metabelian, supersoluble, monomial
Aliases: C24⋊1S3, C3⋊1D24, C32⋊5D8, C6.8D12, C12.47D6, C8⋊1(C3⋊S3), (C3×C24)⋊1C2, (C3×C6).23D4, C12⋊S3⋊1C2, C2.4(C12⋊S3), (C3×C12).33C22, C4.9(C2×C3⋊S3), SmallGroup(144,88)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C32⋊5D8
G = < a,b,c,d | a3=b3=c8=d2=1, ab=ba, ac=ca, dad=a-1, bc=cb, dbd=b-1, dcd=c-1 >
Subgroups: 354 in 66 conjugacy classes, 27 normal (9 characteristic)
C1, C2, C2, C3, C4, C22, S3, C6, C8, D4, C32, C12, D6, D8, C3⋊S3, C3×C6, C24, D12, C3×C12, C2×C3⋊S3, D24, C3×C24, C12⋊S3, C32⋊5D8
Quotients: C1, C2, C22, S3, D4, D6, D8, C3⋊S3, D12, C2×C3⋊S3, D24, C12⋊S3, C32⋊5D8
►On 72 points
Generators in S72
(1 59 47)(2 60 48)(3 61 41)(4 62 42)(5 63 43)(6 64 44)(7 57 45)(8 58 46)(9 26 51)(10 27 52)(11 28 53)(12 29 54)(13 30 55)(14 31 56)(15 32 49)(16 25 50)(17 70 38)(18 71 39)(19 72 40)(20 65 33)(21 66 34)(22 67 35)(23 68 36)(24 69 37)
(1 25 18)(2 26 19)(3 27 20)(4 28 21)(5 29 22)(6 30 23)(7 31 24)(8 32 17)(9 40 48)(10 33 41)(11 34 42)(12 35 43)(13 36 44)(14 37 45)(15 38 46)(16 39 47)(49 70 58)(50 71 59)(51 72 60)(52 65 61)(53 66 62)(54 67 63)(55 68 64)(56 69 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)
(1 8)(2 7)(3 6)(4 5)(9 69)(10 68)(11 67)(12 66)(13 65)(14 72)(15 71)(16 70)(17 25)(18 32)(19 31)(20 30)(21 29)(22 28)(23 27)(24 26)(33 55)(34 54)(35 53)(36 52)(37 51)(38 50)(39 49)(40 56)(41 64)(42 63)(43 62)(44 61)(45 60)(46 59)(47 58)(48 57)
G:=sub<Sym(72)| (1,59,47)(2,60,48)(3,61,41)(4,62,42)(5,63,43)(6,64,44)(7,57,45)(8,58,46)(9,26,51)(10,27,52)(11,28,53)(12,29,54)(13,30,55)(14,31,56)(15,32,49)(16,25,50)(17,70,38)(18,71,39)(19,72,40)(20,65,33)(21,66,34)(22,67,35)(23,68,36)(24,69,37), (1,25,18)(2,26,19)(3,27,20)(4,28,21)(5,29,22)(6,30,23)(7,31,24)(8,32,17)(9,40,48)(10,33,41)(11,34,42)(12,35,43)(13,36,44)(14,37,45)(15,38,46)(16,39,47)(49,70,58)(50,71,59)(51,72,60)(52,65,61)(53,66,62)(54,67,63)(55,68,64)(56,69,57), (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), (1,8)(2,7)(3,6)(4,5)(9,69)(10,68)(11,67)(12,66)(13,65)(14,72)(15,71)(16,70)(17,25)(18,32)(19,31)(20,30)(21,29)(22,28)(23,27)(24,26)(33,55)(34,54)(35,53)(36,52)(37,51)(38,50)(39,49)(40,56)(41,64)(42,63)(43,62)(44,61)(45,60)(46,59)(47,58)(48,57)>;
G:=Group( (1,59,47)(2,60,48)(3,61,41)(4,62,42)(5,63,43)(6,64,44)(7,57,45)(8,58,46)(9,26,51)(10,27,52)(11,28,53)(12,29,54)(13,30,55)(14,31,56)(15,32,49)(16,25,50)(17,70,38)(18,71,39)(19,72,40)(20,65,33)(21,66,34)(22,67,35)(23,68,36)(24,69,37), (1,25,18)(2,26,19)(3,27,20)(4,28,21)(5,29,22)(6,30,23)(7,31,24)(8,32,17)(9,40,48)(10,33,41)(11,34,42)(12,35,43)(13,36,44)(14,37,45)(15,38,46)(16,39,47)(49,70,58)(50,71,59)(51,72,60)(52,65,61)(53,66,62)(54,67,63)(55,68,64)(56,69,57), (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), (1,8)(2,7)(3,6)(4,5)(9,69)(10,68)(11,67)(12,66)(13,65)(14,72)(15,71)(16,70)(17,25)(18,32)(19,31)(20,30)(21,29)(22,28)(23,27)(24,26)(33,55)(34,54)(35,53)(36,52)(37,51)(38,50)(39,49)(40,56)(41,64)(42,63)(43,62)(44,61)(45,60)(46,59)(47,58)(48,57) );
