?
G = C6.342+ (1+4) order 192 = 26·3
34th non-split extension by C6 of 2+ (1+4) acting via 2+ (1+4)/C2×D4=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C6.342+ (1+4), C4⋊D4⋊8S3, C4⋊C4.90D6, (D4×Dic3)⋊17C2, (C2×D4).153D6, C22⋊C4.48D6, Dic3⋊4D4⋊7C2, C23.9D6⋊16C2, Dic3.Q8⋊11C2, (C2×C12).36C23, (C2×C6).145C24, D6⋊C4.13C22, (C22×C4).236D6, C23.14D6⋊29C2, C2.36(D4⋊6D6), (C6×D4).119C22, C23.16D6⋊5C2, C23.8D6⋊15C2, (C22×C6).16C23, Dic3.22(C4○D4), C22.1(D4⋊2S3), C23.23D6⋊20C2, Dic3⋊C4.16C22, (C22×S3).63C23, C4⋊Dic3.206C22, C23.189(C22×S3), C22.166(S3×C23), (C2×Dic3).66C23, (C4×Dic3).92C22, (C22×C12).378C22, C3⋊6(C22.47C24), C6.D4.22C22, (C22×Dic3).106C22, (C3×C4⋊D4)⋊9C2, (C4×C3⋊D4)⋊53C2, C4⋊C4⋊S3⋊12C2, C2.36(S3×C4○D4), C6.81(C2×C4○D4), (C2×Dic3⋊C4)⋊40C2, (C2×C6).21(C4○D4), C2.33(C2×D4⋊2S3), (S3×C2×C4).208C22, (C3×C4⋊C4).141C22, (C2×C4).293(C22×S3), (C2×C3⋊D4).26C22, (C3×C22⋊C4).10C22, SmallGroup(192,1160)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 576 in 238 conjugacy classes, 97 normal (91 characteristic)
C1, C2 [×3], C2 [×5], C3, C4 [×12], C22, C22 [×2], C22 [×11], S3, C6 [×3], C6 [×4], C2×C4 [×4], C2×C4 [×15], D4 [×10], C23 [×3], C23, Dic3 [×2], Dic3 [×6], C12 [×4], D6 [×3], C2×C6, C2×C6 [×2], C2×C6 [×8], C42 [×3], C22⋊C4 [×2], C22⋊C4 [×8], C4⋊C4, C4⋊C4 [×9], C22×C4, C22×C4 [×5], C2×D4 [×3], C2×D4 [×3], C4×S3, C2×Dic3 [×7], C2×Dic3 [×6], C3⋊D4 [×5], C2×C12 [×4], C2×C12, C3×D4 [×5], C22×S3, C22×C6 [×3], C2×C4⋊C4, C42⋊C2, C4×D4 [×4], C4⋊D4, C4⋊D4 [×3], C22.D4 [×2], C42.C2, C42⋊2C2 [×2], C4×Dic3 [×3], Dic3⋊C4 [×7], C4⋊Dic3 [×2], D6⋊C4 [×3], C6.D4 [×5], C3×C22⋊C4 [×2], C3×C4⋊C4, S3×C2×C4, C22×Dic3 [×4], C2×C3⋊D4 [×3], C22×C12, C6×D4 [×3], C22.47C24, C23.16D6, C23.8D6, Dic3⋊4D4, C23.9D6, Dic3.Q8, C4⋊C4⋊S3, C2×Dic3⋊C4, C4×C3⋊D4, D4×Dic3 [×2], C23.23D6, C23.14D6 [×3], C3×C4⋊D4, C6.342+ (1+4)
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), D4⋊2S3 [×2], S3×C23, C22.47C24, C2×D4⋊2S3, D4⋊6D6, S3×C4○D4, C6.342+ (1+4)
Generators and relations
G = < a,b,c,d,e | a6=b4=e2=1, c2=a3, d2=b2, bab-1=cac-1=a-1, ad=da, ae=ea, cbc-1=b-1, dbd-1=ebe=a3b, cd=dc, ce=ec, ede=a3b2d >
►On 96 points
(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 65 14 57)(2 64 15 56)(3 63 16 55)(4 62 17 60)(5 61 18 59)(6 66 13 58)(7 49 91 47)(8 54 92 46)(9 53 93 45)(10 52 94 44)(11 51 95 43)(12 50 96 48)(19 73 27 71)(20 78 28 70)(21 77 29 69)(22 76 30 68)(23 75 25 67)(24 74 26 72)(31 88 39 80)(32 87 40 79)(33 86 41 84)(34 85 42 83)(35 90 37 82)(36 89 38 81)
