?
G = C6.72+ (1+4) order 192 = 26·3
7th non-split extension by C6 of 2+ (1+4) acting via 2+ (1+4)/C2×D4=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C6.72+ (1+4), C6.12- (1+4), C12⋊Q8⋊8C2, (C2×C12)⋊3Q8, (C2×C4)⋊3Dic6, C4⋊C4.261D6, C12.67(C2×Q8), C6.9(C22×Q8), (C2×C6).44C24, C12.3Q8⋊8C2, C4.32(C2×Dic6), (C22×C4).191D6, C2.11(D4⋊6D6), (C2×C12).135C23, C12.48D4.6C2, Dic3⋊C4.1C22, C22.82(S3×C23), C22.11(C2×Dic6), C2.11(C22×Dic6), C4⋊Dic3.358C22, C23.233(C22×S3), (C22×C12).74C22, (C22×C6).393C23, C2.5(Q8.15D6), (C2×Dic3).14C23, (C2×Dic6).21C22, (C4×Dic3).62C22, C3⋊1(C23.41C23), C6.D4.86C22, C23.26D6.20C2, (C6×C4⋊C4).20C2, (C2×C4⋊C4).27S3, (C2×C6).52(C2×Q8), (C3×C4⋊C4).293C22, (C2×C4).570(C22×S3), SmallGroup(192,1059)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 440 in 206 conjugacy classes, 111 normal (17 characteristic)
C1, C2 [×3], C2 [×2], C3, C4 [×4], C4 [×12], C22, C22 [×2], C22 [×2], C6 [×3], C6 [×2], C2×C4 [×10], C2×C4 [×10], Q8 [×4], C23, Dic3 [×8], C12 [×4], C12 [×4], C2×C6, C2×C6 [×2], C2×C6 [×2], C42 [×4], C22⋊C4 [×4], C4⋊C4 [×4], C4⋊C4 [×16], C22×C4, C22×C4 [×2], C2×Q8 [×4], Dic6 [×4], C2×Dic3 [×8], C2×C12 [×10], C2×C12 [×2], C22×C6, C2×C4⋊C4, C42⋊C2 [×2], C22⋊Q8 [×4], C42.C2 [×4], C4⋊Q8 [×4], C4×Dic3 [×4], Dic3⋊C4 [×8], C4⋊Dic3 [×8], C6.D4 [×4], C3×C4⋊C4 [×4], C2×Dic6 [×4], C22×C12, C22×C12 [×2], C23.41C23, C12⋊Q8 [×4], C12.3Q8 [×4], C12.48D4 [×4], C23.26D6 [×2], C6×C4⋊C4, C6.72+ (1+4)
Quotients:
C1, C2 [×15], C22 [×35], S3, Q8 [×4], C23 [×15], D6 [×7], C2×Q8 [×6], C24, Dic6 [×4], C22×S3 [×7], C22×Q8, 2+ (1+4), 2- (1+4), C2×Dic6 [×6], S3×C23, C23.41C23, C22×Dic6, D4⋊6D6, Q8.15D6, C6.72+ (1+4)
Generators and relations
G = < a,b,c,d,e | a6=b4=1, c2=e2=a3, d2=b2, bab-1=dad-1=eae-1=a-1, ac=ca, cbc-1=b-1, dbd-1=a3b, be=eb, dcd-1=ece-1=a3c, 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 95 14 7)(2 94 15 12)(3 93 16 11)(4 92 17 10)(5 91 18 9)(6 96 13 8)(19 83 29 88)(20 82 30 87)(21 81 25 86)(22 80 26 85)(23 79 27 90)(24 84 28 89)(31 68 41 73)(32 67 42 78)(33 72 37 77)(34 71 38 76)(35 70 39 75)(36 69 40 74)(43 59 54 64)(44 58 49 63)(45 57 50 62)(46 56 51 61)(47 55 52 66)(48 60 53 65)
(1 19 4 22)(2 20 5 23)(3 21 6 24)(7 83 10 80)(8 84 11 81)(9 79 12 82)(13 28 16 25)(14 29 17 26)(15 30 18 27)(31 54 34 51)(32 49 35 52)(33 50 36 53)(37 45 40 48)(38 46 41 43)(39 47 42 44)(55 67 58 70)(56 68 59 71)(57 69 60 72)(61 73 64 76)(62 74 65 77)(63 75 66 78)(85 95 88 92)(86 96 89 93)(87 91 90 94)
(1 59 14 64)(2 58 15 63)(3 57 16 62)(4 56 17 61)(5 55 18 66)(6 60 13 65)(7 46 95 51)(8 45 96 50)(9 44 91 49)(10 43 92 54)(11 48 93 53)(12 47 94 52)(19 68 29 73)(20 67 30 78)(21 72 25 77)(22 71 26 76)(23 70 27 75)(24 69 28 74)(31 80 41 85)(32 79 42 90)(33 84 37 89)(34 83 38 88)(35 82 39 87)(36 81 40 86)
(1 71 4 68)(2 70 5 67)(3 69 6 72)(7 34 10 31)(8 33 11 36)(9 32 12 35)(13 77 16 74)(14 76 17 73)(15 75 18 78)(19 59 22 56)(20 58 23 55)(21 57 24 60)(25 62 28 65)(26 61 29 64)(27 66 30 63)(37 93 40 96)(38 92 41 95)(39 91 42 94)(43 85 46 88)(44 90 47 87)(45 89 48 86)(49 79 52 82)(50 84 53 81)(51 83 54 80)
