?
G = C6.702- (1+4) order 192 = 26·3
25th non-split extension by C6 of 2- (1+4) acting via 2- (1+4)/C4○D4=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C6.702- (1+4), C6.352+ (1+4), C4⋊C4.91D6, C4⋊D4.8S3, C22⋊C4.6D6, (D4×Dic3)⋊18C2, (C2×D4).154D6, Dic3.Q8⋊12C2, (C2×C6).146C24, C2.28(Q8○D12), (C22×C4).237D6, C2.37(D4⋊6D6), C12.48D4⋊43C2, (C2×C12).625C23, (C6×D4).120C22, C23.8D6⋊16C2, C4⋊Dic3.45C22, C23.22(C22×S3), (C22×C6).17C23, Dic3.D4⋊17C2, C23.23D6⋊21C2, C22.6(D4⋊2S3), C22.167(S3×C23), (C4×Dic3).93C22, (C2×Dic6).33C22, (C2×Dic3).67C23, Dic3⋊C4.159C22, (C22×C12).311C22, C3⋊3(C22.33C24), C6.D4.23C22, (C22×Dic3).107C22, C6.82(C2×C4○D4), (C3×C4⋊D4).8C2, (C2×Dic3⋊C4)⋊29C2, (C2×C6).22(C4○D4), C2.34(C2×D4⋊2S3), (C3×C4⋊C4).142C22, (C2×C4).174(C22×S3), (C3×C22⋊C4).11C22, SmallGroup(192,1161)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 496 in 218 conjugacy classes, 95 normal (31 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×12], C22, C22 [×2], C22 [×8], C6 [×3], C6 [×4], C2×C4 [×2], C2×C4 [×2], C2×C4 [×14], D4 [×5], Q8, C23, C23 [×2], Dic3 [×8], C12 [×4], C2×C6, C2×C6 [×2], C2×C6 [×8], C42 [×2], C22⋊C4 [×2], C22⋊C4 [×8], C4⋊C4, C4⋊C4 [×13], C22×C4, C22×C4 [×4], C2×D4, C2×D4 [×2], C2×Q8, Dic6, C2×Dic3 [×8], C2×Dic3 [×5], C2×C12 [×2], C2×C12 [×2], C2×C12, C3×D4 [×5], C22×C6, C22×C6 [×2], C2×C4⋊C4, C4×D4 [×2], C4⋊D4, C22⋊Q8 [×3], C22.D4 [×4], C42.C2 [×2], C42⋊2C2 [×2], C4×Dic3 [×2], Dic3⋊C4 [×10], C4⋊Dic3, C4⋊Dic3 [×2], C6.D4 [×8], C3×C22⋊C4 [×2], C3×C4⋊C4, C2×Dic6, C22×Dic3 [×2], C22×Dic3 [×2], C22×C12, C6×D4, C6×D4 [×2], C22.33C24, Dic3.D4 [×2], C23.8D6 [×2], Dic3.Q8 [×2], C2×Dic3⋊C4, C12.48D4, D4×Dic3 [×2], C23.23D6 [×4], C3×C4⋊D4, C6.702- (1+4)
Quotients:
C1, C2 [×15], C22 [×35], S3, C23 [×15], D6 [×7], C4○D4 [×2], C24, C22×S3 [×7], C2×C4○D4, 2+ (1+4), 2- (1+4), D4⋊2S3 [×2], S3×C23, C22.33C24, C2×D4⋊2S3, D4⋊6D6, Q8○D12, C6.702- (1+4)
Generators and relations
G = < a,b,c,d,e | a6=b4=c2=1, d2=a3b2, e2=b2, ab=ba, ac=ca, dad-1=eae-1=a-1, cbc=b-1, bd=db, ebe-1=a3b, cd=dc, ce=ec, ede-1=b2d >
►On 96 points
Generators in S96
(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 51 17 43)(2 52 18 44)(3 53 13 45)(4 54 14 46)(5 49 15 47)(6 50 16 48)(7 64 92 56)(8 65 93 57)(9 66 94 58)(10 61 95 59)(11 62 96 60)(12 63 91 55)(19 31 26 38)(20 32 27 39)(21 33 28 40)(22 34 29 41)(23 35 30 42)(24 36 25 37)(67 87 75 79)(68 88 76 80)(69 89 77 81)(70 90 78 82)(71 85 73 83)(72 86 74 84)