(1 38 4 41)(2 37 5 40)(3 42 6 39)(7 73 10 76)(8 78 11 75)(9 77 12 74)(13 31 16 34)(14 36 17 33)(15 35 18 32)(19 52 22 49)(20 51 23 54)(21 50 24 53)(25 46 28 43)(26 45 29 48)(27 44 30 47)(55 83 58 80)(56 82 59 79)(57 81 60 84)(61 87 64 90)(62 86 65 89)(63 85 66 88)(67 92 70 95)(68 91 71 94)(69 96 72 93)
(1 72 14 74)(2 67 15 75)(3 68 16 76)(4 69 17 77)(5 70 18 78)(6 71 13 73)(7 42 91 34)(8 37 92 35)(9 38 93 36)(10 39 94 31)(11 40 95 32)(12 41 96 33)(19 55 27 63)(20 56 28 64)(21 57 29 65)(22 58 30 66)(23 59 25 61)(24 60 26 62)(43 90 51 82)(44 85 52 83)(45 86 53 84)(46 87 54 79)(47 88 49 80)(48 89 50 81)
(1 12)(2 7)(3 8)(4 9)(5 10)(6 11)(13 95)(14 96)(15 91)(16 92)(17 93)(18 94)(19 87)(20 88)(21 89)(22 90)(23 85)(24 86)(25 83)(26 84)(27 79)(28 80)(29 81)(30 82)(31 67)(32 68)(33 69)(34 70)(35 71)(36 72)(37 73)(38 74)(39 75)(40 76)(41 77)(42 78)(43 55)(44 56)(45 57)(46 58)(47 59)(48 60)(49 61)(50 62)(51 63)(52 64)(53 65)(54 66)
G:=sub<Sym(96)| (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), (1,65,14,57)(2,64,15,56)(3,63,16,55)(4,62,17,60)(5,61,18,59)(6,66,13,58)(7,49,91,47)(8,54,92,46)(9,53,93,45)(10,52,94,44)(11,51,95,43)(12,50,96,48)(19,73,27,71)(20,78,28,70)(21,77,29,69)(22,76,30,68)(23,75,25,67)(24,74,26,72)(31,88,39,80)(32,87,40,79)(33,86,41,84)(34,85,42,83)(35,90,37,82)(36,89,38,81), (1,38,4,41)(2,37,5,40)(3,42,6,39)(7,73,10,76)(8,78,11,75)(9,77,12,74)(13,31,16,34)(14,36,17,33)(15,35,18,32)(19,52,22,49)(20,51,23,54)(21,50,24,53)(25,46,28,43)(26,45,29,48)(27,44,30,47)(55,83,58,80)(56,82,59,79)(57,81,60,84)(61,87,64,90)(62,86,65,89)(63,85,66,88)(67,92,70,95)(68,91,71,94)(69,96,72,93), (1,72,14,74)(2,67,15,75)(3,68,16,76)(4,69,17,77)(5,70,18,78)(6,71,13,73)(7,42,91,34)(8,37,92,35)(9,38,93,36)(10,39,94,31)(11,40,95,32)(12,41,96,33)(19,55,27,63)(20,56,28,64)(21,57,29,65)(22,58,30,66)(23,59,25,61)(24,60,26,62)(43,90,51,82)(44,85,52,83)(45,86,53,84)(46,87,54,79)(47,88,49,80)(48,89,50,81), (1,12)(2,7)(3,8)(4,9)(5,10)(6,11)(13,95)(14,96)(15,91)(16,92)(17,93)(18,94)(19,87)(20,88)(21,89)(22,90)(23,85)(24,86)(25,83)(26,84)(27,79)(28,80)(29,81)(30,82)(31,67)(32,68)(33,69)(34,70)(35,71)(36,72)(37,73)(38,74)(39,75)(40,76)(41,77)(42,78)(43,55)(44,56)(45,57)(46,58)(47,59)(48,60)(49,61)(50,62)(51,63)(52,64)(53,65)(54,66)>;
