metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.140D6, C6.892- (1+4), C6.722+ (1+4), (C2×D4).109D6, (C2×Q8).107D6, C22⋊C4.34D6, C4.4D4.9S3, (C2×C12).78C23, (C2×C6).216C24, C2.50(Q8○D12), C12.6Q8⋊28C2, Dic3⋊Q8⋊23C2, C2.74(D4⋊6D6), (C4×C12).221C22, (C6×D4).209C22, C23.8D6⋊38C2, C4⋊Dic3.50C22, (C22×C6).46C23, C23.48(C22×S3), (C6×Q8).125C22, Dic3.D4⋊39C2, Dic3⋊C4.83C22, C22.237(S3×C23), C23.23D6.6C2, C3⋊3(C22.57C24), (C2×Dic6).176C22, (C2×Dic3).111C23, (C4×Dic3).132C22, C6.D4.53C22, (C22×Dic3).141C22, (C3×C4.4D4).7C2, (C2×C4).192(C22×S3), (C3×C22⋊C4).63C22, SmallGroup(192,1231)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 448 in 196 conjugacy classes, 91 normal (17 characteristic)
C1, C2, C2 [×2], C2 [×2], C3, C4 [×13], C22, C22 [×6], C6, C6 [×2], C6 [×2], C2×C4, C2×C4 [×4], C2×C4 [×10], D4, Q8 [×3], C23 [×2], Dic3 [×8], C12 [×5], C2×C6, C2×C6 [×6], C42, C42 [×2], C22⋊C4 [×4], C22⋊C4 [×6], C4⋊C4 [×16], C22×C4 [×2], C2×D4, C2×Q8, C2×Q8 [×2], Dic6 [×2], C2×Dic3 [×8], C2×Dic3 [×2], C2×C12, C2×C12 [×4], C3×D4, C3×Q8, C22×C6 [×2], C22⋊Q8 [×4], C22.D4 [×2], C4.4D4, C42.C2 [×2], C42⋊2C2 [×4], C4⋊Q8 [×2], C4×Dic3 [×2], Dic3⋊C4 [×12], C4⋊Dic3 [×4], C6.D4 [×6], C4×C12, C3×C22⋊C4 [×4], C2×Dic6 [×2], C22×Dic3 [×2], C6×D4, C6×Q8, C22.57C24, C12.6Q8 [×2], Dic3.D4 [×4], C23.8D6 [×4], C23.23D6 [×2], Dic3⋊Q8 [×2], C3×C4.4D4, C42.140D6
Quotients:
C1, C2 [×15], C22 [×35], S3, C23 [×15], D6 [×7], C24, C22×S3 [×7], 2+ (1+4), 2- (1+4) [×2], S3×C23, C22.57C24, D4⋊6D6, Q8○D12 [×2], C42.140D6
Generators and relations
G = < a,b,c,d | a4=b4=c6=1, d2=b2, ab=ba, cac-1=a-1b2, dad-1=a-1, cbc-1=b-1, dbd-1=a2b-1, dcd-1=c-1 >
►On 96 points
(1 49 10 35)(2 53 11 33)(3 51 12 31)(4 32 7 52)(5 36 8 50)(6 34 9 54)(13 74 22 69)(14 78 23 67)(15 76 24 71)(16 68 19 73)(17 72 20 77)(18 70 21 75)(25 89 63 86)(26 82 64 79)(27 85 65 88)(28 84 66 81)(29 87 61 90)(30 80 62 83)(37 44 40 94)(38 92 41 48)(39 46 42 96)(43 58 93 55)(45 60 95 57)(47 56 91 59)
(1 45 4 48)(2 43 5 46)(3 47 6 44)(7 92 10 95)(8 96 11 93)(9 94 12 91)(13 66 17 63)(14 64 18 61)(15 62 16 65)(19 27 24 30)(20 25 22 28)(21 29 23 26)(31 59 54 37)(32 38 49 60)(33 55 50 39)(34 40 51 56)(35 57 52 41)(36 42 53 58)(67 82 75 87)(68 88 76 83)(69 84 77 89)(70 90 78 79)(71 80 73 85)(72 86 74 81)
(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 14 4 18)(2 13 5 17)(3 15 6 16)(7 21 10 23)(8 20 11 22)(9 19 12 24)(25 46 28 43)(26 45 29 48)(27 44 30 47)(31 76 54 68)(32 75 49 67)(33 74 50 72)(34 73 51 71)(35 78 52 70)(36 77 53 69)(37 80 59 85)(38 79 60 90)(39 84 55 89)(40 83 56 88)(41 82 57 87)(42 81 58 86)(61 92 64 95)(62 91 65 94)(63 96 66 93)
