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
Subgroups: 488 in 212 conjugacy classes, 99 normal (29 characteristic)
C1, C2 [×3], C2 [×2], C3, C4 [×4], C4 [×11], C22, C22 [×6], S3 [×2], C6 [×3], C2×C4 [×3], C2×C4 [×4], C2×C4 [×10], D4 [×2], Q8 [×6], C23 [×2], Dic3 [×6], C12 [×4], C12 [×5], D6 [×6], C2×C6, C42, C42 [×2], C42 [×4], C22⋊C4 [×10], C4⋊C4, C4⋊C4 [×2], C4⋊C4 [×9], C22×C4 [×2], C2×D4, C2×Q8, C2×Q8 [×2], Dic6 [×4], C4×S3 [×4], D12 [×2], C2×Dic3 [×6], C2×C12 [×3], C2×C12 [×4], C3×Q8 [×2], C22×S3 [×2], C42⋊C2 [×2], C4×D4, C4×Q8, C4×Q8 [×2], C22⋊Q8 [×2], C4.4D4 [×2], C42⋊2C2 [×4], C4⋊Q8, C4×Dic3 [×4], Dic3⋊C4 [×4], C4⋊Dic3, C4⋊Dic3 [×4], D6⋊C4 [×10], C4×C12, C4×C12 [×2], C3×C4⋊C4, C3×C4⋊C4 [×2], C2×Dic6 [×2], S3×C2×C4 [×2], C2×D12, C6×Q8, C22.50C24, C4×Dic6 [×2], C12⋊2Q8, C4×D12, C42⋊7S3 [×2], C4⋊C4⋊7S3 [×2], C4⋊C4⋊S3 [×4], D6⋊3Q8 [×2], Q8×C12, C42.135D6
Quotients:
C1, C2 [×15], C22 [×35], S3, C23 [×15], D6 [×7], C4○D4 [×4], C24, C22×S3 [×7], C2×C4○D4 [×2], 2- (1+4), C4○D12 [×2], Q8⋊3S3 [×2], S3×C23, C22.50C24, C2×C4○D12, C2×Q8⋊3S3, Q8○D12, C42.135D6
Generators and relations
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 >
►On 96 points
(1 58 44 79)(2 59 45 80)(3 60 46 81)(4 49 47 82)(5 50 48 83)(6 51 37 84)(7 52 38 73)(8 53 39 74)(9 54 40 75)(10 55 41 76)(11 56 42 77)(12 57 43 78)(13 67 94 35)(14 68 95 36)(15 69 96 25)(16 70 85 26)(17 71 86 27)(18 72 87 28)(19 61 88 29)(20 62 89 30)(21 63 90 31)(22 64 91 32)(23 65 92 33)(24 66 93 34)
(1 21 38 96)(2 16 39 91)(3 23 40 86)(4 18 41 93)(5 13 42 88)(6 20 43 95)(7 15 44 90)(8 22 45 85)(9 17 46 92)(10 24 47 87)(11 19 48 94)(12 14 37 89)(25 58 63 73)(26 53 64 80)(27 60 65 75)(28 55 66 82)(29 50 67 77)(30 57 68 84)(31 52 69 79)(32 59 70 74)(33 54 71 81)(34 49 72 76)(35 56 61 83)(36 51 62 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 16 19 22)(14 21 20 15)(17 24 23 18)(25 68 31 62)(26 61 32 67)(27 66 33 72)(28 71 34 65)(29 64 35 70)(30 69 36 63)(37 38 43 44)(39 48 45 42)(40 41 46 47)(49 75 55 81)(50 80 56 74)(51 73 57 79)(52 78 58 84)(53 83 59 77)(54 76 60 82)(85 88 91 94)(86 93 92 87)(89 96 95 90)
G:=sub<Sym(96)| (1,58,44,79)(2,59,45,80)(3,60,46,81)(4,49,47,82)(5,50,48,83)(6,51,37,84)(7,52,38,73)(8,53,39,74)(9,54,40,75)(10,55,41,76)(11,56,42,77)(12,57,43,78)(13,67,94,35)(14,68,95,36)(15,69,96,25)(16,70,85,26)(17,71,86,27)(18,72,87,28)(19,61,88,29)(20,62,89,30)(21,63,90,31)(22,64,91,32)(23,65,92,33)(24,66,93,34), (1,21,38,96)(2,16,39,91)(3,23,40,86)(4,18,41,93)(5,13,42,88)(6,20,43,95)(7,15,44,90)(8,22,45,85)(9,17,46,92)(10,24,47,87)(11,19,48,94)(12,14,37,89)(25,58,63,73)(26,53,64,80)(27,60,65,75)(28,55,66