?
G = C6.792- (1+4) order 192 = 26·3
34th non-split extension by C6 of 2- (1+4) acting via 2- (1+4)/C4○D4=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C6.792- (1+4), C12⋊Q8⋊30C2, C4⋊C4.103D6, D6⋊Q8⋊26C2, (C2×D4).158D6, C22⋊C4.25D6, Dic3.8(C2×D4), C22.42(S3×D4), C6.80(C22×D4), (C2×C12).66C23, (C2×C6).192C24, D6⋊C4.30C22, C2.40(Q8○D12), (C2×Dic3).79D4, C22.D4⋊2S3, (C22×C4).270D6, (C22×Dic6)⋊10C2, (C6×D4).130C22, C23.28D6⋊6C2, (C22×C6).28C23, C23.11D6⋊29C2, Dic3.D4⋊27C2, C22.D12⋊18C2, C23.16D6⋊10C2, C23.23D6⋊13C2, Dic3⋊C4.37C22, (C22×S3).83C23, C4⋊Dic3.223C22, (C22×C12).85C22, C23.208(C22×S3), C22.213(S3×C23), C3⋊3(C23.38C23), (C2×Dic3).242C23, (C2×Dic6).249C22, (C4×Dic3).119C22, C6.D4.38C22, (C22×Dic3).126C22, C2.53(C2×S3×D4), (C2×C6).56(C2×D4), (C2×D4⋊2S3).8C2, (S3×C2×C4).108C22, (C3×C4⋊C4).172C22, (C2×C4).188(C22×S3), (C3×C22.D4)⋊2C2, (C2×C3⋊D4).44C22, (C3×C22⋊C4).47C22, SmallGroup(192,1207)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 656 in 270 conjugacy classes, 103 normal (39 characteristic)
C1, C2, C2 [×2], C2 [×4], C3, C4 [×14], C22, C22 [×2], C22 [×8], S3, C6, C6 [×2], C6 [×3], C2×C4, C2×C4 [×4], C2×C4 [×19], D4 [×6], Q8 [×10], C23 [×2], C23, Dic3 [×4], Dic3 [×5], C12 [×5], D6 [×3], C2×C6, C2×C6 [×2], C2×C6 [×5], C42 [×2], C22⋊C4, C22⋊C4 [×2], C22⋊C4 [×7], C4⋊C4 [×2], C4⋊C4 [×8], C22×C4, C22×C4 [×4], C2×D4, C2×D4 [×2], C2×Q8 [×9], C4○D4 [×4], Dic6 [×10], C4×S3 [×2], C2×Dic3 [×3], C2×Dic3 [×8], C2×Dic3 [×4], C3⋊D4 [×4], C2×C12, C2×C12 [×4], C2×C12 [×2], C3×D4 [×2], C22×S3, C22×C6 [×2], C42⋊C2, C22⋊Q8 [×4], C22.D4, C22.D4 [×3], C4.4D4 [×2], C4⋊Q8 [×2], C22×Q8, C2×C4○D4, C4×Dic3 [×2], Dic3⋊C4 [×6], C4⋊Dic3 [×2], D6⋊C4 [×4], C6.D4, C6.D4 [×2], C3×C22⋊C4, C3×C22⋊C4 [×2], C3×C4⋊C4 [×2], C2×Dic6, C2×Dic6 [×4], C2×Dic6 [×4], S3×C2×C4, D4⋊2S3 [×4], C22×Dic3 [×3], C2×C3⋊D4 [×2], C22×C12, C6×D4, C23.38C23, C23.16D6, Dic3.D4 [×2], C23.11D6 [×2], C22.D12, C12⋊Q8 [×2], D6⋊Q8 [×2], C23.28D6, C23.23D6, C3×C22.D4, C22×Dic6, C2×D4⋊2S3, C6.792- (1+4)
Quotients:
C1, C2 [×15], C22 [×35], S3, D4 [×4], C23 [×15], D6 [×7], C2×D4 [×6], C24, C22×S3 [×7], C22×D4, 2- (1+4) [×2], S3×D4 [×2], S3×C23, C23.38C23, C2×S3×D4, Q8○D12 [×2], C6.792- (1+4)
Generators and relations
G = < a,b,c,d,e | a6=b4=c2=1, d2=e2=b2, ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc=b-1, bd=db, ebe-1=a3b, cd=dc, ce=ec, ede-1=b2d >
►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 89 17 82)(2 90 18 83)(3 85 13 84)(4 86 14 79)(5 87 15 80)(6 88 16 81)(7 27 92 20)(8 28 93 21)(9 29 94 22)(10 30 95 23)(11 25 96 24)(12 26 91 19)(31 62 38 55)(32 63 39 56)(33 64 40 57)(34 65 41 58)(35 66 42 59)(36 61 37 60)(43 74 50 67)(44 75 51 68)(45 76 52 69)(46 77 53 70)(47 78 54 71)(48 73 49 72)