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), (1,95,14,7)(2,94,15,12)(3,93,16,11)(4,92,17,10)(5,91,18,9)(6,96,13,8)(19,83,29,88)(20,82,30,87)(21,81,25,86)(22,80,26,85)(23,79,27,90)(24,84,28,89)(31,68,41,73)(32,67,42,78)(33,72,37,77)(34,71,38,76)(35,70,39,75)(36,69,40,74)(43,59,54,64)(44,58,49,63)(45,57,50,62)(46,56,51,61)(47,55,52,66)(48,60,53,65), (1,19,4,22)(2,20,5,23)(3,21,6,24)(7,83,10,80)(8,84,11,81)(9,79,12,82)(13,28,16,25)(14,29,17,26)(15,30,18,27)(31,54,34,51)(32,49,35,52)(33,50,36,53)(37,45,40,48)(38,46,41,43)(39,47,42,44)(55,67,58,70)(56,68,59,71)(57,69,60,72)(61,73,64,76)(62,74,65,77)(63,75,66,78)(85,95,88,92)(86,96,89,93)(87,91,90,94), (1,59,14,64)(2,58,15,63)(3,57,16,62)(4,56,17,61)(5,55,18,66)(6,60,13,65)(7,46,95,51)(8,45,96,50)(9,44,91,49)(10,43,92,54)(11,48,93,53)(12,47,94,52)(19,68,29,73)(20,67,30,78)(21,72,25,77)(22,71,26,76)(23,70,27,75)(24,69,28,74)(31,80,41,85)(32,79,42,90)(33,84,37,89)(34,83,38,88)(35,82,39,87)(36,81,40,86), (1,71,4,68)(2,70,5,67)(3,69,6,72)(7,34,10,31)(8,33,11,36)(9,32,12,35)(13,77,16,74)(14,76,17,73)(15,75,18,78)(19,59,22,56)(20,58,23,55)(21,57,24,60)(25,62,28,65)(26,61,29,64)(27,66,30,63)(37,93,40,96)(38,92,41,95)(39,91,42,94)(43,85,46,88)(44,90,47,87)(45,89,48,86)(49,79,52,82)(50,84,53,81)(51,83,54,80)>;
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), (1,95,14,7)(2,94,15,12)(3,93,16,11)(4,92,17,10)(5,91,18,9)(6,96,13,8)(19,83,29,88)(20,82,30,87)(21,81,25,86)(22,80,26,85)(23,79,27,90)(24,84,28,89)(31,68,41,73)(32,67,42,78)(33,72,37,77)(34,71,38,76)(35,70,39,75)(36,69,40,74)(43,59,54,64)(44,58,49,63)(45,57,50,62)(46,56,51,61)(47,55,52,66)(48,60,53,65), (1,19,4,22)(2,20,5,23)(3,21,6,24)(7,83,10,80)(8,84,11,81)(9,79,12,82)(13,28,16,25)(14,29,17,26)(15,30,18,27)(31,54,34,51)(32,49,35,52)(33,50,36,53)(37,45,40,48)(38,46,41,43)(39,47,42,44)(55,67,58,70)(56,68,59,71)(57,69,60,72)(61,73,64,76)(62,74,65,77)(63,75,66,78)(85,95,88,92)(86,96,89,93)(87,91,90,94), (1,59,14,64)(2,58,15,63)(3,57,16,62)(4,56,17,61)(5,55,18,66)(6,60,13,65)(7,46,95,51)(8,45,96,50)(9,44,91,49)(10,43,92,54)(11,48,93,53)(12,47,94,52)(19,68,29,73)(20,67,30,78)(21,72,25,77)(22,71,26,76)(23,70,27,75)(24,69,28,74)(31,80,41,85)(32,79,42,90)(33,84,37,89)(34,83,38,88)(35,82,39,87)(36,81,40,86), (1,71,4,68)(2,70,5,67)(3,69,6,72)(7,34,10,31)(8,33,11,36)(9,32,12,35)(13,77,16,74)(14,76,17,73)(15,75,18,78)(19,59,22,56)(20,58,23,55)(21,57,24,60)(25,62,28,65)(26,61,29,64)(27,66,30,63)(37,93,40,96)(38,92,41,95)(39,91,42,94)(43,85,46,88)(44,90,47,87)(45,89,48,86)(49,79,52,82)(50,84,53,81)(51,83,54,80) );
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)], [(1,95,14,7),(2,94,15,12),(3,93,16,11),(4,92,17,10),(5,91,18,9),(6,96,13,8),(19,83,29,88),(20,82,30,87),(21,81,25,86),(22,80,26,85),(23,79,27,90),(24,84,28,89),(31,68,41,73),(32,67,42,78),(33,72,37,77),(34,71,38,76),(35,70,39,75),(36,69,40,74),(43,59,54,64),(44,58,49,63),(45,57,50,62),(46,56,51,61),(47,55,52,66),(48,60,53,65)], [(1,19,4,22),(2,20,5,23),(3,21,6,24),(7,83,10,80),(8,84,11,81),(9,79,12,82),(13,28,16,25),(14,29,17,26),(15,30,18,27),