metabelian, supersoluble, monomial, A-group, rational
Aliases: C34⋊4C2, C33⋊9S3, C3⋊(C33⋊C2), C32⋊4(C3⋊S3), SmallGroup(162,54)
Series: Derived ►Chief ►Lower central ►Upper central
| C34 — C34⋊C2 |
Generators and relations for C34⋊C2
G = < a,b,c,d,e | a3=b3=c3=d3=e2=1, ab=ba, ac=ca, ad=da, eae=a-1, bc=cb, bd=db, ebe=b-1, cd=dc, ece=c-1, ede=d-1 >
Subgroups: 2664 in 424 conjugacy classes, 213 normal (3 characteristic)
C1, C2, C3, S3, C32, C3⋊S3, C33, C33⋊C2, C34, C34⋊C2
Quotients: C1, C2, S3, C3⋊S3, C33⋊C2, C34⋊C2
►On 81 points
Generators in S81
(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)(73 74 75)(76 77 78)(79 80 81)
(1 11 8)(2 12 9)(3 10 7)(4 63 60)(5 61 58)(6 62 59)(13 19 16)(14 20 17)(15 21 18)(22 29 25)(23 30 26)(24 28 27)(31 37 34)(32 38 35)(33 39 36)(40 46 43)(41 47 44)(42 48 45)(49 56 52)(50 57 53)(51 55 54)(64 71 68)(65 72 69)(66 70 67)(73 79 76)(74 80 77)(75 81 78)
(1 59 32)(2 60 33)(3 58 31)(4 39 12)(5 37 10)(6 38 11)(7 61 34)(8 62 35)(9 63 36)(13 67 40)(14 68 41)(15 69 42)(16 70 43)(17 71 44)(18 72 45)(19 66 46)(20 64 47)(21 65 48)(22 73 49)(23 74 50)(24 75 51)(25 76 52)(26 77 53)(27 78 54)(28 81 55)(29 79 56)(30 80 57)
(1 74 14)(2 75 15)(3 73 13)(4 55 65)(5 56 66)(6 57 64)(7 76 16)(8 77 17)(9 78 18)(10 79 19)(11 80 20)(12 81 21)(22 40 31)(23 41 32)(24 42 33)(25 43 34)(26 44 35)(27 45 36)(28 48 39)(29 46 37)(30 47 38)(49 67 58)(50 68 59)(51 69 60)(52 70 61)(53 71 62)(54 72 63)
(2 3)(4 34)(5 36)(6 35)(7 12)(8 11)(9 10)(13 75)(14 74)(15 73)(16 81)(17 80)(18 79)(19 78)(20 77)(21 76)(22 69)(23 68)(24 67)(25 65)(26 64)(27 66)(28 70)(29 72)(30 71)(31 60)(32 59)(33 58)(37 63)(38 62)(39 61)(40 51)(41 50)(42 49)(43 55)(44 57)(45 56)(46 54)(47 53)(48 52)
G:=sub<Sym(81)| (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)(73,74,75)(76,77,78)(79,80,81), (1,11,8)(2,12,9)(3,10,7)(4,63,60)(5,61,58)(6,62,59)(13,19,16)(14,20,17)(15,21,18)(22,29,25)(23,30,26)(24,28,27)(31,37,34)(32,38,35)(33,39,36)(40,46,43)(41,47,44)(42,48,45)(49,56,52)(50,57,53)(51,55,54)(64,71,68)(65,72,69)(66,70,67)(73,79,76)(74,80,77)(75,81,78), (1,59,32)(2,60,33)(3,58,31)(4,39,12)(5,37,10)(6,38,11)(7,61,34)(8,62,35)(9,63,36)(13,67,40)(14,68,41)(15,69,42)(16,70,43)(17,71,44)(18,72,45)(19,66,46)(20,64,47)(21,65,48)(22,73,49)(23,74,50)(24,75,51)(25,76,52)(26,77,53)(27,78,54)(28,81,55)(29,79,56)(30,80,57), (1,74,14)(2,75,15)(3,73,13)(4,55,65)(5,56,66)(6,57,64)(7,76,16)(8,77,17)(9,78,18)(10,79,19)(11,80,20)(12,81,21)(22,40,31)(23,41,32)(24,42,33)(25,43,34)(26,44,35)(27,45,36)(28,48,39)(29,46,37)(30,47,38)(49,67,58)(50,68,59)(51,69,60)(52,70,61)(53,71,62)(54,72,63), (2,3)(4,34)(5,36)(6,35)(7,12)(8,11)(9,10)(13,75)(14,74)(15,73)(16,81)(17,80)(18,79)(19,78)(20,77)(21,76)(22,69)(23,68)(24,67)(25,65)(26,64)(27,66)(28,70)(29,72)(30,71)(31,60)(32,59)(33,58)(37,63)(38,62)(39,61)(40,51)(41,50)(42,49)(43,55)(44,57)(45,56)(46,54)(47,53)(48,52)>;
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)(73,74,75)(76,77,78)(79,80,81), (1,11,8)(2,12,9)(3,10,7)(4,63,60)(5,61,58)(6,62,59)(13,19,16)(14,20,17)(15,21,18)(22,29,25)(23,30,26)(24,28,27)(31,37,34)(32,38,35)(33,39,36)(40,46,43)(41,47,44)(42,48,45)(49,56,52)(50,57,53)(51,55,54)(64,71,68)(65,72,69)(66,70,67)(73,79,76)(74,80,77)(75,81,78), (1,59,32)(2,60,33)(3,58,31)(4,39,12)(5,37,10)(6,38,11)(7,61,34)(8,62,35)(9,63,36)(13,67,40)(14,68,41)(15,69,42)(16,70,43)(17,71,44)(18,72,45)(19,66,46)(20,64,47)(21,65,48)(22,73,49)(23,74,50)(24,75,51)(25,76,52)(26,77,53)(27,78,54)(28,81,55)(29,79,56)(30,80,57), (1,74,14)(2,75,15)(3,73,13)(4,55,65)(5,56,66)(6,57,64)(7,76,16)(8,77,17)(9,78,18)(10,79,19)(11,80,20)(12,81,21)(22,40,31)(23,41,32)(24,42,33)(25,43,34)(26,44,35)(27,45,36)(28,48,39)(29,46,37)(30,47,38)(49,67,58)(50,68,59)(51,69,60)(52,70,61)(53,71,62)(54,72,63), (2,3)(4,34)(5,36)(6,35)(7,12)(8,11)(9,10)(13,75)(14,74)(15,73)(16,81)(17,80)(18,79)(19,78)(20,77)(21,76)(22,69)(23,68)(24,67)(25,65)(26,64)(27,66)(28,70)(29,72)(30,71)(31,60)(32,59)(33,58)(37,63)(38,62)(39,61)(40,51)(41,50)(42,49)(43,55)(44,57)(45,56)(46,54)(47,53)(48,52) );