(1 58)(2 59)(3 60)(4 55)(5 56)(6 57)(7 51)(8 52)(9 53)(10 54)(11 49)(12 50)(13 61)(14 62)(15 63)(16 64)(17 65)(18 66)(19 67)(20 68)(21 69)(22 70)(23 71)(24 72)(25 73)(26 74)(27 75)(28 76)(29 77)(30 78)(31 79)(32 80)(33 81)(34 82)(35 83)(36 84)(37 85)(38 86)(39 87)(40 88)(41 89)(42 90)(43 91)(44 92)(45 93)(46 94)(47 95)(48 96)
(1 82 17 89)(2 81 18 88)(3 80 13 87)(4 79 14 86)(5 84 15 85)(6 83 16 90)(7 28 92 21)(8 27 93 20)(9 26 94 19)(10 25 95 24)(11 30 96 23)(12 29 91 22)(31 62 38 55)(32 61 39 60)(33 66 40 59)(34 65 41 58)(35 64 42 57)(36 63 37 56)(43 70 50 77)(44 69 51 76)(45 68 52 75)(46 67 53 74)(47 72 54 73)(48 71 49 78)
(1 29 17 22)(2 30 18 23)(3 25 13 24)(4 26 14 19)(5 27 15 20)(6 28 16 21)(7 90 92 83)(8 85 93 84)(9 86 94 79)(10 87 95 80)(11 88 96 81)(12 89 91 82)(31 53 38 46)(32 54 39 47)(33 49 40 48)(34 50 41 43)(35 51 42 44)(36 52 37 45)(55 74 62 67)(56 75 63 68)(57 76 64 69)(58 77 65 70)(59 78 66 71)(60 73 61 72)
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), (1,89,17,82)(2,90,18,83)(3,85,13,84)(4,86,14,79)(5,87,15,80)(6,88,16,81)(7,27,92,20)(8,28,93,21)(9,29,94,22)(10,30,95,23)(11,25,96,24)(12,26,91,19)(31,62,38,55)(32,63,39,56)(33,64,40,57)(34,65,41,58)(35,66,42,59)(36,61,37,60)(43,74,50,67)(44,75,51,68)(45,76,52,69)(46,77,53,70)(47,78,54,71)(48,73,49,72), (1,58)(2,59)(3,60)(4,55)(5,56)(6,57)(7,51)(8,52)(9,53)(10,54)(11,49)(12,50)(13,61)(14,62)(15,63)(16,64)(17,65)(18,66)(19,67)(20,68)(21,69)(22,70)(23,71)(24,72)(25,73)(26,74)(27,75)(28,76)(29,77)(30,78)(31,79)(32,80)(33,81)(34,82)(35,83)(36,84)(37,85)(38,86)(39,87)(40,88)(41,89)(42,90)(43,91)(44,92)(45,93)(46,94)(47,95)(48,96), (1,82,17,89)(2,81,18,88)(3,80,13,87)(4,79,14,86)(5,84,15,85)(6,83,16,90)(7,28,92,21)(8,27,93,20)(9,26,94,19)(10,25,95,24)(11,30,96,23)(12,29,91,22)(31,62,38,55)(32,61,39,60)(33,66,40,59)(34,65,41,58)(35,64,42,57)(36,63,37,56)(43,70,50,77)(44,69,51,76)(45,68,52,75)(46,67,53,74)(47,72,54,73)(48,71,49,78), (1,29,17,22)(2,30,18,23)(3,25,13,24)(4,26,14,19)(5,27,15,20)(6,28,16,21)(7,90,92,83)(8,85,93,84)(9,86,94,79)(10,87,95,80)(11,88,96,81)(12,89,91,82)(31,53,38,46)(32,54,39,47)(33,49,40,48)(34,50,41,43)(35,51,42,44)(36,52,37,45)(55,74,62,67)(56,75,63,68)(57,76,64,69)(58,77,65,70)(59,78,66,71)(60,73,61,72)>;