(1 38)(2 39)(3 40)(4 41)(5 42)(6 37)(7 76)(8 77)(9 78)(10 73)(11 74)(12 75)(13 33)(14 34)(15 35)(16 36)(17 31)(18 32)(19 43)(20 44)(21 45)(22 46)(23 47)(24 48)(25 50)(26 51)(27 52)(28 53)(29 54)(30 49)(55 79)(56 80)(57 81)(58 82)(59 83)(60 84)(61 85)(62 86)(63 87)(64 88)(65 89)(66 90)(67 91)(68 92)(69 93)(70 94)(71 95)(72 96)
(1 80 14 85)(2 79 15 90)(3 84 16 89)(4 83 17 88)(5 82 18 87)(6 81 13 86)(7 29 95 19)(8 28 96 24)(9 27 91 23)(10 26 92 22)(11 25 93 21)(12 30 94 20)(31 64 41 59)(32 63 42 58)(33 62 37 57)(34 61 38 56)(35 66 39 55)(36 65 40 60)(43 76 54 71)(44 75 49 70)(45 74 50 69)(46 73 51 68)(47 78 52 67)(48 77 53 72)
(1 73 17 71)(2 78 18 70)(3 77 13 69)(4 76 14 68)(5 75 15 67)(6 74 16 72)(7 34 92 41)(8 33 93 40)(9 32 94 39)(10 31 95 38)(11 36 96 37)(12 35 91 42)(19 64 26 56)(20 63 27 55)(21 62 28 60)(22 61 29 59)(23 66 30 58)(24 65 25 57)(43 88 51 80)(44 87 52 79)(45 86 53 84)(46 85 54 83)(47 90 49 82)(48 89 50 81)
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), (1,51,17,43)(2,52,18,44)(3,53,13,45)(4,54,14,46)(5,49,15,47)(6,50,16,48)(7,64,92,56)(8,65,93,57)(9,66,94,58)(10,61,95,59)(11,62,96,60)(12,63,91,55)(19,31,26,38)(20,32,27,39)(21,33,28,40)(22,34,29,41)(23,35,30,42)(24,36,25,37)(67,87,75,79)(68,88,76,80)(69,89,77,81)(70,90,78,82)(71,85,73,83)(72,86,74,84), (1,38)(2,39)(3,40)(4,41)(5,42)(6,37)(7,76)(8,77)(9,78)(10,73)(11,74)(12,75)(13,33)(14,34)(15,35)(16,36)(17,31)(18,32)(19,43)(20,44)(21,45)(22,46)(23,47)(24,48)(25,50)(26,51)(27,52)(28,53)(29,54)(30,49)(55,79)(56,80)(57,81)(58,82)(59,83)(60,84)(61,85)(62,86)(63,87)(64,88)(65,89)(66,90)(67,91)(68,92)(69,93)(70,94)(71,95)(72,96), (1,80,14,85)(2,79,15,90)(3,84,16,89)(4,83,17,88)(5,82,18,87)(6,81,13,86)(7,29,95,19)(8,28,96,24)(9,27,91,23)(10,26,92,22)(11,25,93,21)(12,30,94,20)(31,64,41,59)(32,63,42,58)(33,62,37,57)(34,61,38,56)(35,66,39,55)(36,65,40,60)(43,76,54,71)(44,75,49,70)(45,74,50,69)(46,73,51,68)(47,78,52,67)(48,77,53,72), (1,73,17,71)(2,78,18,70)(3,77,13,69)(4,76,14,68)(5,75,15,67)(6,74,16,72)(7,34,92,41)(8,33,93,40)(9,32,94,39)(10,31,95,38)(11,36,96,37)(12,35,91,42)(19,64,26,56)(20,63,27,55)(21,62,28,60)(22,61,29,59)(23,66,30,58)(24,65,25,57)(43,88,51,80)(44,87,52,79)(45,86,53,84)(46,85,54,83)(47,90,49,82)(48,89,50,81)>;