G:=Group( (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), (1,65,14,57)(2,64,15,56)(3,63,16,55)(4,62,17,60)(5,61,18,59)(6,66,13,58)(7,49,91,47)(8,54,92,46)(9,53,93,45)(10,52,94,44)(11,51,95,43)(12,50,96,48)(19,73,27,71)(20,78,28,70)(21,77,29,69)(22,76,30,68)(23,75,25,67)(24,74,26,72)(31,88,39,80)(32,87,40,79)(33,86,41,84)(34,85,42,83)(35,90,37,82)(36,89,38,81), (1,38,4,41)(2,37,5,40)(3,42,6,39)(7,73,10,76)(8,78,11,75)(9,77,12,74)(13,31,16,34)(14,36,17,33)(15,35,18,32)(19,52,22,49)(20,51,23,54)(21,50,24,53)(25,46,28,43)(26,45,29,48)(27,44,30,47)(55,83,58,80)(56,82,59,79)(57,81,60,84)(61,87,64,90)(62,86,65,89)(63,85,66,88)(67,92,70,95)(68,91,71,94)(69,96,72,93), (1,72,14,74)(2,67,15,75)(3,68,16,76)(4,69,17,77)(5,70,18,78)(6,71,13,73)(7,42,91,34)(8,37,92,35)(9,38,93,36)(10,39,94,31)(11,40,95,32)(12,41,96,33)(19,55,27,63)(20,56,28,64)(21,57,29,65)(22,58,30,66)(23,59,25,61)(24,60,26,62)(43,90,51,82)(44,85,52,83)(45,86,53,84)(46,87,54,79)(47,88,49,80)(48,89,50,81), (1,12)(2,7)(3,8)(4,9)(5,10)(6,11)(13,95)(14,96)(15,91)(16,92)(17,93)(18,94)(19,87)(20,88)(21,89)(22,90)(23,85)(24,86)(25,83)(26,84)(27,79)(28,80)(29,81)(30,82)(31,67)(32,68)(33,69)(34,70)(35,71)(36,72)(37,73)(38,74)(39,75)(40,76)(41,77)(42,78)(43,55)(44,56)(45,57)(46,58)(47,59)(48,60)(49,61)(50,62)(51,63)(52,64)(53,65)(54,66) );
G=PermutationGroup([(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)], [(1,65,14,57),(2,64,15,56),(3,63,16,55),(4,62,17,60),(5,61,18,59),(6,66,13,58),(7,49,91,47),(8,54,92,46),(9,53,93,45),(10,52,94,44),(11,51,95,43),(12,50,96,48),(19,73,27,71),(20,78,28,70),(21,77,29,69),(22,76,30,68),(23,75,25,67),(24,74,26,72),(31,88,39,80),(32,87,40,79),(33,86,41,84),(34,85,42,83),(35,90,37,82),(36,89,38,81)], [(1,38,4,41),(2,37,5,40),(3,42,6,39),(7,73,10,76),(8,78,11,75),(9,77,12,74),(13,31,16,34),(14,36,17,33),(15,35,18,32),(19,52,22,49),(20,51,23,54),(21,50,24,53),(25,46,28,43),(26,45,29,48),(27,44,30,47),(55,83,58,80),(56,82,59,79),(57,81,60,84),(61,87,64,90),(62,86,65,89),(63,85,66,88),(67,92,70,95),(68,91,71,94),(69,96,72,93)], [(1,72,14,74),(2,67,15,75),(3,68,16,76),(4,69,17,77),(5,70,18,78),(6,71,13,73),(7,42,91,34),(8,37,92,35),(9,38,93,36),(10,39,94,31),(11,40,95,32),(12,41,96,33),(19,55,27,63),(20,56,28,64),(21,57,29,65),(22,58,30,66),(23,59,25,61),(24,60,26,62),(43,90,51,82),(44,85,52,83),(45,86,53,84),(46,87,54,79),(47,88,49,80),(48,89,50,81)], [(1,12),(2,7),(3,8),(4,9),(5,10),(6,11),(13,95),(14,96),(15,91),(16,92),(17,93),(18,94),(19,87),(20,88),(21,89),(22,90),(23,85),(24,86),(25,83),(26,84),(27,79),(28,80),(29,81),(30,82),(31,67),(32,68),(33,69),(34,70),(35,71),(36,72),(37,73),(38,74),(39,75),(40,76),(41,77),(42,78),(43,55),(44,56),(45,57),(46,58),(47,59),(48,60),(49,61),(50,62),(51,63),(52,64),(53,65),(54,66)])
