4 | ··· | 4 | 12 | ··· | 12 | 6 | ··· | 6 |
108 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | ||||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | S3 | D4 | D6 | SD16 | C3×S3 | C3⋊D4 | C3×D4 | S3×C6 | C3×SD16 | C3×C3⋊D4 | D4.S3 | C3×D4.S3 |
| kernel | C32×D4.S3 | C32×C3⋊C8 | C32×Dic6 | D4×C33 | C3×D4.S3 | C3×C3⋊C8 | C3×Dic6 | D4×C32 | D4×C32 | C32×C6 | C3×C12 | C33 | C3×D4 | C3×C6 | C3×C6 | C12 | C32 | C6 | C32 | C3 |
| # reps | 1 | 1 | 1 | 1 | 8 | 8 | 8 | 8 | 1 | 1 | 1 | 2 | 8 | 2 | 8 | 8 | 16 | 16 | 1 | 8 |
Matrix representation of C32×D4.S3 ►in GL6(𝔽73)
| 8 | 0 | 0 | 0 | 0 | 0 |
| 0 | 8 | 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 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 0 | 0 | 0 |
| 0 | 0 | 0 | 8 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 72 | 36 | 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 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 0 | 0 | 0 |
| 0 | 0 | 21 | 64 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 12 | 41 | 0 | 0 | 0 | 0 |
| 25 | 61 | 0 | 0 | 0 | 0 |
| 0 | 0 | 30 | 7 | 0 | 0 |
| 0 | 0 | 28 | 43 | 0 | 0 |
| 0 | 0 | 0 | 0 | 67 | 6 |
| 0 | 0 | 0 | 0 | 6 | 6 |
G:=sub<GL(6,GF(73))| [8,0,0,0,0,0,0,8,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,1,0],[72,0,0,0,0,0,36,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,8,21,0,0,0,0,0,64,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,25,0,0,0,0,41,61,0,0,0,0,0,0,30,28,0,0,0,0,7,43,0,0,0,0,0,0,67,6,0,0,0,0,6,6] >;
C32×D4.S3 in GAP, Magma, Sage, TeX
C_3^2\times D_4.S_3
% in TeX
G:=Group("C3^2xD4.S3"); // GroupNames label
G:=SmallGroup(432,476);
// by ID
G=gap.SmallGroup(432,476);
# by ID
G:=PCGroup([7,-2,-2,-3,-3,-2,-2,-3,504,533,3784,1901,102,14118]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^3=b^3=c^4=d^2=e^3=1,f^2=c^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,d*c*d=f*c*f^-1=c^-1,c*e=e*c,d*e=e*d,f*d*f^-1=c*d,f*e*f^-1=e^-1>;
// generators/relations