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), (1,51,17,43)(2,52,18,44)(3,53,13,45)(4,54,14,46)(5,49,15,47)(6,50,16,48)(7,64,92,56)(8,65,93,57)(9,66,94,58)(10,61,95,59)(11,62,96,60)(12,63,91,55)(19,31,26,38)(20,32,27,39)(21,33,28,40)(22,34,29,41)(23,35,30,42)(24,36,25,37)(67,87,75,79)(68,88,76,80)(69,89,77,81)(70,90,78,82)(71,85,73,83)(72,86,74,84), (1,38)(2,39)(3,40)(4,41)(5,42)(6,37)(7,76)(8,77)(9,78)(10,73)(11,74)(12,75)(13,33)(14,34)(15,35)(16,36)(17,31)(18,32)(19,43)(20,44)(21,45)(22,46)(23,47)(24,48)(25,50)(26,51)(27,52)(28,53)(29,54)(30,49)(55,79)(56,80)(57,81)(58,82)(59,83)(60,84)(61,85)(62,86)(63,87)(64,88)(65,89)(66,90)(67,91)(68,92)(69,93)(70,94)(71,95)(72,96), (1,80,14,85)(2,79,15,90)(3,84,16,89)(4,83,17,88)(5,82,18,87)(6,81,13,86)(7,29,95,19)(8,28,96,24)(9,27,91,23)(10,26,92,22)(11,25,93,21)(12,30,94,20)(31,64,41,59)(32,63,42,58)(33,62,37,57)(34,61,38,56)(35,66,39,55)(36,65,40,60)(43,76,54,71)(44,75,49,70)(45,74,50,69)(46,73,51,68)(47,78,52,67)(48,77,53,72), (1,73,17,71)(2,78,18,70)(3,77,13,69)(4,76,14,68)(5,75,15,67)(6,74,16,72)(7,34,92,41)(8,33,93,40)(9,32,94,39)(10,31,95,38)(11,36,96,37)(12,35,91,42)(19,64,26,56)(20,63,27,55)(21,62,28,60)(22,61,29,59)(23,66,30,58)(24,65,25,57)(43,88,51,80)(44,87,52,79)(45,86,53,84)(46,85,54,83)(47,90,49,82)(48,89,50,81) );
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)], [(1,51,17,43),(2,52,18,44),(3,53,13,45),(4,54,14,46),(5,49,15,47),(6,50,16,48),(7,64,92,56),(8,65,93,57),(9,66,94,58),(10,61,95,59),(11,62,96,60),(12,63,91,55),(19,31,26,38),(20,32,27,39),(21,33,28,40),(22,34,29,41),(23,35,30,42),(24,36,25,37),(67,87,75,79),(68,88,76,80),(69,89,77,81),(70,90,78,82),(71,85,73,83),(72,86,74,84)], [(1,38),(2,39),(3,40),(4,41),(5,42),(6,37),(7,76),(8,77),(9,78),(10,73),(11,74),(12,75),(13,33),(14,34),(15,35),(16,36),(17,31),(18,32),(19,43),(20,44),(21,45),(22,46),(23,47),(24,48),(25,50),(26,51),(27,52),(28,53),(29,54),(30,49),(55,79),(56,80),(57,81),(58,82),(59,83),(60,84),(61,85),(62,86),(63,87),(64,88),(65,89),(66,90),(67,91),(68,92),(69,93),(70,94),(71,95),(72,96)], [(1,80,14,85),(2,79,15,90),(3,84,16,89),(4,83,17,88),(5,82,18,87),(6,81,13,86),(7,29,95,19),(8,28,96,24),(9,27,91,23),(10,26,92,22),(11,25,93,21),(12,30,94,20),(31,64,41,59),(32,63,42,58),(33,62,37,57),(34,61,38,56),(35,66,39,55),(36,65,40,60),(43,76,54,71),(44,75,49,70),(45,74,50,69),(46,73,51,68),(47,78,52,67),(48,77,53,72)], [(1,73,17,71),(2,78,18,70),(3,77,13,69),(4,76,14,68),(5,75,15,67),(6,74,16,72),(7,34,92,41),(8,33,93,40),(9,32,94,39),(10,31,95,38),(11,36,96,37),(12,35,91,42),(19,64,26,56),(20,63,27,55),(21,62,28,60),(22,61,29,59),(23,66,30,58),(24,65,25,57),(43,88,51,80),(44,87,52,79),(45,86,53,84),(46,85,54,83),(47,90,49,82),(48,89,50,81)])
Matrix representation ►G ⊆ GL8(𝔽13)
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 12 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 10 |