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), (1,89,17,82)(2,90,18,83)(3,85,13,84)(4,86,14,79)(5,87,15,80)(6,88,16,81)(7,27,92,20)(8,28,93,21)(9,29,94,22)(10,30,95,23)(11,25,96,24)(12,26,91,19)(31,62,38,55)(32,63,39,56)(33,64,40,57)(34,65,41,58)(35,66,42,59)(36,61,37,60)(43,74,50,67)(44,75,51,68)(45,76,52,69)(46,77,53,70)(47,78,54,71)(48,73,49,72), (1,58)(2,59)(3,60)(4,55)(5,56)(6,57)(7,51)(8,52)(9,53)(10,54)(11,49)(12,50)(13,61)(14,62)(15,63)(16,64)(17,65)(18,66)(19,67)(20,68)(21,69)(22,70)(23,71)(24,72)(25,73)(26,74)(27,75)(28,76)(29,77)(30,78)(31,79)(32,80)(33,81)(34,82)(35,83)(36,84)(37,85)(38,86)(39,87)(40,88)(41,89)(42,90)(43,91)(44,92)(45,93)(46,94)(47,95)(48,96), (1,82,17,89)(2,81,18,88)(3,80,13,87)(4,79,14,86)(5,84,15,85)(6,83,16,90)(7,28,92,21)(8,27,93,20)(9,26,94,19)(10,25,95,24)(11,30,96,23)(12,29,91,22)(31,62,38,55)(32,61,39,60)(33,66,40,59)(34,65,41,58)(35,64,42,57)(36,63,37,56)(43,70,50,77)(44,69,51,76)(45,68,52,75)(46,67,53,74)(47,72,54,73)(48,71,49,78), (1,29,17,22)(2,30,18,23)(3,25,13,24)(4,26,14,19)(5,27,15,20)(6,28,16,21)(7,90,92,83)(8,85,93,84)(9,86,94,79)(10,87,95,80)(11,88,96,81)(12,89,91,82)(31,53,38,46)(32,54,39,47)(33,49,40,48)(34,50,41,43)(35,51,42,44)(36,52,37,45)(55,74,62,67)(56,75,63,68)(57,76,64,69)(58,77,65,70)(59,78,66,71)(60,73,61,72) );
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)], [(1,89,17,82),(2,90,18,83),(3,85,13,84),(4,86,14,79),(5,87,15,80),(6,88,16,81),(7,27,92,20),(8,28,93,21),(9,29,94,22),(10,30,95,23),(11,25,96,24),(12,26,91,19),(31,62,38,55),(32,63,39,56),(33,64,40,57),(34,65,41,58),(35,66,42,59),(36,61,37,60),(43,74,50,67),(44,75,51,68),(45,76,52,69),(46,77,53,70),(47,78,54,71),(48,73,49,72)], [(1,58),(2,59),(3,60),(4,55),(5,56),(6,57),(7,51),(8,52),(9,53),(10,54),(11,49),(12,50),(13,61),(14,62),(15,63),(16,64),(17,65),(18,66),(19,67),(20,68),(21,69),(22,70),(23,71),(24,72),(25,73),(26,74),(27,75),(28,76),(29,77),(30,78),(31,79),(32,80),(33,81),(34,82),(35,83),(36,84),(37,85),(38,86),(39,87),(40,88),(41,89),(42,90),(43,91),(44,92),(45,93),(46,94),(47,95),(48,96)], [(1,82,17,89),(2,81,18,88),(3,80,13,87),(4,79,14,86),(5,84,15,85),(6,83,16,90),(7,28,92,21),(8,27,93,20),(9,26,94,19),(10,25,95,24),(11,30,96,23),(12,29,91,22),(31,62,38,55),(32,61,39,60),(33,66,40,59),(34,65,41,58),(35,64,42,57),(36,63,37,56),(43,70,50,77),(44,69,51,76),(45,68,52,75),(46,67,53,74),(47,72,54,73),(48,71,49,78)], [(1,29,17,22),(2,30,18,23),(3,25,13,24),(4,26,14,19),(5,27,15,20),(6,28,16,21),(7,90,92,83),(8,85,93,84),(9,86,94,79),(10,87,95,80),(11,88,96,81),(12,89,91,82),(31,53,38,46),(32,54,39,47),(33,49,40,48),(34,50,41,43),(35,51,42,44),(36,52,37,45),(55,74,62,67),(56,75,63,68),(57,76,64,69),(58,77,65,70),(59,78,66,71),(60,73,61,72)])