(31,54,34,51),(32,49,35,52),(33,50,36,53),(37,45,40,48),(38,46,41,43),(39,47,42,44),(55,67,58,70),(56,68,59,71),(57,69,60,72),(61,73,64,76),(62,74,65,77),(63,75,66,78),(85,95,88,92),(86,96,89,93),(87,91,90,94)], [(1,59,14,64),(2,58,15,63),(3,57,16,62),(4,56,17,61),(5,55,18,66),(6,60,13,65),(7,46,95,51),(8,45,96,50),(9,44,91,49),(10,43,92,54),(11,48,93,53),(12,47,94,52),(19,68,29,73),(20,67,30,78),(21,72,25,77),(22,71,26,76),(23,70,27,75),(24,69,28,74),(31,80,41,85),(32,79,42,90),(33,84,37,89),(34,83,38,88),(35,82,39,87),(36,81,40,86)], [(1,71,4,68),(2,70,5,67),(3,69,6,72),(7,34,10,31),(8,33,11,36),(9,32,12,35),(13,77,16,74),(14,76,17,73),(15,75,18,78),(19,59,22,56),(20,58,23,55),(21,57,24,60),(25,62,28,65),(26,61,29,64),(27,66,30,63),(37,93,40,96),(38,92,41,95),(39,91,42,94),(43,85,46,88),(44,90,47,87),(45,89,48,86),(49,79,52,82),(50,84,53,81),(51,83,54,80)])
Matrix representation ►G ⊆ GL6(𝔽13)
| 10 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 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 |
| 0 | 5 | 0 | 0 | 0 | 0 |
| 5 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 5 |
| 0 | 0 | 0 | 0 | 6 | 6 |
| 0 | 0 | 6 | 8 | 0 | 0 |
| 0 | 0 | 7 | 7 | 0 | 0 |
| 8 | 0 | 0 | 0 | 0 | 0 |
| 0 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 11 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 11 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 8 | 0 | 0 | 0 | 0 |
| 8 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 11 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 12 | 11 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
G:=sub<GL(6,GF(13))| [10,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,5,0,0,0,0,5,0,0,0,0,0,0,0,0,0,6,7,0,0,0,0,8,7,0,0,7,6,0,0,0,0,5,6,0,0],[8,0,0,0,0,0,0,5,0,0,0,0,0,0,12,0,0,0,0,0,11,1,0,0,0,0,0,0,12,0,0,0,0,0,11,1],[0,1,0,0,0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,12,0,0,0,0,0,0,12,0,0],[0,8,0,0,0,0,8,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,11,1,0,0,12,0,0,0,0,0,11,1,0,0] >;
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4P | 6A | ··· | 6G | 12A | ··· | 12L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 12 | ··· | 12 | 2 | ··· | 2 | 4 | ··· | 4 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | + | + | - | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | Q8 | D6 | D6 | Dic6 | 2+ (1+4) | 2- (1+4) | D4⋊6D6 | Q8.15D6 |
| kernel | C6.72+ (1+4) | C12⋊Q8 | C12.3Q8 | C12.48D4 | C23.26D6 | C6×C4⋊C4 | C2×C4⋊C4 | C2×C12 | C4⋊C4 | C22×C4 | C2×C4 | C6 | C6 | C2 | C2 |
| # reps | 1 | 4 | 4 | 4 | 2 | 1 | 1 | 4 | 4 | 3 | 8 | 1 | 1 | 2 | 2 |
In GAP, Magma, Sage, TeX
C_6._72_+^{(1+4)} % in TeX
G:=Group("C6.7ES+(2,2)"); // GroupNames label
G:=SmallGroup(192,1059);
// by ID
G=gap.SmallGroup(192,1059);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,232,758,675,570,80,6278]);
// Polycyclic
G:=Group<a,b,c,d,e|a^6=b^4=1,c^2=e^2=a^3,d^2=b^2,b*a*b^-1=d*a*d^-1=e*a*e^-1=a^-1,a*c=c*a,c*b*c^-1=b^-1,d*b*d^-1=a^3*b,b*e=e*b,d*c*d^-1=e*c*e^-1=a^3*c,e*d*e^-1=b^2*d>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