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),(73,74,75),(76,77,78),(79,80,81)], [(1,11,8),(2,12,9),(3,10,7),(4,63,60),(5,61,58),(6,62,59),(13,19,16),(14,20,17),(15,21,18),(22,29,25),(23,30,26),(24,28,27),(31,37,34),(32,38,35),(33,39,36),(40,46,43),(41,47,44),(42,48,45),(49,56,52),(50,57,53),(51,55,54),(64,71,68),(65,72,69),(66,70,67),(73,79,76),(74,80,77),(75,81,78)], [(1,59,32),(2,60,33),(3,58,31),(4,39,12),(5,37,10),(6,38,11),(7,61,34),(8,62,35),(9,63,36),(13,67,40),(14,68,41),(15,69,42),(16,70,43),(17,71,44),(18,72,45),(19,66,46),(20,64,47),(21,65,48),(22,73,49),(23,74,50),(24,75,51),(25,76,52),(26,77,53),(27,78,54),(28,81,55),(29,79,56),(30,80,57)], [(1,74,14),(2,75,15),(3,73,13),(4,55,65),(5,56,66),(6,57,64),(7,76,16),(8,77,17),(9,78,18),(10,79,19),(11,80,20),(12,81,21),(22,40,31),(23,41,32),(24,42,33),(25,43,34),(26,44,35),(27,45,36),(28,48,39),(29,46,37),(30,47,38),(49,67,58),(50,68,59),(51,69,60),(52,70,61),(53,71,62),(54,72,63)], [(2,3),(4,34),(5,36),(6,35),(7,12),(8,11),(9,10),(13,75),(14,74),(15,73),(16,81),(17,80),(18,79),(19,78),(20,77),(21,76),(22,69),(23,68),(24,67),(25,65),(26,64),(27,66),(28,70),(29,72),(30,71),(31,60),(32,59),(33,58),(37,63),(38,62),(39,61),(40,51),(41,50),(42,49),(43,55),(44,57),(45,56),(46,54),(47,53),(48,52)]])
C34⋊C2 is a maximal subgroup of
C34⋊4C4 S3×C33⋊C2 C3⋊S32 C34⋊4C6 C34⋊5C6 C34⋊10C6 C33⋊9D9 C35⋊C2
C34⋊C2 is a maximal quotient of
C34⋊8C4 C33⋊9D9 C34⋊13S3 3+ 1+4⋊3C2 C35⋊C2
42 conjugacy classes
| class | 1 | 2 | 3A | ··· | 3AN |
| order | 1 | 2 | 3 | ··· | 3 |
| size | 1 | 81 | 2 | ··· | 2 |
42 irreducible representations
| dim | 1 | 1 | 2 |
| type | + | + | + |
| image | C1 | C2 | S3 |
| kernel | C34⋊C2 | C34 | C33 |
| # reps | 1 | 1 | 40 |
Matrix representation of C34⋊C2 ►in GL8(ℤ)
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | -1 | -1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | -1 | -1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | -1 | -1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| -1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| -1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | -1 | -1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | -1 | -1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 |
| -1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | -1 | -1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | -1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 |
G:=sub<GL(8,Integers())| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,-1,0],[-1,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[-1,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,-1],[-1,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,1] >;
C34⋊C2 in GAP, Magma, Sage, TeX
C_3^4\rtimes C_2
% in TeX
G:=Group("C3^4:C2"); // GroupNames label
G:=SmallGroup(162,54);
// by ID
G=gap.SmallGroup(162,54);
# by ID
G:=PCGroup([5,-2,-3,-3,-3,-3,41,182,723,2704]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^3=c^3=d^3=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,e*a*e=a^-1,b*c=c*b,b*d=d*b,e*b*e=b^-1,c*d=d*c,e*c*e=c^-1,e*d*e=d^-1>;
// generators/relations