Matrix representation ►G ⊆ GL6(𝔽13)
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 12 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 12 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 2 | 11 | 9 |
| 0 | 0 | 11 | 2 | 4 | 2 |
| 0 | 0 | 4 | 2 | 9 | 11 |
| 0 | 0 | 11 | 2 | 2 | 11 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 11 | 6 | 0 |
| 0 | 0 | 2 | 11 | 0 | 6 |
| 0 | 0 | 9 | 11 | 4 | 2 |
| 0 | 0 | 2 | 11 | 11 | 2 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 11 | 2 | 4 |
| 0 | 0 | 2 | 4 | 2 | 11 |
| 0 | 0 | 0 | 0 | 11 | 2 |
| 0 | 0 | 0 | 0 | 4 | 2 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 10 | 7 | 0 | 0 |
| 0 | 0 | 6 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 10 | 7 |
| 0 | 0 | 0 | 0 | 6 | 3 |
G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,1,0,0,0,0,12,0,0,0,0,0,0,0,12,1,0,0,0,0,12,0],[12,0,0,0,0,0,0,1,0,0,0,0,0,0,4,11,4,11,0,0,2,2,2,2,0,0,11,4,9,2,0,0,9,2,11,11],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,9,2,9,2,0,0,11,11,11,11,0,0,6,0,4,11,0,0,0,6,2,2],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,9,2,0,0,0,0,11,4,0,0,0,0,2,2,11,4,0,0,4,11,2,2],[0,12,0,0,0,0,12,0,0,0,0,0,0,0,10,6,0,0,0,0,7,3,0,0,0,0,0,0,10,6,0,0,0,0,7,3] >;
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | ··· | 4E | 4F | 4G | 4H | 4I | 4J | ··· | 4N | 6A | 6B | 6C | 6D | 6E | 6F | 12A | 12B | 12C | 12D | 12E | 12F | 12G |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 12 | 2 | 4 | ··· | 4 | 6 | 6 | 6 | 6 | 12 | ··· | 12 | 2 | 2 | 2 | 4 | 4 | 8 | 4 | 4 | 4 | 4 | 8 | 8 | 8 |
36 irreducible representations
| dim | 1 | 1 | 1 | 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 | C2 | C2 | C2 | S3 | D4 | D6 | D6 | D6 | D6 | 2- (1+4) | S3×D4 | Q8○D12 |
| kernel | C6.792- (1+4) | C23.16D6 | Dic3.D4 | C23.11D6 | C22.D12 | C12⋊Q8 | D6⋊Q8 | C23.28D6 | C23.23D6 | C3×C22.D4 | C22×Dic6 | C2×D4⋊2S3 | C22.D4 | C2×Dic3 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C6 | C22 | C2 |
| # reps | 1 | 1 | 2 | 2 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 3 | 2 | 1 | 1 | 2 | 2 | 4 |
In GAP, Magma, Sage, TeX
C_6._{79}2_-^{(1+4)} % in TeX
G:=Group("C6.79ES-(2,2)"); // GroupNames label
G:=SmallGroup(192,1207);
// by ID
G=gap.SmallGroup(192,1207);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,387,1123,185,136,6278]);
// Polycyclic
G:=Group<a,b,c,d,e|a^6=b^4=c^2=1,d^2=e^2=b^2,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,a*e=e*a,c*b*c=b^-1,b*d=d*b,e*b*e^-1=a^3*b,c*d=d*c,c*e=e*c,e*d*e^-1=b^2*d>;
// generators/relations