metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.135D6, C6.672- 1+4, (C4×Q8)⋊21S3, C4⋊C4.302D6, (Q8×C12)⋊19C2, D6⋊3Q8⋊11C2, C12⋊2Q8⋊29C2, (C4×Dic6)⋊41C2, (C4×D12).23C2, (C2×Q8).207D6, C4.69(C4○D12), (C2×C6).128C24, D6⋊C4.56C22, C2.25(Q8○D12), C12.122(C4○D4), (C4×C12).180C22, (C2×C12).591C23, C42⋊7S3.11C2, C4.50(Q8⋊3S3), (C6×Q8).228C22, (C2×D12).219C22, (C22×S3).50C23, C4⋊Dic3.400C22, C22.149(S3×C23), Dic3⋊C4.157C22, C3⋊4(C22.50C24), (C2×Dic6).243C22, (C4×Dic3).210C22, (C2×Dic3).218C23, C4⋊C4⋊7S3⋊18C2, C4⋊C4⋊S3⋊11C2, C2.67(C2×C4○D12), C6.113(C2×C4○D4), (S3×C2×C4).78C22, C2.13(C2×Q8⋊3S3), (C3×C4⋊C4).356C22, (C2×C4).172(C22×S3), SmallGroup(192,1143)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.135D6
G = < a,b,c,d | a4=b4=1, c6=d2=a2b2, ab=ba, ac=ca, dad-1=a-1, cbc-1=a2b-1, bd=db, dcd-1=c5 >
Subgroups: 488 in 212 conjugacy classes, 99 normal (29 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C2×C4, C2×C4, C2×C4, D4, Q8, C23, Dic3, C12, C12, D6, C2×C6, C42, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, C2×Q8, Dic6, C4×S3, D12, C2×Dic3, C2×C12, C2×C12, C3×Q8, C22×S3, C42⋊C2, C4×D4, C4×Q8, C4×Q8, C22⋊Q8, C4.4D4, C42⋊2C2, C4⋊Q8, C4×Dic3, Dic3⋊C4, C4⋊Dic3, C4⋊Dic3, D6⋊C4, C4×C12, C4×C12, C3×C4⋊C4, C3×C4⋊C4, C2×Dic6, S3×C2×C4, C2×D12, C6×Q8, C22.50C24, C4×Dic6, C12⋊2Q8, C4×D12, C42⋊7S3, C4⋊C4⋊7S3, C4⋊C4⋊S3, D6⋊3Q8, Q8×C12, C42.135D6
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C24, C22×S3, C2×C4○D4, 2- 1+4, C4○D12, Q8⋊3S3, S3×C23, C22.50C24, C2×C4○D12, C2×Q8⋊3S3, Q8○D12, C42.135D6
►On 96 points
(1 29 41 85)(2 30 42 86)(3 31 43 87)(4 32 44 88)(5 33 45 89)(6 34 46 90)(7 35 47 91)(8 36 48 92)(9 25 37 93)(10 26 38 94)(11 27 39 95)(12 28 40 96)(13 80 54 67)(14 81 55 68)(15 82 56 69)(16 83 57 70)(17 84 58 71)(18 73 59 72)(19 74 60 61)(20 75 49 62)(21 76 50 63)(22 77 51 64)(23 78 52 65)(24 79 53 66)
(1 57 47 22)(2 52 48 17)(3 59 37 24)(4 54 38 19)(5 49 39 14)(6 56 40 21)(7 51 41 16)(8 58 42 23)(9 53 43 18)(10 60 44 13)(11 55 45 20)(12 50 46 15)(25 66 87 73)(26 61 88 80)(27 68 89 75)(28 63 90 82)(29 70 91 77)(30 65 92 84)(31 72 93 79)(32 67 94 74)(33 62 95 81)(34 69 96 76)(35 64 85 83)(36 71 86 78)
(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 24 19 18)(14 17 20 23)(15 22 21 16)(25 94 31 88)(26 87 32 93)(27 92 33 86)(28 85 34 91)(29 90 35 96)(30 95 36 89)(37 38 43 44)(39 48 45 42)(40 41 46 47)(49 52 55 58)(50 57 56 51)(53 60 59 54)(61 73 67 79)(62 78 68 84)(63 83 69 77)(64 76 70 82)(65 81 71 75)(66 74 72 80)
G:=sub<Sym(96)| (1,29,41,85)(2,30,42,86)(3,31,43,87)(4,32,44,88)(5,33,45,89)(6,34,46,90)(7,35,47,91)(8,36,48,92)(9,25,37,93)(10,26,38,94)(11,27,39,95)(12,28,40,96)(13,80,54,67)(14,81,55,68)(15,82,56,69)(16,83,57,70)(17,84,58,71)(18,73,59,72)(19,74,60,61)(20,75,49,62)(21,76,50,63)(22,77,51,64)(23,78,52,65)(24,79,53,66), (1,57,47,22)(2,52,48,17)(3,59,37,24)(4,54,38,19)(5,49,39,14)(6,56,40,21)(7,51,41,16)(8,58,42,23)(9,53,43,18)(10,60,44,13)(11,55,45,20)(12,50,46,15)(25,66,87,73)(26,61,88,80)(27,68,89,75)(28,63,90,82)(29,70,91,77)(30,65,92,84)(31,72,93,79)(32,67,94,74)(33,62,95,81)(34,69,96,76)(35,64,85,83)(36,71,86,78), (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)(13,24,19,18)(14,17,20,23)(15,22,21,16)(25,94,31,88)(26,87,32,93)(27,92,33,86)(28,85,34,91)(29,90,35,96)(30,95,36,89)(37,38,43,44)(39,48,45,42)(40,41,46,47)(49,52,55,58)(50,57,56,51)(53,60,59,54)(61,73,67,79)(62,78,68,84)(63,83,69,77)(64,76,70,82)(65,81,71,75)(66,74,72,80)>;