,82)(29,50,67,77)(30,57,68,84)(31,52,69,79)(32,59,70,74)(33,54,71,81)(34,49,72,76)(35,56,61,83)(36,51,62,78), (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)(13,16,19,22)(14,21,20,15)(17,24,23,18)(25,68,31,62)(26,61,32,67)(27,66,33,72)(28,71,34,65)(29,64,35,70)(30,69,36,63)(37,38,43,44)(39,48,45,42)(40,41,46,47)(49,75,55,81)(50,80,56,74)(51,73,57,79)(52,78,58,84)(53,83,59,77)(54,76,60,82)(85,88,91,94)(86,93,92,87)(89,96,95,90)>;
G:=Group( (1,58,44,79)(2,59,45,80)(3,60,46,81)(4,49,47,82)(5,50,48,83)(6,51,37,84)(7,52,38,73)(8,53,39,74)(9,54,40,75)(10,55,41,76)(11,56,42,77)(12,57,43,78)(13,67,94,35)(14,68,95,36)(15,69,96,25)(16,70,85,26)(17,71,86,27)(18,72,87,28)(19,61,88,29)(20,62,89,30)(21,63,90,31)(22,64,91,32)(23,65,92,33)(24,66,93,34), (1,21,38,96)(2,16,39,91)(3,23,40,86)(4,18,41,93)(5,13,42,88)(6,20,43,95)(7,15,44,90)(8,22,45,85)(9,17,46,92)(10,24,47,87)(11,19,48,94)(12,14,37,89)(25,58,63,73)(26,53,64,80)(27,60,65,75)(28,55,66,82)(29,50,67,77)(30,57,68,84)(31,52,69,79)(32,59,70,74)(33,54,71,81)(34,49,72,76)(35,56,61,83)(36,51,62,78), (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)(13,16,19,22)(14,21,20,15)(17,24,23,18)(25,68,31,62)(26,61,32,67)(27,66,33,72)(28,71,34,65)(29,64,35,70)(30,69,36,63)(37,38,43,44)(39,48,45,42)(40,41,46,47)(49,75,55,81)(50,80,56,74)(51,73,57,79)(52,78,58,84)(53,83,59,77)(54,76,60,82)(85,88,91,94)(86,93,92,87)(89,96,95,90) );
G=PermutationGroup([(1,58,44,79),(2,59,45,80),(3,60,46,81),(4,49,47,82),(5,50,48,83),(6,51,37,84),(7,52,38,73),(8,53,39,74),(9,54,40,75),(10,55,41,76),(11,56,42,77),(12,57,43,78),(13,67,94,35),(14,68,95,36),(15,69,96,25),(16,70,85,26),(17,71,86,27),(18,72,87,28),(19,61,88,29),(20,62,89,30),(21,63,90,31),(22,64,91,32),(23,65,92,33),(24,66,93,34)], [(1,21,38,96),(2,16,39,91),(3,23,40,86),(4,18,41,93),(5,13,42,88),(6,20,43,95),(7,15,44,90),(8,22,45,85),(9,17,46,92),(10,24,47,87),(11,19,48,94),(12,14,37,89),(25,58,63,73),(26,53,64,80),(27,60,65,75),(28,55,66,82),(29,50,67,77),(30,57,68,84),(31,52,69,79),(32,59,70,74),(33,54,71,81),(34,49,72,76),(35,56,61,83),(36,51,62,78)], [(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),(13,16,19,22),(14,21,20,15),(17,24,23,18),(25,68,31,62),(26,61,32,67),(27,66,33,72),(28,71,34,65),(29,64,35,70),(30,69,36,63),(37,38,43,44),(39,48,45,42),(40,41,46,47),(49,75,55,81),(50,80,56,74),(51,73,57,79),(52,78,58,84),(53,83,59,77),(54,76,60,82),(85,88,91,94),(86,93,92,87),(89,96,95,90)])
Matrix representation ►G ⊆ 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] >;
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 |
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