direct product, metabelian, supersoluble, monomial, A-group
Aliases: C3×C32⋊4C8, C33⋊6C8, C32⋊6C24, C32⋊5(C3⋊C8), (C3×C6).9C12, C12.14(C3×S3), (C3×C12).16C6, (C3×C12).19S3, (C32×C6).4C4, C12.12(C3⋊S3), C6.7(C3×Dic3), (C32×C12).5C2, C6.6(C3⋊Dic3), (C3×C6).10Dic3, C3⋊(C3×C3⋊C8), C4.2(C3×C3⋊S3), C2.(C3×C3⋊Dic3), SmallGroup(216,83)
Series: Derived ►Chief ►Lower central ►Upper central
| C32 — C3×C32⋊4C8 |
Generators and relations for C3×C32⋊4C8
G = < a,b,c,d | a3=b3=c3=d8=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c-1 >
Subgroups: 128 in 72 conjugacy classes, 38 normal (14 characteristic)
C1, C2, C3, C3, C3, C4, C6, C6, C6, C8, C32, C32, C32, C12, C12, C12, C3×C6, C3×C6, C3×C6, C3⋊C8, C24, C33, C3×C12, C3×C12, C3×C12, C32×C6, C3×C3⋊C8, C32⋊4C8, C32×C12, C3×C32⋊4C8
Quotients: C1, C2, C3, C4, S3, C6, C8, Dic3, C12, C3×S3, C3⋊S3, C3⋊C8, C24, C3×Dic3, C3⋊Dic3, C3×C3⋊S3, C3×C3⋊C8, C32⋊4C8, C3×C3⋊Dic3, C3×C32⋊4C8
►On 72 points
(1 12 54)(2 13 55)(3 14 56)(4 15 49)(5 16 50)(6 9 51)(7 10 52)(8 11 53)(17 36 69)(18 37 70)(19 38 71)(20 39 72)(21 40 65)(22 33 66)(23 34 67)(24 35 68)(25 41 57)(26 42 58)(27 43 59)(28 44 60)(29 45 61)(30 46 62)(31 47 63)(32 48 64)
(1 68 47)(2 48 69)(3 70 41)(4 42 71)(5 72 43)(6 44 65)(7 66 45)(8 46 67)(9 60 21)(10 22 61)(11 62 23)(12 24 63)(13 64 17)(14 18 57)(15 58 19)(16 20 59)(25 56 37)(26 38 49)(27 50 39)(28 40 51)(29 52 33)(30 34 53)(31 54 35)(32 36 55)
(1 24 31)(2 32 17)(3 18 25)(4 26 19)(5 20 27)(6 28 21)(7 22 29)(8 30 23)(9 44 40)(10 33 45)(11 46 34)(12 35 47)(13 48 36)(14 37 41)(15 42 38)(16 39 43)(49 58 71)(50 72 59)(51 60 65)(52 66 61)(53 62 67)(54 68 63)(55 64 69)(56 70 57)
(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)
G:=sub<Sym(72)| (1,12,54)(2,13,55)(3,14,56)(4,15,49)(5,16,50)(6,9,51)(7,10,52)(8,11,53)(17,36,69)(18,37,70)(19,38,71)(20,39,72)(21,40,65)(22,33,66)(23,34,67)(24,35,68)(25,41,57)(26,42,58)(27,43,59)(28,44,60)(29,45,61)(30,46,62)(31,47,63)(32,48,64), (1,68,47)(2,48,69)(3,70,41)(4,42,71)(5,72,43)(6,44,65)(7,66,45)(8,46,67)(9,60,21)(10,22,61)(11,62,23)(12,24,63)(13,64,17)(14,18,57)(15,58,19)(16,20,59)(25,56,37)(26,38,49)(27,50,39)(28,40,51)(29,52,33)(30,34,53)(31,54,35)(32,36,55), (1,24,31)(2,32,17)(3,18,25)(4,26,19)(5,20,27)(6,28,21)(7,22,29)(8,30,23)(9,44,40)(10,33,45)(11,46,34)(12,35,47)(13,48,36)(14,37,41)(15,42,38)(16,39,43)(49,58,71)(50,72,59)(51,60,65)(52,66,61)(53,62,67)(54,68,63)(55,64,69)(56,70,57), (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)>;
G:=Group( (1,12,54)(2,13,55)(3,14,56)(4,15,49)(5,16,50)(6,9,51)(7,10,52)(8,11,53)(17,36,69)(18,37,70)(19,38,71)(20,39,72)(21,40,65)(22,33,66)(23,34,67)(24,35,68)(25,41,57)(26,42,58)(27,43,59)(28,44,60)(29,45,61)(30,46,62)(31,47,63)(32,48,64), (1,68,47)(2,48,69)(3,70,41)(4,42,71)(5,72,43)(6,44,65)(7,66,45)(8,46,67)(9,60,21)(10,22,61)(11,62,23)(12,24,63)(13,64,17)(14,18,57)(15,58,19)(16,20,59)(25,56,37)(26,38,49)(27,50,39)(28,40,51)(29,52,33)(30,34,53)(31,54,35)(32,36,55), (1,24,31)(2,32,17)(3,18,25)(4,26,19)(5,20,27)(6,28,21)(7,22,29)(8,30,23)(9,44,40)(10,33,45)(11,46,34)(12,35,47)(13,48,36)(14,37,41)(15,42,38)(16,39,43)(49,58,71)(50,72,59)(51,60,65)(52,66,61)(53,62,67)(54,68,63)(55,64,69)(56,70,57), (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) );