G:=Group( (1,29,41,85)(2,30,42,86)(3,31,43,87)(4,32,44,88)(5,33,45,89)(6,34,46,90)(7,35,47,91)(8,36,48,92)(9,25,37,93)(10,26,38,94)(11,27,39,95)(12,28,40,96)(13,80,54,67)(14,81,55,68)(15,82,56,69)(16,83,57,70)(17,84,58,71)(18,73,59,72)(19,74,60,61)(20,75,49,62)(21,76,50,63)(22,77,51,64)(23,78,52,65)(24,79,53,66), (1,57,47,22)(2,52,48,17)(3,59,37,24)(4,54,38,19)(5,49,39,14)(6,56,40,21)(7,51,41,16)(8,58,42,23)(9,53,43,18)(10,60,44,13)(11,55,45,20)(12,50,46,15)(25,66,87,73)(26,61,88,80)(27,68,89,75)(28,63,90,82)(29,70,91,77)(30,65,92,84)(31,72,93,79)(32,67,94,74)(33,62,95,81)(34,69,96,76)(35,64,85,83)(36,71,86,78), (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)(13,24,19,18)(14,17,20,23)(15,22,21,16)(25,94,31,88)(26,87,32,93)(27,92,33,86)(28,85,34,91)(29,90,35,96)(30,95,36,89)(37,38,43,44)(39,48,45,42)(40,41,46,47)(49,52,55,58)(50,57,56,51)(53,60,59,54)(61,73,67,79)(62,78,68,84)(63,83,69,77)(64,76,70,82)(65,81,71,75)(66,74,72,80) );
G=PermutationGroup([[(1,29,41,85),(2,30,42,86),(3,31,43,87),(4,32,44,88),(5,33,45,89),(6,34,46,90),(7,35,47,91),(8,36,48,92),(9,25,37,93),(10,26,38,94),(11,27,39,95),(12,28,40,96),(13,80,54,67),(14,81,55,68),(15,82,56,69),(16,83,57,70),(17,84,58,71),(18,73,59,72),(19,74,60,61),(20,75,49,62),(21,76,50,63),(22,77,51,64),(23,78,52,65),(24,79,53,66)], [(1,57,47,22),(2,52,48,17),(3,59,37,24),(4,54,38,19),(5,49,39,14),(6,56,40,21),(7,51,41,16),(8,58,42,23),(9,53,43,18),(10,60,44,13),(11,55,45,20),(12,50,46,15),(25,66,87,73),(26,61,88,80),(27,68,89,75),(28,63,90,82),(29,70,91,77),(30,65,92,84),(31,72,93,79),(32,67,94,74),(33,62,95,81),(34,69,96,76),(35,64,85,83),(36,71,86,78)], [(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),(13,24,19,18),(14,17,20,23),(15,22,21,16),(25,94,31,88),(26,87,32,93),(27,92,33,86),(28,85,34,91),(29,90,35,96),(30,95,36,89),(37,38,43,44),(39,48,45,42),(40,41,46,47),(49,52,55,58),(50,57,56,51),(53,60,59,54),(61,73,67,79),(62,78,68,84),(63,83,69,77),(64,76,70,82),(65,81,71,75),(66,74,72,80)]])
45 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | ··· | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | 4Q | 4R | 4S | 6A | 6B | 6C | 12A | 12B | 12C | 12D | 12E | ··· | 12P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 12 | 12 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
45 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D6 | D6 | D6 | C4○D4 | C4○D12 | 2- 1+4 | Q8⋊3S3 | Q8○D12 |
| kernel | C42.135D6 | C4×Dic6 | C12⋊2Q8 | C4×D12 | C42⋊7S3 | C4⋊C4⋊7S3 | C4⋊C4⋊S3 | D6⋊3Q8 | Q8×C12 | C4×Q8 | C42 | C4⋊C4 | C2×Q8 | C12 | C4 | C6 | C4 | C2 |
| # reps | 1 | 2 | 1 | 1 | 2 | 2 | 4 | 2 | 1 | 1 | 3 | 3 | 1 | 8 | 8 | 1 | 2 | 2 |
Matrix representation of C42.135D6 ►in GL4(𝔽13) generated by
| 3 | 7 | 0 | 0 |
| 6 | 10 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 |
| 8 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 |
| 0 | 0 | 1 | 12 |
| 0 | 0 | 2 | 12 |
| 0 | 12 | 0 | 0 |
| 1 | 12 | 0 | 0 |
| 0 | 0 | 5 | 8 |
| 0 | 0 | 0 | 8 |
| 1 | 12 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 5 | 0 |
| 0 | 0 | 0 | 5 |
G:=sub<GL(4,GF(13))| [3,6,0,0,7,10,0,0,0,0,12,0,0,0,0,12],[8,0,0,0,0,8,0,0,0,0,1,2,0,0,12,12],[0,1,0,0,12,12,0,0,0,0,5,0,0,0,8,8],[1,0,0,0,12,12,0,0,0,0,5,0,0,0,0,5] >;
C42.135D6 in GAP, Magma, Sage, TeX
C_4^2._{135}D_6 % in TeX
G:=Group("C4^2.135D6"); // GroupNames label
G:=SmallGroup(192,1143);
// by ID
G=gap.SmallGroup(192,1143);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,477,232,100,675,185,192,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^6=d^2=a^2*b^2,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,c*b*c^-1=a^2*b^-1,b*d=d*b,d*c*d^-1=c^5>;
// generators/relations