G=PermutationGroup([[(1,59,47),(2,60,48),(3,61,41),(4,62,42),(5,63,43),(6,64,44),(7,57,45),(8,58,46),(9,26,51),(10,27,52),(11,28,53),(12,29,54),(13,30,55),(14,31,56),(15,32,49),(16,25,50),(17,70,38),(18,71,39),(19,72,40),(20,65,33),(21,66,34),(22,67,35),(23,68,36),(24,69,37)], [(1,25,18),(2,26,19),(3,27,20),(4,28,21),(5,29,22),(6,30,23),(7,31,24),(8,32,17),(9,40,48),(10,33,41),(11,34,42),(12,35,43),(13,36,44),(14,37,45),(15,38,46),(16,39,47),(49,70,58),(50,71,59),(51,72,60),(52,65,61),(53,66,62),(54,67,63),(55,68,64),(56,69,57)], [(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)], [(1,8),(2,7),(3,6),(4,5),(9,69),(10,68),(11,67),(12,66),(13,65),(14,72),(15,71),(16,70),(17,25),(18,32),(19,31),(20,30),(21,29),(22,28),(23,27),(24,26),(33,55),(34,54),(35,53),(36,52),(37,51),(38,50),(39,49),(40,56),(41,64),(42,63),(43,62),(44,61),(45,60),(46,59),(47,58),(48,57)]])
C32⋊5D8 is a maximal subgroup of
C3⋊D48 C24.49D6 C32⋊5D16 C6.D24 C32⋊7D16 C32⋊10SD32 S3×D24 C24⋊1D6 D6.3D12 C24.78D6 C24⋊3D6 D8×C3⋊S3 C24⋊7D6 C24.28D6 He3⋊4D8 C72⋊1S3 C33⋊8D8 C33⋊12D8
C32⋊5D8 is a maximal quotient of
C32⋊5D16 C6.D24 C32⋊5Q32 C24⋊1Dic3 C62.84D4 C72⋊1S3 He3⋊5D8 C33⋊8D8 C33⋊12D8
39 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 4 | 6A | 6B | 6C | 6D | 8A | 8B | 12A | ··· | 12H | 24A | ··· | 24P |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 6 | 6 | 6 | 6 | 8 | 8 | 12 | ··· | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 36 | 36 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
39 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | S3 | D4 | D6 | D8 | D12 | D24 |
| kernel | C32⋊5D8 | C3×C24 | C12⋊S3 | C24 | C3×C6 | C12 | C32 | C6 | C3 |
| # reps | 1 | 1 | 2 | 4 | 1 | 4 | 2 | 8 | 16 |
Matrix representation of C32⋊5D8 ►in GL4(𝔽73) generated by
| 72 | 72 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 72 | 72 |
| 0 | 1 | 0 | 0 |
| 72 | 72 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 72 | 72 |
| 50 | 68 | 0 | 0 |
| 5 | 55 | 0 | 0 |
| 0 | 0 | 18 | 68 |
| 0 | 0 | 5 | 23 |
| 23 | 5 | 0 | 0 |
| 55 | 50 | 0 | 0 |
| 0 | 0 | 23 | 5 |
| 0 | 0 | 55 | 50 |
G:=sub<GL(4,GF(73))| [72,1,0,0,72,0,0,0,0,0,0,72,0,0,1,72],[0,72,0,0,1,72,0,0,0,0,0,72,0,0,1,72],[50,5,0,0,68,55,0,0,0,0,18,5,0,0,68,23],[23,55,0,0,5,50,0,0,0,0,23,55,0,0,5,50] >;
C32⋊5D8 in GAP, Magma, Sage, TeX
C_3^2\rtimes_5D_8
% in TeX
G:=Group("C3^2:5D8"); // GroupNames label
G:=SmallGroup(144,88);
// by ID
G=gap.SmallGroup(144,88);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-3,73,79,218,50,964,3461]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^3=c^8=d^2=1,a*b=b*a,a*c=c*a,d*a*d=a^-1,b*c=c*b,d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations