metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.238D6, (C4×S3)⋊8D4, D6.7(C2×D4), C4.35(S3×D4), C12⋊4(C4○D4), C4⋊1D4⋊11S3, C12.66(C2×D4), D6⋊3D4⋊35C2, C4⋊2(D4⋊2S3), (S3×C42)⋊13C2, C12⋊2Q8⋊34C2, (D4×Dic3)⋊34C2, (C2×D4).178D6, C6.94(C22×D4), (C2×C6).260C24, Dic3.66(C2×D4), C23.12D6⋊27C2, (C2×C12).508C23, (C4×C12).203C22, (C6×D4).161C22, C23.76(C22×S3), (C22×C6).74C23, C4⋊Dic3.248C22, C22.281(S3×C23), C3⋊5(C22.26C24), (C22×S3).228C23, (C2×Dic3).135C23, (C2×Dic6).185C22, (C4×Dic3).256C22, C6.D4.72C22, (C22×Dic3).157C22, C2.67(C2×S3×D4), (C3×C4⋊1D4)⋊7C2, C6.96(C2×C4○D4), (C2×D4⋊2S3)⋊21C2, C2.60(C2×D4⋊2S3), (S3×C2×C4).251C22, (C2×C4).597(C22×S3), (C2×C3⋊D4).77C22, SmallGroup(192,1275)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.238D6
G = < a,b,c,d | a4=b4=c6=1, d2=b2, ab=ba, cac-1=dad-1=a-1, cbc-1=dbd-1=b-1, dcd-1=b2c-1 >
Subgroups: 752 in 310 conjugacy classes, 111 normal (19 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C6, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, Dic3, Dic3, C12, D6, D6, C2×C6, C2×C6, C42, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, Dic6, C4×S3, C4×S3, C2×Dic3, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×D4, C22×S3, C22×C6, C2×C42, C4×D4, C4⋊D4, C4.4D4, C4⋊1D4, C4⋊Q8, C2×C4○D4, C4×Dic3, C4×Dic3, C4⋊Dic3, C6.D4, C4×C12, C2×Dic6, S3×C2×C4, S3×C2×C4, D4⋊2S3, C22×Dic3, C2×C3⋊D4, C6×D4, C22.26C24, C12⋊2Q8, S3×C42, D4×Dic3, C23.12D6, D6⋊3D4, C3×C4⋊1D4, C2×D4⋊2S3, C42.238D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, C24, C22×S3, C22×D4, C2×C4○D4, S3×D4, D4⋊2S3, S3×C23, C22.26C24, C2×S3×D4, C2×D4⋊2S3, C42.238D6
►On 96 points
(1 64 19 67)(2 68 20 65)(3 66 21 69)(4 70 22 61)(5 62 23 71)(6 72 24 63)(7 59 35 80)(8 81 36 60)(9 55 31 82)(10 83 32 56)(11 57 33 84)(12 79 34 58)(13 40 86 54)(14 49 87 41)(15 42 88 50)(16 51 89 37)(17 38 90 52)(18 53 85 39)(25 94 46 74)(26 75 47 95)(27 96 48 76)(28 77 43 91)(29 92 44 78)(30 73 45 93)
(1 28 7 42)(2 37 8 29)(3 30 9 38)(4 39 10 25)(5 26 11 40)(6 41 12 27)(13 71 95 84)(14 79 96 72)(15 67 91 80)(16 81 92 68)(17 69 93 82)(18 83 94 70)(19 43 35 50)(20 51 36 44)(21 45 31 52)(22 53 32 46)(23 47 33 54)(24 49 34 48)(55 90 66 73)(56 74 61 85)(57 86 62 75)(58 76 63 87)(59 88 64 77)(60 78 65 89)
(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 6 7 12)(2 11 8 5)(3 4 9 10)(13 16 95 92)(14 91 96 15)(17 18 93 94)(19 24 35 34)(20 33 36 23)(21 22 31 32)(25 38 39 30)(26 29 40 37)(27 42 41 28)(43 48 50 49)(44 54 51 47)(45 46 52 53)(55 56 66 61)(57 60 62 65)(58 64 63 59)(67 72 80 79)(68 84 81 71)(69 70 82 83)(73 74 90 85)(75 78 86 89)(76 88 87 77)
G:=sub<Sym(96)| (1,64,19,67)(2,68,20,65)(3,66,21,69)(4,70,22,61)(5,62,23,71)(6,72,24,63)(7,59,35,80)(8,81,36,60)(9,55,31,82)(10,83,32,56)(11,57,33,84)(12,79,34,58)(13,40,86,54)(14,49,87,41)(15,42,88,50)(16,51,89,37)(17,38,90,52)(18,53,85,39)(25,94,46,74)(26,75,47,95)(27,96,48,76)(28,77,43,91)(29,92,44,78)(30,73,45,93), (1,28,7,42)(2,37,8,29)(3,30,9,38)(4,39,10,25)(5,26,11,40)(6,41,12,27)(13,71,95,84)(14,79,96,72)(15,67,91,80)(16,81,92,68)(17,69,93,82)(18,83,94,70)(19,43,35,50)(20,51,36,44)(21,45,31,52)(22,53,32,46)(23,47,33,54)(24,49,34,48)(55,90,66,73)(56,74,61,85)(57,86,62,75)(58,76,63,87)(59,88,64,77)(60,78,65,89), (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), (1,6,7,12)(2,11,8,5)(3,4,9,10)(13,16,95,92)(14,91,96,15)(17,18,93,94)(19,24,35,34)(20,33,36,23)(21,22,31,32)(25,38,39,30)(26,29,40,37)(27,42,41,28)(43,48,50,49)(44,54,51,47)(45,46,52,53)(55,56,66,61)(57,60,62,65)(58,64,63,59)(67,72,80,79)(68,84,81,71)(69,70,82,83)(73,74,90,85)(75,78,86,89)(76,88,87,77)>;