| 0 | 0 | 0 | 0 | 0 | 0 | 12 | 12 |
| 0 | 0 | 0 | 0 | 8 | 1 | 0 | 1 |
| 0 | 0 | 0 | 0 | 5 | 0 | 0 | 12 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 3 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 1 | 0 |
| 9 | 2 | 0 | 0 | 0 | 0 | 0 | 0 |
| 11 | 4 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 | 0 | 11 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 8 | 5 |
| 0 | 0 | 0 | 0 | 0 | 0 | 5 | 0 |
| 0 | 0 | 0 | 0 | 1 | 5 | 8 | 0 |
| 0 | 0 | 9 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 4 | 0 | 0 | 0 | 0 |
| 4 | 11 | 0 | 0 | 0 | 0 | 0 | 0 |
| 2 | 9 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 | 11 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 8 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 5 | 0 | 8 |
| 0 | 0 | 0 | 0 | 0 | 8 | 8 | 0 |
G:=sub<GL(8,GF(13))| [0,12,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,8,5,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,10,12,1,12],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,3,1,12,1,0,0,0,0,0,1,0,0],[9,11,0,0,0,0,0,0,2,4,0,0,0,0,0,0,0,0,9,11,0,0,0,0,0,0,2,4,0,0,0,0,0,0,0,0,8,1,0,1,0,0,0,0,0,0,0,5,0,0,0,0,11,8,5,8,0,0,0,0,0,5,0,0],[0,0,4,2,0,0,0,0,0,0,11,9,0,0,0,0,9,11,0,0,0,0,0,0,2,4,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,11,8,5,8,0,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0] >;
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4N | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 12A | 12B | 12C | 12D | 12E | 12F |
| order | 1 | 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 | 2 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 12 | ··· | 12 | 2 | 2 | 2 | 4 | 4 | 8 | 8 | 4 | 4 | 4 | 4 | 8 | 8 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D6 | D6 | D6 | D6 | C4○D4 | 2+ (1+4) | 2- (1+4) | D4⋊2S3 | D4⋊6D6 | Q8○D12 |
| kernel | C6.702- (1+4) | Dic3.D4 | C23.8D6 | Dic3.Q8 | C2×Dic3⋊C4 | C12.48D4 | D4×Dic3 | C23.23D6 | C3×C4⋊D4 | C4⋊D4 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C2×C6 | C6 | C6 | C22 | C2 | C2 |
| # reps | 1 | 2 | 2 | 2 | 1 | 1 | 2 | 4 | 1 | 1 | 2 | 1 | 1 | 3 | 4 | 1 | 1 | 2 | 2 | 2 |
In GAP, Magma, Sage, TeX
C_6._{70}2_-^{(1+4)} % in TeX
G:=Group("C6.70ES-(2,2)"); // GroupNames label
G:=SmallGroup(192,1161);
// by ID
G=gap.SmallGroup(192,1161);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,219,675,297,136,6278]);
// Polycyclic
G:=Group<a,b,c,d,e|a^6=b^4=c^2=1,d^2=a^3*b^2,e^2=b^2,a*b=b*a,a*c=c*a,d*a*d^-1=e*a*e^-1=a^-1,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
×
⋊
ℤ
𝔽
○
≀