Aliases: (C6×C12).6C4, C32⋊2C8⋊6C4, (C3×C6).4C42, C62.5(C2×C4), C32⋊2(C8⋊C4), (C3×C6).9M4(2), C2.1(C62.C4), C2.2(C32⋊M4(2)), C2.4(C4×C32⋊C4), (C2×C4).2(C32⋊C4), (C2×C3⋊Dic3).15C4, (C4×C3⋊Dic3).16C2, C3⋊Dic3.28(C2×C4), (C2×C32⋊2C8).9C2, C22.10(C2×C32⋊C4), (C2×C3⋊Dic3).109C22, SmallGroup(288,425)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C32 — C3×C6 — C3⋊Dic3 — C2×C3⋊Dic3 — C2×C32⋊2C8 — C32⋊2C8⋊C4 |
Generators and relations for C32⋊2C8⋊C4
G = < a,b,c,d | a3=b3=c8=d4=1, cbc-1=ab=ba, cac-1=a-1b, ad=da, bd=db, dcd-1=c5 >
Subgroups: 272 in 70 conjugacy classes, 24 normal (16 characteristic)
C1, C2, C3, C4, C22, C6, C8, C2×C4, C2×C4, C32, Dic3, C12, C2×C6, C42, C2×C8, C3×C6, C2×Dic3, C2×C12, C8⋊C4, C3⋊Dic3, C3⋊Dic3, C3×C12, C62, C4×Dic3, C32⋊2C8, C2×C3⋊Dic3, C6×C12, C4×C3⋊Dic3, C2×C32⋊2C8, C32⋊2C8⋊C4
Quotients: C1, C2, C4, C22, C2×C4, C42, M4(2), C8⋊C4, C32⋊C4, C2×C32⋊C4, C32⋊M4(2), C4×C32⋊C4, C62.C4, C32⋊2C8⋊C4
►On 96 points
Generators in S96
(2 44 55)(4 49 46)(6 48 51)(8 53 42)(10 85 34)(12 36 87)(14 81 38)(16 40 83)(18 75 96)(20 90 77)(22 79 92)(24 94 73)(25 64 66)(27 68 58)(29 60 70)(31 72 62)
(1 43 54)(2 44 55)(3 56 45)(4 49 46)(5 47 50)(6 48 51)(7 52 41)(8 53 42)(9 84 33)(10 85 34)(11 35 86)(12 36 87)(13 88 37)(14 81 38)(15 39 82)(16 40 83)(17 74 95)(18 75 96)(19 89 76)(20 90 77)(21 78 91)(22 79 92)(23 93 80)(24 94 73)(25 64 66)(26 67 57)(27 68 58)(28 59 69)(29 60 70)(30 71 61)(31 72 62)(32 63 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)(73 74 75 76 77 78 79 80)(81 82 83 84 85 86 87 88)(89 90 91 92 93 94 95 96)
(1 15 95 28)(2 12 96 25)(3 9 89 30)(4 14 90 27)(5 11 91 32)(6 16 92 29)(7 13 93 26)(8 10 94 31)(17 59 43 39)(18 64 44 36)(19 61 45 33)(20 58 46 38)(21 63 47 35)(22 60 48 40)(23 57 41 37)(24 62 42 34)(49 81 77 68)(50 86 78 65)(51 83 79 70)(52 88 80 67)(53 85 73 72)(54 82 74 69)(55 87 75 66)(56 84 76 71)
G:=sub<Sym(96)| (2,44,55)(4,49,46)(6,48,51)(8,53,42)(10,85,34)(12,36,87)(14,81,38)(16,40,83)(18,75,96)(20,90,77)(22,79,92)(24,94,73)(25,64,66)(27,68,58)(29,60,70)(31,72,62), (1,43,54)(2,44,55)(3,56,45)(4,49,46)(5,47,50)(6,48,51)(7,52,41)(8,53,42)(9,84,33)(10,85,34)(11,35,86)(12,36,87)(13,88,37)(14,81,38)(15,39,82)(16,40,83)(17,74,95)(18,75,96)(19,89,76)(20,90,77)(21,78,91)(22,79,92)(23,93,80)(24,94,73)(25,64,66)(26,67,57)(27,68,58)(28,59,69)(29,60,70)(30,71,61)(31,72,62)(32,63,65), (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)(73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88)(89,90,91,92,93,94,95,96), (1,15,95,28)(2,12,96,25)(3,9,89,30)(4,14,90,27)(5,11,91,32)(6,16,92,29)(7,13,93,26)(8,10,94,31)(17,59,43,39)(18,64,44,36)(19,61,45,33)(20,58,46,38)(21,63,47,35)(22,60,48,40)(23,57,41,37)(24,62,42,34)(49,81,77,68)(50,86,78,65)(51,83,79,70)(52,88,80,67)(53,85,73,72)(54,82,74,69)(55,87,75,66)(56,84,76,71)>;
G:=Group( (2,44,55)(4,49,46)(6,48,51)(8,53,42)(10,85,34)(12,36,87)(14,81,38)(16,40,83)(18,75,96)(20,90,77)(22,79,92)(24,94,73)(25,64,66)(27,68,58)(29,60,70)(31,72,62), (1,43,54)(2,44,55)(3,56,45)(4,49,46)(5,47,50)(6,48,51)(7,52,41)(8,53,42)(9,84,33)(10,85,34)(11,35,86)(12,36,87)(13,88,37)(14,81,38)(15,39,82)(16,40,83)(17,74,95)(18,75,96)(19,89,76)(20,90,77)(21,78,91)(22,79,92)(23,93,80)(24,94,73)(25,64,66)(26,67,57)(27,68,58)(28,59,69)(29,60,70)(30,71,61)(31,72,62)(32,63,65), (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)(73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88)(89,90,91,92,93,94,95,96), (1,15,95,28)(2,12,96,25)(3,9,89,30)(4,14,90,27)(5,11,91,32)(6,16,92,29)(7,13,93,26)(8,10,94,31)(17,59,43,39)(18,64,44,36)(19,61,45,33)(20,58,46,38)(21,63,47,35)(22,60,48,40)(23,57,41,37)(24,62,42,34)(49,81,77,68)(50,86,78,65)(51,83,79,70)(52,88,80,67)(53,85,73,72)(54,82,74,69)(55,87,75,66)(56,84,76,71) );
