?
G = C6.422- (1+4) order 192 = 26·3
42nd non-split extension by C6 of 2- (1+4) acting via 2- (1+4)/C2×Q8=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C6.422- (1+4), (C6×Q8)⋊12C4, (C2×Q8)⋊11Dic3, (Q8×Dic3)⋊25C2, C6.47(C23×C4), (C2×Q8).231D6, C12.96(C22×C4), (C2×C6).304C24, Q8.13(C2×Dic3), (C22×C4).292D6, (C22×Q8).15S3, C2.9(C23×Dic3), (C2×C12).551C23, C22.47(S3×C23), (C6×Q8).233C22, C4.19(C22×Dic3), C4⋊Dic3.389C22, C23.247(C22×S3), (C22×C6).422C23, C2.4(Q8.15D6), (C22×C12).284C22, C3⋊3(C23.32C23), (C2×Dic3).288C23, (C4×Dic3).169C22, C23.26D6.25C2, C22.10(C22×Dic3), C6.D4.145C22, (Q8×C2×C6).9C2, (C3×Q8).26(C2×C4), (C2×C12).134(C2×C4), (C2×C4).30(C2×Dic3), (C2×C4).632(C22×S3), (C2×C6).210(C22×C4), SmallGroup(192,1371)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 424 in 266 conjugacy classes, 191 normal (11 characteristic)
C1, C2, C2 [×2], C2 [×2], C3, C4 [×12], C4 [×8], C22, C22 [×2], C22 [×2], C6, C6 [×2], C6 [×2], C2×C4 [×18], C2×C4 [×8], Q8 [×16], C23, Dic3 [×8], C12 [×12], C2×C6, C2×C6 [×2], C2×C6 [×2], C42 [×12], C22⋊C4 [×4], C4⋊C4 [×12], C22×C4 [×3], C2×Q8 [×12], C2×Dic3 [×8], C2×C12 [×18], C3×Q8 [×16], C22×C6, C42⋊C2 [×6], C4×Q8 [×8], C22×Q8, C4×Dic3 [×12], C4⋊Dic3 [×12], C6.D4 [×4], C22×C12 [×3], C6×Q8 [×12], C23.32C23, C23.26D6 [×6], Q8×Dic3 [×8], Q8×C2×C6, C6.422- (1+4)
Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], S3, C2×C4 [×28], C23 [×15], Dic3 [×8], D6 [×7], C22×C4 [×14], C24, C2×Dic3 [×28], C22×S3 [×7], C23×C4, 2- (1+4) [×2], C22×Dic3 [×14], S3×C23, C23.32C23, Q8.15D6 [×2], C23×Dic3, C6.422- (1+4)
Generators and relations
G = < a,b,c,d,e | a6=b4=1, c2=a3, d2=a3b2, e2=b2, ab=ba, cac-1=dad-1=a-1, ae=ea, cbc-1=b-1, bd=db, be=eb, 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 38 17 33)(2 39 18 34)(3 40 13 35)(4 41 14 36)(5 42 15 31)(6 37 16 32)(7 69 96 73)(8 70 91 74)(9 71 92 75)(10 72 93 76)(11 67 94 77)(12 68 95 78)(19 52 30 46)(20 53 25 47)(21 54 26 48)(22 49 27 43)(23 50 28 44)(24 51 29 45)(55 82 65 86)(56 83 66 87)(57 84 61 88)(58 79 62 89)(59 80 63 90)(60 81 64 85)
(1 85 4 88)(2 90 5 87)(3 89 6 86)(7 24 10 21)(8 23 11 20)(9 22 12 19)(13 79 16 82)(14 84 17 81)(15 83 18 80)(25 91 28 94)(26 96 29 93)(27 95 30 92)(31 56 34 59)(32 55 35 58)(33 60 36 57)(37 65 40 62)(38 64 41 61)(39 63 42 66)(43 68 46 71)(44 67 47 70)(45 72 48 69)(49 78 52 75)(50 77 53 74)(51 76 54 73)
(1 69 14 76)(2 68 15 75)(3 67 16 74)(4 72 17 73)(5 71 18 78)(6 70 13 77)(7 41 93 33)(8 40 94 32)(9 39 95 31)(10 38 96 36)(11 37 91 35)(12 42 92 34)(19 66 27 59)(20 65 28 58)(21 64 29 57)(22 63 30 56)(23 62 25 55)(24 61 26 60)(43 80 52 87)(44 79 53 86)(45 84 54 85)(46 83 49 90)(47 82 50 89)(48 81 51 88)
(1 41 17 36)(2 42 18 31)(3 37 13 32)(4 38 14 33)(5 39 15 34)(6 40 16 35)(7 76 96 72)(8 77 91 67)(9 78 92 68)(10 73 93 69)(11 74 94 70)(12 75 95 71)(19 49 30 43)(20 50 25 44)(21 51 26 45)(22 52 27 46)(23 53 28 47)(24 54 29 48)(55 89 65 79)(56 90 66 80)(57 85 61 81)(58 86 62 82)(59 87 63 83)(60 88 64 84)
