metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.234D6, (S3×C42)⋊10C2, D6⋊3Q8⋊32C2, (Q8×Dic3)⋊19C2, D6.9(C4○D4), (D4×Dic3)⋊30C2, (C2×D4).175D6, C4.4D4⋊19S3, (C2×Q8).162D6, C22⋊C4.74D6, C23.9D6⋊45C2, D6⋊3D4.12C2, (C2×C6).224C24, D6⋊C4.36C22, C12.6Q8⋊20C2, Dic3⋊4D4⋊33C2, C12.125(C4○D4), C4.38(D4⋊2S3), (C4×C12).187C22, (C2×C12).504C23, (C6×D4).157C22, C23.8D6⋊41C2, (C22×C6).54C23, C23.56(C22×S3), (C6×Q8).128C22, Dic3.43(C4○D4), C23.16D6⋊19C2, Dic3⋊C4.70C22, C4⋊Dic3.234C22, C22.245(S3×C23), (C22×S3).218C23, C3⋊9(C23.36C23), (C2×Dic3).310C23, (C4×Dic3).134C22, C6.D4.57C22, (C22×Dic3).144C22, C2.80(S3×C4○D4), C6.191(C2×C4○D4), C2.56(C2×D4⋊2S3), (C3×C4.4D4)⋊16C2, (S3×C2×C4).298C22, (C2×C4).301(C22×S3), (C2×C3⋊D4).62C22, (C3×C22⋊C4).66C22, SmallGroup(192,1239)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 528 in 234 conjugacy classes, 99 normal (43 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×2], C4 [×12], C22, C22 [×10], S3 [×2], C6 [×3], C6 [×2], C2×C4 [×3], C2×C4 [×2], C2×C4 [×17], D4 [×6], Q8 [×2], C23 [×2], C23, Dic3 [×2], Dic3 [×6], C12 [×2], C12 [×4], D6 [×2], D6 [×2], C2×C6, C2×C6 [×6], C42, C42 [×5], C22⋊C4 [×4], C22⋊C4 [×6], C4⋊C4 [×10], C22×C4 [×5], C2×D4, C2×D4 [×2], C2×Q8, C4×S3 [×6], C2×Dic3 [×3], C2×Dic3 [×4], C2×Dic3 [×4], C3⋊D4 [×4], C2×C12 [×3], C2×C12 [×2], C3×D4 [×2], C3×Q8 [×2], C22×S3, C22×C6 [×2], C2×C42, C42⋊C2 [×2], C4×D4 [×3], C4×Q8, C4⋊D4, C22⋊Q8, C22.D4 [×2], C4.4D4, C42.C2, C42⋊2C2 [×2], C4×Dic3 [×3], C4×Dic3 [×2], Dic3⋊C4 [×6], C4⋊Dic3 [×2], C4⋊Dic3 [×2], D6⋊C4 [×2], C6.D4 [×4], C4×C12, C3×C22⋊C4 [×4], S3×C2×C4 [×3], C22×Dic3 [×2], C2×C3⋊D4 [×2], C6×D4, C6×Q8, C23.36C23, C12.6Q8, S3×C42, C23.16D6 [×2], C23.8D6 [×2], Dic3⋊4D4 [×2], C23.9D6 [×2], D4×Dic3, D6⋊3D4, Q8×Dic3, D6⋊3Q8, C3×C4.4D4, C42.234D6
Quotients:
C1, C2 [×15], C22 [×35], S3, C23 [×15], D6 [×7], C4○D4 [×6], C24, C22×S3 [×7], C2×C4○D4 [×3], D4⋊2S3 [×2], S3×C23, C23.36C23, C2×D4⋊2S3, S3×C4○D4 [×2], C42.234D6
Generators and relations
G = < a,b,c,d | a4=b4=c6=1, d2=a2, ab=ba, cac-1=dad-1=ab2, cbc-1=dbd-1=a2b, dcd-1=a2c-1 >
►On 96 points
(1 80 19 64)(2 68 20 51)(3 82 21 66)(4 70 22 53)(5 84 23 62)(6 72 24 49)(7 67 26 50)(8 81 27 65)(9 69 28 52)(10 83 29 61)(11 71 30 54)(12 79 25 63)(13 47 86 38)(14 58 87 34)(15 43 88 40)(16 60 89 36)(17 45 90 42)(18 56 85 32)(31 93 55 73)(33 95 57 75)(35 91 59 77)(37 94 46 74)(39 96 48 76)(41 92 44 78)
(1 59 7 43)(2 36 8 41)(3 55 9 45)(4 32 10 37)(5 57 11 47)(6 34 12 39)(13 62 95 54)(14 79 96 72)(15 64 91 50)(16 81 92 68)(17 66 93 52)(18 83 94 70)(19 35 26 40)(20 60 27 44)(21 31 28 42)(22 56 29 46)(23 33 30 38)(24 58 25 48)(49 87 63 76)(51 89 65 78)(53 85 61 74)(67 88 80 77)(69 90 82 73)(71 86 84 75)