G:=sub<Sym(96)| (1,49,10,35)(2,53,11,33)(3,51,12,31)(4,32,7,52)(5,36,8,50)(6,34,9,54)(13,74,22,69)(14,78,23,67)(15,76,24,71)(16,68,19,73)(17,72,20,77)(18,70,21,75)(25,89,63,86)(26,82,64,79)(27,85,65,88)(28,84,66,81)(29,87,61,90)(30,80,62,83)(37,44,40,94)(38,92,41,48)(39,46,42,96)(43,58,93,55)(45,60,95,57)(47,56,91,59), (1,45,4,48)(2,43,5,46)(3,47,6,44)(7,92,10,95)(8,96,11,93)(9,94,12,91)(13,66,17,63)(14,64,18,61)(15,62,16,65)(19,27,24,30)(20,25,22,28)(21,29,23,26)(31,59,54,37)(32,38,49,60)(33,55,50,39)(34,40,51,56)(35,57,52,41)(36,42,53,58)(67,82,75,87)(68,88,76,83)(69,84,77,89)(70,90,78,79)(71,80,73,85)(72,86,74,81), (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), (1,14,4,18)(2,13,5,17)(3,15,6,16)(7,21,10,23)(8,20,11,22)(9,19,12,24)(25,46,28,43)(26,45,29,48)(27,44,30,47)(31,76,54,68)(32,75,49,67)(33,74,50,72)(34,73,51,71)(35,78,52,70)(36,77,53,69)(37,80,59,85)(38,79,60,90)(39,84,55,89)(40,83,56,88)(41,82,57,87)(42,81,58,86)(61,92,64,95)(62,91,65,94)(63,96,66,93)>;
G:=Group( (1,49,10,35)(2,53,11,33)(3,51,12,31)(4,32,7,52)(5,36,8,50)(6,34,9,54)(13,74,22,69)(14,78,23,67)(15,76,24,71)(16,68,19,73)(17,72,20,77)(18,70,21,75)(25,89,63,86)(26,82,64,79)(27,85,65,88)(28,84,66,81)(29,87,61,90)(30,80,62,83)(37,44,40,94)(38,92,41,48)(39,46,42,96)(43,58,93,55)(45,60,95,57)(47,56,91,59), (1,45,4,48)(2,43,5,46)(3,47,6,44)(7,92,10,95)(8,96,11,93)(9,94,12,91)(13,66,17,63)(14,64,18,61)(15,62,16,65)(19,27,24,30)(20,25,22,28)(21,29,23,26)(31,59,54,37)(32,38,49,60)(33,55,50,39)(34,40,51,56)(35,57,52,41)(36,42,53,58)(67,82,75,87)(68,88,76,83)(69,84,77,89)(70,90,78,79)(71,80,73,85)(72,86,74,81), (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), (1,14,4,18)(2,13,5,17)(3,15,6,16)(7,21,10,23)(8,20,11,22)(9,19,12,24)(25,46,28,43)(26,45,29,48)(27,44,30,47)(31,76,54,68)(32,75,49,67)(33,74,50,72)(34,73,51,71)(35,78,52,70)(36,77,53,69)(37,80,59,85)(38,79,60,90)(39,84,55,89)(40,83,56,88)(41,82,57,87)(42,81,58,86)(61,92,64,95)(62,91,65,94)(63,96,66,93) );
G=PermutationGroup([(1,49,10,35),(2,53,11,33),(3,51,12,31),(4,32,7,52),(5,36,8,50),(6,34,9,54),(13,74,22,69),(14,78,23,67),(15,76,24,71),(16,68,19,73),(17,72,20,77),(18,70,21,75),(25,89,63,86),(26,82,64,79),(27,85,65,88),(28,84,66,81),(29,87,61,90),(30,80,62,83),(37,44,40,94),(38,92,41,48),(39,46,42,96),(43,58,93,55),(45,60,95,57),(47,56,91,59)], [(1,45,4,48),(2,43,5,46),(3,47,6,44),(7,92,10,95),(8,96,11,93),(9,94,12,91),(13,66,17,63),(14,64,18,61),(15,62,16,65),(19,27,24,30),(20,25,22,28),(21,29,23,26),(31,59,54,37),(32,38,49,60),(33,55,50,39),(34,40,51,56),(35,57,52,41),(36,42,53,58),(67,82,75,87),(68,88,76,83),(69,84,77,89),(70,90,78,79),(71,80,73,85),(72,86,74,81)], [(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)], [(1,14,4,18),(2,13,5,17),(3,15,6,16),(7,21,10,23),(8,20,11,22),(9,19,12,24),(25,46,28,43),(26,45,29,48),(27,44,30,47),(31,76,54,68),(32,75,49,67),(33,74,50,72),(34,73,51,71),(35,78,52,70),(36,77,53,69),(37,80,59,85),(38,79,60,90),(39,84,55,89),(40,83,56,88),(41,82,57,87),(42,81,58,86),(61,92,64,95),(62,91,65,94),(63,96,66,93)])