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), (1,38,17,33)(2,39,18,34)(3,40,13,35)(4,41,14,36)(5,42,15,31)(6,37,16,32)(7,69,96,73)(8,70,91,74)(9,71,92,75)(10,72,93,76)(11,67,94,77)(12,68,95,78)(19,52,30,46)(20,53,25,47)(21,54,26,48)(22,49,27,43)(23,50,28,44)(24,51,29,45)(55,82,65,86)(56,83,66,87)(57,84,61,88)(58,79,62,89)(59,80,63,90)(60,81,64,85), (1,85,4,88)(2,90,5,87)(3,89,6,86)(7,24,10,21)(8,23,11,20)(9,22,12,19)(13,79,16,82)(14,84,17,81)(15,83,18,80)(25,91,28,94)(26,96,29,93)(27,95,30,92)(31,56,34,59)(32,55,35,58)(33,60,36,57)(37,65,40,62)(38,64,41,61)(39,63,42,66)(43,68,46,71)(44,67,47,70)(45,72,48,69)(49,78,52,75)(50,77,53,74)(51,76,54,73), (1,69,14,76)(2,68,15,75)(3,67,16,74)(4,72,17,73)(5,71,18,78)(6,70,13,77)(7,41,93,33)(8,40,94,32)(9,39,95,31)(10,38,96,36)(11,37,91,35)(12,42,92,34)(19,66,27,59)(20,65,28,58)(21,64,29,57)(22,63,30,56)(23,62,25,55)(24,61,26,60)(43,80,52,87)(44,79,53,86)(45,84,54,85)(46,83,49,90)(47,82,50,89)(48,81,51,88), (1,41,17,36)(2,42,18,31)(3,37,13,32)(4,38,14,33)(5,39,15,34)(6,40,16,35)(7,76,96,72)(8,77,91,67)(9,78,92,68)(10,73,93,69)(11,74,94,70)(12,75,95,71)(19,49,30,43)(20,50,25,44)(21,51,26,45)(22,52,27,46)(23,53,28,47)(24,54,29,48)(55,89,65,79)(56,90,66,80)(57,85,61,81)(58,86,62,82)(59,87,63,83)(60,88,64,84)>;
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), (1,38,17,33)(2,39,18,34)(3,40,13,35)(4,41,14,36)(5,42,15,31)(6,37,16,32)(7,69,96,73)(8,70,91,74)(9,71,92,75)(10,72,93,76)(11,67,94,77)(12,68,95,78)(19,52,30,46)(20,53,25,47)(21,54,26,48)(22,49,27,43)(23,50,28,44)(24,51,29,45)(55,82,65,86)(56,83,66,87)(57,84,61,88)(58,79,62,89)(59,80,63,90)(60,81,64,85), (1,85,4,88)(2,90,5,87)(3,89,6,86)(7,24,10,21)(8,23,11,20)(9,22,12,19)(13,79,16,82)(14,84,17,81)(15,83,18,80)(25,91,28,94)(26,96,29,93)(27,95,30,92)(31,56,34,59)(32,55,35,58)(33,60,36,57)(37,65,40,62)(38,64,41,61)(39,63,42,66)(43,68,46,71)(44,67,47,70)(45,72,48,69)(49,78,52,75)(50,77,53,74)(51,76,54,73), (1,69,14,76)(2,68,15,75)(3,67,16,74)(4,72,17,73)(5,71,18,78)(6,70,13,77)(7,41,93,33)(8,40,94,32)(9,39,95,31)(10,38,96,36)(11,37,91,35)(12,42,92,34)(19,66,27,59)(20,65,28,58)(21,64,29,57)(22,63,30,56)(23,62,25,55)(24,61,26,60)(43,80,52,87)(44,79,53,86)(45,84,54,85)(46,83,49,90)(47,82,50,89)(48,81,51,88), (1,41,17,36)(2,42,18,31)(3,37,13,32)(4,38,14,33)(5,39,15,34)(6,40,16,35)(7,76,96,72)(8,77,91,67)(9,78,92,68)(10,73,93,69)(11,74,94,70)(12,75,95,71)(19,49,30,43)(20,50,25,44)(21,51,26,45)(22,52,27,46)(23,53,28,47)(24,54,29,48)(55,89,65,79)(56,90,66,80)(57,85,61,81)(58,86,62,82)(59,87,63,83)(60,88,64,84) );
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)], [(1,38,17,33),(2,39,18,34),(3,40,13,35),(4,41,14,36),(5,42,15,31),(6,37,16,32),(7,69,96,73),(8,70,91,74),(9,71,92,75),(10,72,93,76),(11,67,94,77),(12,68,95,78),(19,52,30,46),(20,53,25,47),(21,54,26,48),(22,49,27,43),(23,50,28,44),(24,51,29,45),(55,82,65,86),(56,83,66,87),(57,84,61,88),(58,79,62,89),(59,80,63,90),(60,81,64,85)], [(1,85,4,88),(2,90,5,87),(3,89,6,86),(7,24,10,21),(8,23,11,20),(9,22,12,19),(13,79,16,82),(14,84,17,81),(15,83,18,80),(25,91,28,94),(26,96,29,93),(27,95,30,92),(31,56,34,59),(32,55,35,58),(33,60,36,57),(37,65,40,62),(38,64,41,61),