direct product, metacyclic, nilpotent (class 2), monomial
Aliases: C32×M4(2), C24⋊7C6, C12.6C12, C62.3C4, C4.6C62, C8⋊3(C3×C6), C4.(C3×C12), (C3×C24)⋊11C2, C2.3(C6×C12), (C2×C6).4C12, C22.(C3×C12), (C6×C12).14C2, C6.15(C2×C12), (C2×C12).14C6, (C3×C12).10C4, C12.29(C2×C6), (C3×C12).54C22, (C2×C4).2(C3×C6), (C3×C6).37(C2×C4), SmallGroup(144,105)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C32×M4(2)
G = < a,b,c,d | a3=b3=c8=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c5 >
Subgroups: 66 in 60 conjugacy classes, 54 normal (14 characteristic)
C1, C2, C2, C3, C4, C22, C6, C6, C8, C2×C4, C32, C12, C2×C6, M4(2), C3×C6, C3×C6, C24, C2×C12, C3×C12, C62, C3×M4(2), C3×C24, C6×C12, C32×M4(2)
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, C32, C12, C2×C6, M4(2), C3×C6, C2×C12, C3×C12, C62, C3×M4(2), C6×C12, C32×M4(2)
►On 72 points
(1 34 59)(2 35 60)(3 36 61)(4 37 62)(5 38 63)(6 39 64)(7 40 57)(8 33 58)(9 65 48)(10 66 41)(11 67 42)(12 68 43)(13 69 44)(14 70 45)(15 71 46)(16 72 47)(17 29 53)(18 30 54)(19 31 55)(20 32 56)(21 25 49)(22 26 50)(23 27 51)(24 28 52)
(1 31 11)(2 32 12)(3 25 13)(4 26 14)(5 27 15)(6 28 16)(7 29 9)(8 30 10)(17 48 57)(18 41 58)(19 42 59)(20 43 60)(21 44 61)(22 45 62)(23 46 63)(24 47 64)(33 54 66)(34 55 67)(35 56 68)(36 49 69)(37 50 70)(38 51 71)(39 52 72)(40 53 65)
(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)
(2 6)(4 8)(10 14)(12 16)(18 22)(20 24)(26 30)(28 32)(33 37)(35 39)(41 45)(43 47)(50 54)(52 56)(58 62)(60 64)(66 70)(68 72)
G:=sub<Sym(72)| (1,34,59)(2,35,60)(3,36,61)(4,37,62)(5,38,63)(6,39,64)(7,40,57)(8,33,58)(9,65,48)(10,66,41)(11,67,42)(12,68,43)(13,69,44)(14,70,45)(15,71,46)(16,72,47)(17,29,53)(18,30,54)(19,31,55)(20,32,56)(21,25,49)(22,26,50)(23,27,51)(24,28,52), (1,31,11)(2,32,12)(3,25,13)(4,26,14)(5,27,15)(6,28,16)(7,29,9)(8,30,10)(17,48,57)(18,41,58)(19,42,59)(20,43,60)(21,44,61)(22,45,62)(23,46,63)(24,47,64)(33,54,66)(34,55,67)(35,56,68)(36,49,69)(37,50,70)(38,51,71)(39,52,72)(40,53,65), (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), (2,6)(4,8)(10,14)(12,16)(18,22)(20,24)(26,30)(28,32)(33,37)(35,39)(41,45)(43,47)(50,54)(52,56)(58,62)(60,64)(66,70)(68,72)>;
G:=Group( (1,34,59)(2,35,60)(3,36,61)(4,37,62)(5,38,63)(6,39,64)(7,40,57)(8,33,58)(9,65,48)(10,66,41)(11,67,42)(12,68,43)(13,69,44)(14,70,45)(15,71,46)(16,72,47)(17,29,53)(18,30,54)(19,31,55)(20,32,56)(21,25,49)(22,26,50)(23,27,51)(24,28,52), (1,31,11)(2,32,12)(3,25,13)(4,26,14)(5,27,15)(6,28,16)(7,29,9)(8,30,10)(17,48,57)(18,41,58)(19,42,59)(20,43,60)(21,44,61)(22,45,62)(23,46,63)(24,47,64)(33,54,66)(34,55,67)(35,56,68)(36,49,69)(37,50,70)(38,51,71)(39,52,72)(40,53,65), (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), (2,6)(4,8)(10,14)(12,16)(18,22)(20,24)(26,30)(28,32)(33,37)(35,39)(41,45)(43,47)(50,54)(52,56)(58,62)(60,64)(66,70)(68,72) );