Matrix representation ►G ⊆ GL6(𝔽13)
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 1 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 1 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 5 |
| 0 | 0 | 0 | 0 | 5 | 0 |
| 5 | 0 | 0 | 0 | 0 | 0 |
| 0 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 | 0 |
| 0 | 0 | 0 | 0 | 0 | 8 |
| 1 | 11 | 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 | 8 | 0 |
| 0 | 0 | 0 | 0 | 0 | 5 |
| 5 | 3 | 0 | 0 | 0 | 0 |
| 5 | 8 | 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 | 12 |
G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,12,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[1,1,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,12,0,0,0,0,0,0,0,0,5,0,0,0,0,5,0],[5,0,0,0,0,0,0,5,0,0,0,0,0,0,0,12,0,0,0,0,12,0,0,0,0,0,0,0,5,0,0,0,0,0,0,8],[1,0,0,0,0,0,11,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,8,0,0,0,0,0,0,5],[5,5,0,0,0,0,3,8,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12] >;
39 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 3 | 4A | 4B | 4C | 4D | 4E | 4F | ··· | 4M | 4N | 4O | 4P | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 12A | 12B | 12C | 12D | 12E | 12F |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 12 | 2 | 2 | 2 | 4 | 4 | 4 | 6 | ··· | 6 | 12 | 12 | 12 | 2 | 2 | 2 | 4 | 4 | 8 | 8 | 4 | 4 | 4 | 4 | 8 | 8 |
39 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D6 | D6 | D6 | D6 | C4○D4 | C4○D4 | 2+ (1+4) | D4⋊2S3 | D4⋊6D6 | S3×C4○D4 |
| kernel | C6.342+ (1+4) | C23.16D6 | C23.8D6 | Dic3⋊4D4 | C23.9D6 | Dic3.Q8 | C4⋊C4⋊S3 | C2×Dic3⋊C4 | C4×C3⋊D4 | D4×Dic3 | C23.23D6 | C23.14D6 | C3×C4⋊D4 | C4⋊D4 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | Dic3 | C2×C6 | C6 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 3 | 1 | 1 | 2 | 1 | 1 | 3 | 4 | 4 | 1 | 2 | 2 | 2 |
In GAP, Magma, Sage, TeX
C_6._{34}2_+^{(1+4)} % in TeX
G:=Group("C6.34ES+(2,2)"); // GroupNames label
G:=SmallGroup(192,1160);
// by ID
G=gap.SmallGroup(192,1160);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,758,100,794,297,6278]);
// Polycyclic
G:=Group<a,b,c,d,e|a^6=b^4=e^2=1,c^2=a^3,d^2=b^2,b*a*b^-1=c*a*c^-1=a^-1,a*d=d*a,a*e=e*a,c*b*c^-1=b^-1,d*b*d^-1=e*b*e=a^3*b,c*d=d*c,c*e=e*c,e*d*e=a^3*b^2*d>;
// generators/relations