(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 19 24)(2 23 20 5)(3 4 21 22)(7 12 26 25)(8 30 27 11)(9 10 28 29)(13 16 86 89)(14 88 87 15)(17 18 90 85)(31 32 55 56)(33 36 57 60)(34 59 58 35)(37 45 46 42)(38 41 47 44)(39 43 48 40)(49 67 72 50)(51 71 68 54)(52 53 69 70)(61 82 83 66)(62 65 84 81)(63 80 79 64)(73 74 93 94)(75 78 95 92)(76 91 96 77)
G:=sub<Sym(96)| (1,80,19,64)(2,68,20,51)(3,82,21,66)(4,70,22,53)(5,84,23,62)(6,72,24,49)(7,67,26,50)(8,81,27,65)(9,69,28,52)(10,83,29,61)(11,71,30,54)(12,79,25,63)(13,47,86,38)(14,58,87,34)(15,43,88,40)(16,60,89,36)(17,45,90,42)(18,56,85,32)(31,93,55,73)(33,95,57,75)(35,91,59,77)(37,94,46,74)(39,96,48,76)(41,92,44,78), (1,59,7,43)(2,36,8,41)(3,55,9,45)(4,32,10,37)(5,57,11,47)(6,34,12,39)(13,62,95,54)(14,79,96,72)(15,64,91,50)(16,81,92,68)(17,66,93,52)(18,83,94,70)(19,35,26,40)(20,60,27,44)(21,31,28,42)(22,56,29,46)(23,33,30,38)(24,58,25,48)(49,87,63,76)(51,89,65,78)(53,85,61,74)(67,88,80,77)(69,90,82,73)(71,86,84,75), (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), (1,6,19,24)(2,23,20,5)(3,4,21,22)(7,12,26,25)(8,30,27,11)(9,10,28,29)(13,16,86,89)(14,88,87,15)(17,18,90,85)(31,32,55,56)(33,36,57,60)(34,59,58,35)(37,45,46,42)(38,41,47,44)(39,43,48,40)(49,67,72,50)(51,71,68,54)(52,53,69,70)(61,82,83,66)(62,65,84,81)(63,80,79,64)(73,74,93,94)(75,78,95,92)(76,91,96,77)>;
G:=Group( (1,80,19,64)(2,68,20,51)(3,82,21,66)(4,70,22,53)(5,84,23,62)(6,72,24,49)(7,67,26,50)(8,81,27,65)(9,69,28,52)(10,83,29,61)(11,71,30,54)(12,79,25,63)(13,47,86,38)(14,58,87,34)(15,43,88,40)(16,60,89,36)(17,45,90,42)(18,56,85,32)(31,93,55,73)(33,95,57,75)(35,91,59,77)(37,94,46,74)(39,96,48,76)(41,92,44,78), (1,59,7,43)(2,36,8,41)(3,55,9,45)(4,32,10,37)(5,57,11,47)(6,34,12,39)(13,62,95,54)(14,79,96,72)(15,64,91,50)(16,81,92,68)(17,66,93,52)(18,83,94,70)(19,35,26,40)(20,60,27,44)(21,31,28,42)(22,56,29,46)(23,33,30,38)(24,58,25,48)(49,87,63,76)(51,89,65,78)(53,85,61,74)(67,88,80,77)(69,90,82,73)(71,86,84,75), (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), (1,6,19,24)(2,23,20,5)(3,4,21,22)(7,12,26,25)(8,30,27,11)(9,10,28,29)(13,16,86,89)(14,88,87,15)(17,18,90,85)(31,32,55,56)(33,36,57,60)(34,59,58,35)(37,45,46,42)(38,41,47,44)(39,43,48,40)(49,67,72,50)(51,71,68,54)(52,53,69,70)(61,82,83,66)(62,65,84,81)(63,80,79,64)(73,74,93,94)(75,78,95,92)(76,91,96,77) );
G=PermutationGroup([(1,80,19,64),(2,68,20,51),(3,82,21,66),(4,70,22,53),(5,84,23,62),(6,72,24,49),(7,67,26,50),(8,81,27,65),(9,69,28,52),(10,83,29,61),(11,71,30,54),(12,79,25,63),(13,47,86,38),(14,58,87,34),(15,43,88,40),(16,60,89,36),(17,45,90,42),(18,56,85,32),(31,93,55,73),(33,95,57,75),(35,91,59,77),(37,94,46,74),(39,96,48,76),(41,92,44,78)], [(1,59,7,43),(2,36,8,41),(3,55,9,45),(4,32,10,37),(5,57,11,47),(6,34,12,39),(13,62,95,54),(14,79,96,72),(15,64,91,50),(