G=PermutationGroup([[(1,34,59),(2,35,60),(3,36,61),(4,37,62),(5,38,63),(6,39,64),(7,40,57),(8,33,58),(9,65,48),(10,66,41),(11,67,42),(12,68,43),(13,69,44),(14,70,45),(15,71,46),(16,72,47),(17,29,53),(18,30,54),(19,31,55),(20,32,56),(21,25,49),(22,26,50),(23,27,51),(24,28,52)], [(1,31,11),(2,32,12),(3,25,13),(4,26,14),(5,27,15),(6,28,16),(7,29,9),(8,30,10),(17,48,57),(18,41,58),(19,42,59),(20,43,60),(21,44,61),(22,45,62),(23,46,63),(24,47,64),(33,54,66),(34,55,67),(35,56,68),(36,49,69),(37,50,70),(38,51,71),(39,52,72),(40,53,65)], [(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)], [(2,6),(4,8),(10,14),(12,16),(18,22),(20,24),(26,30),(28,32),(33,37),(35,39),(41,45),(43,47),(50,54),(52,56),(58,62),(60,64),(66,70),(68,72)]])
C32×M4(2) is a maximal subgroup of
C62.8Q8 C12.19D12 C12.20D12 C62.37D4 C24.47D6 C24⋊3D6 C24.5D6
90 conjugacy classes
| class | 1 | 2A | 2B | 3A | ··· | 3H | 4A | 4B | 4C | 6A | ··· | 6H | 6I | ··· | 6P | 8A | 8B | 8C | 8D | 12A | ··· | 12P | 12Q | ··· | 12X | 24A | ··· | 24AF |
| order | 1 | 2 | 2 | 3 | ··· | 3 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 8 | 8 | 8 | 8 | 12 | ··· | 12 | 12 | ··· | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 2 | 1 | ··· | 1 | 1 | 1 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
90 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | |||||||||
| image | C1 | C2 | C2 | C3 | C4 | C4 | C6 | C6 | C12 | C12 | M4(2) | C3×M4(2) |
| kernel | C32×M4(2) | C3×C24 | C6×C12 | C3×M4(2) | C3×C12 | C62 | C24 | C2×C12 | C12 | C2×C6 | C32 | C3 |
| # reps | 1 | 2 | 1 | 8 | 2 | 2 | 16 | 8 | 16 | 16 | 2 | 16 |
Matrix representation of C32×M4(2) ►in GL3(𝔽73) generated by
| 1 | 0 | 0 |
| 0 | 64 | 0 |
| 0 | 0 | 64 |
| 8 | 0 | 0 |
| 0 | 8 | 0 |
| 0 | 0 | 8 |
| 72 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 27 | 0 |
| 72 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 72 |
G:=sub<GL(3,GF(73))| [1,0,0,0,64,0,0,0,64],[8,0,0,0,8,0,0,0,8],[72,0,0,0,0,27,0,1,0],[72,0,0,0,1,0,0,0,72] >;
C32×M4(2) in GAP, Magma, Sage, TeX
C_3^2\times M_4(2)
% in TeX
G:=Group("C3^2xM4(2)"); // GroupNames label
G:=SmallGroup(144,105);
// by ID
G=gap.SmallGroup(144,105);
# by ID
G:=PCGroup([6,-2,-2,-3,-3,-2,-2,216,889,88]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^3=c^8=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^5>;
// generators/relations