(39,63,42,66),(43,68,46,71),(44,67,47,70),(45,72,48,69),(49,78,52,75),(50,77,53,74),(51,76,54,73)], [(1,69,14,76),(2,68,15,75),(3,67,16,74),(4,72,17,73),(5,71,18,78),(6,70,13,77),(7,41,93,33),(8,40,94,32),(9,39,95,31),(10,38,96,36),(11,37,91,35),(12,42,92,34),(19,66,27,59),(20,65,28,58),(21,64,29,57),(22,63,30,56),(23,62,25,55),(24,61,26,60),(43,80,52,87),(44,79,53,86),(45,84,54,85),(46,83,49,90),(47,82,50,89),(48,81,51,88)], [(1,41,17,36),(2,42,18,31),(3,37,13,32),(4,38,14,33),(5,39,15,34),(6,40,16,35),(7,76,96,72),(8,77,91,67),(9,78,92,68),(10,73,93,69),(11,74,94,70),(12,75,95,71),(19,49,30,43),(20,50,25,44),(21,51,26,45),(22,52,27,46),(23,53,28,47),(24,54,29,48),(55,89,65,79),(56,90,66,80),(57,85,61,81),(58,86,62,82),(59,87,63,83),(60,88,64,84)])
Matrix representation ►G ⊆ GL6(𝔽13)
| 1 | 1 | 0 | 0 | 0 | 0 |
| 12 | 0 | 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 | 1 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 5 | 0 | 0 |
| 0 | 0 | 5 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 5 | 2 |
| 0 | 0 | 8 | 8 | 0 | 8 |
| 2 | 4 | 0 | 0 | 0 | 0 |
| 2 | 11 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 5 | 2 |
| 0 | 0 | 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 5 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 11 | 9 | 0 | 0 | 0 | 0 |
| 11 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 8 | 1 | 3 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 5 | 12 | 5 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 5 | 0 | 0 |
| 0 | 0 | 5 | 0 | 0 | 0 |
| 0 | 0 | 12 | 12 | 8 | 11 |
| 0 | 0 | 0 | 0 | 0 | 5 |
G:=sub<GL(6,GF(13))| [1,12,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,5,1,8,0,0,5,0,1,8,0,0,0,0,5,0,0,0,0,0,2,8],[2,2,0,0,0,0,4,11,0,0,0,0,0,0,1,0,0,0,0,0,1,0,5,0,0,0,5,8,0,0,0,0,2,0,0,12],[11,11,0,0,0,0,9,2,0,0,0,0,0,0,8,0,0,0,0,0,8,0,12,5,0,0,1,1,0,12,0,0,3,0,0,5],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,5,12,0,0,0,5,0,12,0,0,0,0,0,8,0,0,0,0,0,11,5] >;
54 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | ··· | 4L | 4M | ··· | 4AB | 6A | ··· | 6G | 12A | ··· | 12L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | ··· | 4 | 4 | ··· | 4 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 6 | ··· | 6 | 2 | ··· | 2 | 4 | ··· | 4 |
54 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | - | + | - | ||
| image | C1 | C2 | C2 | C2 | C4 | S3 | D6 | Dic3 | D6 | 2- (1+4) | Q8.15D6 |
| kernel | C6.422- (1+4) | C23.26D6 | Q8×Dic3 | Q8×C2×C6 | C6×Q8 | C22×Q8 | C22×C4 | C2×Q8 | C2×Q8 | C6 | C2 |
| # reps | 1 | 6 | 8 | 1 | 16 | 1 | 3 | 8 | 4 | 2 | 4 |
In GAP, Magma, Sage, TeX
C_6._{42}2_-^{(1+4)} % in TeX
G:=Group("C6.42ES-(2,2)"); // GroupNames label
G:=SmallGroup(192,1371);
// by ID
G=gap.SmallGroup(192,1371);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,232,387,184,1123,6278]);
// Polycyclic
G:=Group<a,b,c,d,e|a^6=b^4=1,c^2=a^3,d^2=a^3*b^2,e^2=b^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1,a*e=e*a,c*b*c^-1=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=b^2*d>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