Matrix representation ►G ⊆ GL8(𝔽13)
| 3 | 7 | 0 | 0 | 0 | 0 | 0 | 0 |
| 6 | 10 | 0 | 0 | 0 | 0 | 0 | 0 |
| 7 | 0 | 10 | 7 | 0 | 0 | 0 | 0 |
| 0 | 6 | 6 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 | 0 | 12 |
| 0 | 0 | 0 | 0 | 1 | 12 | 1 | 0 |
| 0 | 0 | 12 | 1 | 0 | 0 | 0 | 0 |
| 12 | 12 | 11 | 12 | 0 | 0 | 0 | 0 |
| 5 | 5 | 1 | 0 | 0 | 0 | 0 | 0 |
| 4 | 5 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 | 0 | 12 |
| 0 | 12 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 12 | 0 | 0 | 0 | 0 | 0 | 0 |
| 9 | 10 | 1 | 1 | 0 | 0 | 0 | 0 |
| 8 | 9 | 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 10 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 0 | 9 | 0 |
| 0 | 0 | 0 | 0 | 7 | 0 | 0 | 4 |
| 9 | 2 | 0 | 0 | 0 | 0 | 0 | 0 |
| 11 | 4 | 0 | 0 | 0 | 0 | 0 | 0 |
| 4 | 0 | 2 | 4 | 0 | 0 | 0 | 0 |
| 2 | 11 | 2 | 11 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 | 8 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 | 4 | 4 |
| 0 | 0 | 0 | 0 | 3 | 0 | 9 | 0 |
| 0 | 0 | 0 | 0 | 6 | 10 | 9 | 0 |
G:=sub<GL(8,GF(13))| [3,6,7,0,0,0,0,0,7,10,0,6,0,0,0,0,0,0,10,6,0,0,0,0,0,0,7,3,0,0,0,0,0,0,0,0,12,12,0,1,0,0,0,0,2,1,12,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0],[0,12,5,4,0,0,0,0,0,12,5,5,0,0,0,0,12,11,1,1,0,0,0,0,1,12,0,0,0,0,0,0,0,0,0,0,1,0,12,12,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12],[0,1,9,8,0,0,0,0,12,12,10,9,0,0,0,0,0,0,1,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,3,3,3,7,0,0,0,0,0,10,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,4],[9,11,4,2,0,0,0,0,2,4,0,11,0,0,0,0,0,0,2,2,0,0,0,0,0,0,4,11,0,0,0,0,0,0,0,0,4,4,3,6,0,0,0,0,0,0,0,10,0,0,0,0,8,4,9,9,0,0,0,0,0,4,0,0] >;
33 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | ··· | 4E | 4F | ··· | 4M | 6A | 6B | 6C | 6D | 6E | 12A | ··· | 12F | 12G | 12H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | ··· | 4 | 4 | ··· | 4 | 6 | 6 | 6 | 6 | 6 | 12 | ··· | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 2 | 4 | ··· | 4 | 12 | ··· | 12 | 2 | 2 | 2 | 8 | 8 | 4 | ··· | 4 | 8 | 8 |
33 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D6 | D6 | D6 | D6 | 2+ (1+4) | 2- (1+4) | D4⋊6D6 | Q8○D12 |
| kernel | C42.140D6 | C12.6Q8 | Dic3.D4 | C23.8D6 | C23.23D6 | Dic3⋊Q8 | C3×C4.4D4 | C4.4D4 | C42 | C22⋊C4 | C2×D4 | C2×Q8 | C6 | C6 | C2 | C2 |
| # reps | 1 | 2 | 4 | 4 | 2 | 2 | 1 | 1 | 1 | 4 | 1 | 1 | 1 | 2 | 2 | 4 |
In GAP, Magma, Sage, TeX
C_4^2._{140}D_6 % in TeX
G:=Group("C4^2.140D6"); // GroupNames label
G:=SmallGroup(192,1231);
// by ID
G=gap.SmallGroup(192,1231);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,758,219,184,1571,570,297,136,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^6=1,d^2=b^2,a*b=b*a,c*a*c^-1=a^-1*b^2,d*a*d^-1=a^-1,c*b*c^-1=b^-1,d*b*d^-1=a^2*b^-1,d*c*d^-1=c^-1>;
// generators/relations