16,81,92,68),(17,66,93,52),(18,83,94,70),(19,35,26,40),(20,60,27,44),(21,31,28,42),(22,56,29,46),(23,33,30,38),(24,58,25,48),(49,87,63,76),(51,89,65,78),(53,85,61,74),(67,88,80,77),(69,90,82,73),(71,86,84,75)], [(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)], [(1,6,19,24),(2,23,20,5),(3,4,21,22),(7,12,26,25),(8,30,27,11),(9,10,28,29),(13,16,86,89),(14,88,87,15),(17,18,90,85),(31,32,55,56),(33,36,57,60),(34,59,58,35),(37,45,46,42),(38,41,47,44),(39,43,48,40),(49,67,72,50),(51,71,68,54),(52,53,69,70),(61,82,83,66),(62,65,84,81),(63,80,79,64),(73,74,93,94),(75,78,95,92),(76,91,96,77)])
Matrix representation ►G ⊆ GL6(𝔽13)
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 0 | 0 | 0 |
| 0 | 0 | 5 | 5 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 | 0 |
| 0 | 0 | 0 | 0 | 0 | 5 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 0 | 0 | 0 |
| 0 | 0 | 5 | 5 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 12 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 11 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 12 | 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 2 | 0 | 0 |
| 0 | 0 | 12 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 0 | 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,8,5,0,0,0,0,0,5,0,0,0,0,0,0,5,0,0,0,0,0,0,5],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,8,5,0,0,0,0,0,5,0,0,0,0,0,0,12,0,0,0,0,0,0,1],[0,12,0,0,0,0,1,1,0,0,0,0,0,0,12,0,0,0,0,0,11,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[12,0,0,0,0,0,1,1,0,0,0,0,0,0,1,12,0,0,0,0,2,12,0,0,0,0,0,0,0,1,0,0,0,0,12,0] >;
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | 4Q | 4R | 4S | 4T | 6A | 6B | 6C | 6D | 6E | 12A | ··· | 12F | 12G | 12H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 12 | ··· | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 6 | 6 | 2 | 2 | ··· | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 8 | 8 | 4 | ··· | 4 | 8 | 8 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D6 | D6 | D6 | D6 | C4○D4 | C4○D4 | C4○D4 | D4⋊2S3 | S3×C4○D4 |
| kernel | C42.234D6 | C12.6Q8 | S3×C42 | C23.16D6 | C23.8D6 | Dic3⋊4D4 | C23.9D6 | D4×Dic3 | D6⋊3D4 | Q8×Dic3 | D6⋊3Q8 | C3×C4.4D4 | C4.4D4 | C42 | C22⋊C4 | C2×D4 | C2×Q8 | Dic3 | C12 | D6 | C4 | C2 |
| # reps | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 4 | 4 | 4 | 2 | 4 |
In GAP, Magma, Sage, TeX
C_4^2._{234}D_6 % in TeX
G:=Group("C4^2.234D6"); // GroupNames label
G:=SmallGroup(192,1239);
// by ID
G=gap.SmallGroup(192,1239);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,100,1123,346,297,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^6=1,d^2=a^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a*b^2,c*b*c^-1=d*b*d^-1=a^2*b,d*c*d^-1=a^2*c^-1>;
// generators/relations