G=PermutationGroup([[(1,12,54),(2,13,55),(3,14,56),(4,15,49),(5,16,50),(6,9,51),(7,10,52),(8,11,53),(17,36,69),(18,37,70),(19,38,71),(20,39,72),(21,40,65),(22,33,66),(23,34,67),(24,35,68),(25,41,57),(26,42,58),(27,43,59),(28,44,60),(29,45,61),(30,46,62),(31,47,63),(32,48,64)], [(1,68,47),(2,48,69),(3,70,41),(4,42,71),(5,72,43),(6,44,65),(7,66,45),(8,46,67),(9,60,21),(10,22,61),(11,62,23),(12,24,63),(13,64,17),(14,18,57),(15,58,19),(16,20,59),(25,56,37),(26,38,49),(27,50,39),(28,40,51),(29,52,33),(30,34,53),(31,54,35),(32,36,55)], [(1,24,31),(2,32,17),(3,18,25),(4,26,19),(5,20,27),(6,28,21),(7,22,29),(8,30,23),(9,44,40),(10,33,45),(11,46,34),(12,35,47),(13,48,36),(14,37,41),(15,42,38),(16,39,43),(49,58,71),(50,72,59),(51,60,65),(52,66,61),(53,62,67),(54,68,63),(55,64,69),(56,70,57)], [(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)]])
C3×C32⋊4C8 is a maximal subgroup of
C33⋊4C16 C3×S3×C3⋊C8 C12.69S32 C33⋊7M4(2) C33⋊9M4(2) C33⋊7D8 C33⋊14SD16 C33⋊15SD16 C33⋊7Q16 C12.93S32 C33⋊10M4(2) C33⋊9D8 C33⋊18SD16 C33⋊9Q16 C3⋊S3×C24
72 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | ··· | 3N | 4A | 4B | 6A | 6B | 6C | ··· | 6N | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | ··· | 12AB | 24A | ··· | 24H |
| order | 1 | 2 | 3 | 3 | 3 | ··· | 3 | 4 | 4 | 6 | 6 | 6 | ··· | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 9 | 9 | 9 | 9 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 9 | ··· | 9 |
72 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | ||||||||||
| image | C1 | C2 | C3 | C4 | C6 | C8 | C12 | C24 | S3 | Dic3 | C3×S3 | C3⋊C8 | C3×Dic3 | C3×C3⋊C8 |
| kernel | C3×C32⋊4C8 | C32×C12 | C32⋊4C8 | C32×C6 | C3×C12 | C33 | C3×C6 | C32 | C3×C12 | C3×C6 | C12 | C32 | C6 | C3 |
| # reps | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 8 | 4 | 4 | 8 | 8 | 8 | 16 |
Matrix representation of C3×C32⋊4C8 ►in GL4(𝔽73) generated by
| 64 | 0 | 0 | 0 |
| 0 | 64 | 0 | 0 |
| 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 8 |
| 8 | 17 | 0 | 0 |
| 0 | 64 | 0 | 0 |
| 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 64 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 64 | 0 |
| 0 | 0 | 0 | 8 |
| 51 | 29 | 0 | 0 |
| 22 | 22 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 72 | 0 |
G:=sub<GL(4,GF(73))| [64,0,0,0,0,64,0,0,0,0,8,0,0,0,0,8],[8,0,0,0,17,64,0,0,0,0,8,0,0,0,0,64],[1,0,0,0,0,1,0,0,0,0,64,0,0,0,0,8],[51,22,0,0,29,22,0,0,0,0,0,72,0,0,1,0] >;
C3×C32⋊4C8 in GAP, Magma, Sage, TeX
C_3\times C_3^2\rtimes_4C_8
% in TeX
G:=Group("C3xC3^2:4C8"); // GroupNames label
G:=SmallGroup(216,83);
// by ID
G=gap.SmallGroup(216,83);
# by ID
G:=PCGroup([6,-2,-3,-2,-2,-3,-3,36,50,1444,5189]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^3=c^3=d^8=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^-1>;
// generators/relations