G:=Group( (1,64,19,67)(2,68,20,65)(3,66,21,69)(4,70,22,61)(5,62,23,71)(6,72,24,63)(7,59,35,80)(8,81,36,60)(9,55,31,82)(10,83,32,56)(11,57,33,84)(12,79,34,58)(13,40,86,54)(14,49,87,41)(15,42,88,50)(16,51,89,37)(17,38,90,52)(18,53,85,39)(25,94,46,74)(26,75,47,95)(27,96,48,76)(28,77,43,91)(29,92,44,78)(30,73,45,93), (1,28,7,42)(2,37,8,29)(3,30,9,38)(4,39,10,25)(5,26,11,40)(6,41,12,27)(13,71,95,84)(14,79,96,72)(15,67,91,80)(16,81,92,68)(17,69,93,82)(18,83,94,70)(19,43,35,50)(20,51,36,44)(21,45,31,52)(22,53,32,46)(23,47,33,54)(24,49,34,48)(55,90,66,73)(56,74,61,85)(57,86,62,75)(58,76,63,87)(59,88,64,77)(60,78,65,89), (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), (1,6,7,12)(2,11,8,5)(3,4,9,10)(13,16,95,92)(14,91,96,15)(17,18,93,94)(19,24,35,34)(20,33,36,23)(21,22,31,32)(25,38,39,30)(26,29,40,37)(27,42,41,28)(43,48,50,49)(44,54,51,47)(45,46,52,53)(55,56,66,61)(57,60,62,65)(58,64,63,59)(67,72,80,79)(68,84,81,71)(69,70,82,83)(73,74,90,85)(75,78,86,89)(76,88,87,77) );
G=PermutationGroup([[(1,64,19,67),(2,68,20,65),(3,66,21,69),(4,70,22,61),(5,62,23,71),(6,72,24,63),(7,59,35,80),(8,81,36,60),(9,55,31,82),(10,83,32,56),(11,57,33,84),(12,79,34,58),(13,40,86,54),(14,49,87,41),(15,42,88,50),(16,51,89,37),(17,38,90,52),(18,53,85,39),(25,94,46,74),(26,75,47,95),(27,96,48,76),(28,77,43,91),(29,92,44,78),(30,73,45,93)], [(1,28,7,42),(2,37,8,29),(3,30,9,38),(4,39,10,25),(5,26,11,40),(6,41,12,27),(13,71,95,84),(14,79,96,72),(15,67,91,80),(16,81,92,68),(17,69,93,82),(18,83,94,70),(19,43,35,50),(20,51,36,44),(21,45,31,52),(22,53,32,46),(23,47,33,54),(24,49,34,48),(55,90,66,73),(56,74,61,85),(57,86,62,75),(58,76,63,87),(59,88,64,77),(60,78,65,89)], [(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)], [(1,6,7,12),(2,11,8,5),(3,4,9,10),(13,16,95,92),(14,91,96,15),(17,18,93,94),(19,24,35,34),(20,33,36,23),(21,22,31,32),(25,38,39,30),(26,29,40,37),(27,42,41,28),(43,48,50,49),(44,54,51,47),(45,46,52,53),(55,56,66,61),(57,60,62,65),(58,64,63,59),(67,72,80,79),(68,84,81,71),(69,70,82,83),(73,74,90,85),(75,78,86,89),(76,88,87,77)]])
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 3 | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | 4Q | 4R | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 12A | ··· | 12F |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 6 | 6 | 2 | 2 | ··· | 2 | 3 | 3 | 3 | 3 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 8 | 8 | 8 | 8 | 4 | ··· | 4 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D6 | D6 | C4○D4 | S3×D4 | D4⋊2S3 |
| kernel | C42.238D6 | C12⋊2Q8 | S3×C42 | D4×Dic3 | C23.12D6 | D6⋊3D4 | C3×C4⋊1D4 | C2×D4⋊2S3 | C4⋊1D4 | C4×S3 | C42 | C2×D4 | C12 | C4 | C4 |
| # reps | 1 | 1 | 1 | 4 | 2 | 4 | 1 | 2 | 1 | 4 | 1 | 6 | 8 | 2 | 4 |
Matrix representation of C42.238D6 ►in GL6(𝔽13)
| 8 | 0 | 0 | 0 | 0 | 0 |
| 0 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 5 | 0 | 0 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 12 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 12 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(13))| [8,0,0,0,0,0,0,5,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,1,0],[5,0,0,0,0,0,0,8,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[0,12,0,0,0,0,12,0,0,0,0,0,0,0,12,1,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,1],[0,12,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,12,1,0,0,0,0,0,0,12,0,0,0,0,0,0,1] >;
C42.238D6 in GAP, Magma, Sage, TeX
C_4^2._{238}D_6 % in TeX
G:=Group("C4^2.238D6"); // GroupNames label
G:=SmallGroup(192,1275);
// by ID
G=gap.SmallGroup(192,1275);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,100,675,570,185,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^6=1,d^2=b^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=b^2*c^-1>;
// generators/relations