G=PermutationGroup([[(2,44,55),(4,49,46),(6,48,51),(8,53,42),(10,85,34),(12,36,87),(14,81,38),(16,40,83),(18,75,96),(20,90,77),(22,79,92),(24,94,73),(25,64,66),(27,68,58),(29,60,70),(31,72,62)], [(1,43,54),(2,44,55),(3,56,45),(4,49,46),(5,47,50),(6,48,51),(7,52,41),(8,53,42),(9,84,33),(10,85,34),(11,35,86),(12,36,87),(13,88,37),(14,81,38),(15,39,82),(16,40,83),(17,74,95),(18,75,96),(19,89,76),(20,90,77),(21,78,91),(22,79,92),(23,93,80),(24,94,73),(25,64,66),(26,67,57),(27,68,58),(28,59,69),(29,60,70),(30,71,61),(31,72,62),(32,63,65)], [(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),(73,74,75,76,77,78,79,80),(81,82,83,84,85,86,87,88),(89,90,91,92,93,94,95,96)], [(1,15,95,28),(2,12,96,25),(3,9,89,30),(4,14,90,27),(5,11,91,32),(6,16,92,29),(7,13,93,26),(8,10,94,31),(17,59,43,39),(18,64,44,36),(19,61,45,33),(20,58,46,38),(21,63,47,35),(22,60,48,40),(23,57,41,37),(24,62,42,34),(49,81,77,68),(50,86,78,65),(51,83,79,70),(52,88,80,67),(53,85,73,72),(54,82,74,69),(55,87,75,66),(56,84,76,71)]])
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | ··· | 6F | 8A | ··· | 8H | 12A | ··· | 12H |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 8 | ··· | 8 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 2 | 2 | 9 | 9 | 9 | 9 | 18 | 18 | 4 | ··· | 4 | 18 | ··· | 18 | 4 | ··· | 4 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | - | ||||||
| image | C1 | C2 | C2 | C4 | C4 | C4 | M4(2) | C32⋊C4 | C2×C32⋊C4 | C32⋊M4(2) | C4×C32⋊C4 | C62.C4 |
| kernel | C32⋊2C8⋊C4 | C4×C3⋊Dic3 | C2×C32⋊2C8 | C32⋊2C8 | C2×C3⋊Dic3 | C6×C12 | C3×C6 | C2×C4 | C22 | C2 | C2 | C2 |
| # reps | 1 | 1 | 2 | 8 | 2 | 2 | 4 | 2 | 2 | 4 | 4 | 4 |
Matrix representation of C32⋊2C8⋊C4 ►in GL6(𝔽73)
| 1 | 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 | 0 | 1 |
| 0 | 0 | 46 | 46 | 72 | 72 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 1 | 72 | 0 | 0 |
| 0 | 0 | 0 | 27 | 0 | 1 |
| 0 | 0 | 46 | 0 | 72 | 72 |
| 22 | 0 | 0 | 0 | 0 | 0 |
| 10 | 51 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 72 |
| 0 | 0 | 27 | 27 | 2 | 1 |
| 0 | 0 | 35 | 43 | 46 | 0 |
| 0 | 0 | 43 | 69 | 46 | 0 |
| 27 | 71 | 0 | 0 | 0 | 0 |
| 72 | 46 | 0 | 0 | 0 | 0 |
| 0 | 0 | 27 | 0 | 0 | 0 |
| 0 | 0 | 0 | 27 | 0 | 0 |
| 0 | 0 | 25 | 25 | 46 | 0 |
| 0 | 0 | 25 | 25 | 0 | 46 |
G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,46,0,0,0,1,0,46,0,0,0,0,0,72,0,0,0,0,1,72],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,46,0,0,72,72,27,0,0,0,0,0,0,72,0,0,0,0,1,72],[22,10,0,0,0,0,0,51,0,0,0,0,0,0,0,27,35,43,0,0,0,27,43,69,0,0,1,2,46,46,0,0,72,1,0,0],[27,72,0,0,0,0,71,46,0,0,0,0,0,0,27,0,25,25,0,0,0,27,25,25,0,0,0,0,46,0,0,0,0,0,0,46] >;
C32⋊2C8⋊C4 in GAP, Magma, Sage, TeX
C_3^2\rtimes_2C_8\rtimes C_4
% in TeX
G:=Group("C3^2:2C8:C4"); // GroupNames label
G:=SmallGroup(288,425);
// by ID
G=gap.SmallGroup(288,425);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,28,253,64,100,9413,691,12550,2372]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^3=c^8=d^4=1,c*b*c^-1=a*b=b*a,c*a*c^-1=a^-1*b,a*d=d*a,b*d=d*b,d*c*d^-1=c^5>;
// generators/relations