metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.164D6, C6.1022- (1+4), C6.1412+ (1+4), C12⋊Q8⋊41C2, C4⋊C4.119D6, C42⋊7S3⋊9C2, C42⋊2C2⋊7S3, C4.D12⋊42C2, D6⋊Q8⋊45C2, (C4×Dic6)⋊16C2, Dic3⋊D4.5C2, C22⋊C4.82D6, D6.D4⋊43C2, C2.66(D4○D12), (C4×C12).36C22, (C2×C6).254C24, D6⋊C4.47C22, C2.66(Q8○D12), Dic3⋊4D4⋊38C2, (C2×C12).605C23, (C2×D12).37C22, (C22×C6).68C23, C23.70(C22×S3), Dic3.16(C4○D4), C23.11D6⋊47C2, Dic3.D4⋊47C2, C22.D12⋊31C2, C4⋊Dic3.319C22, C22.275(S3×C23), Dic3⋊C4.127C22, (C22×S3).113C23, (C2×Dic6).256C22, (C2×Dic3).267C23, (C4×Dic3).219C22, C6.D4.69C22, C3⋊10(C22.36C24), (C22×Dic3).154C22, C4⋊C4⋊7S3⋊42C2, C4⋊C4⋊S3⋊44C2, C2.101(S3×C4○D4), C6.212(C2×C4○D4), (C3×C42⋊2C2)⋊9C2, (S3×C2×C4).136C22, (C3×C4⋊C4).206C22, (C2×C4).210(C22×S3), (C2×C3⋊D4).74C22, (C3×C22⋊C4).79C22, SmallGroup(192,1269)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 560 in 216 conjugacy classes, 93 normal (91 characteristic)
C1, C2 [×3], C2 [×3], C3, C4 [×13], C22, C22 [×9], S3 [×2], C6 [×3], C6, C2×C4 [×6], C2×C4 [×10], D4 [×4], Q8 [×4], C23, C23 [×2], Dic3 [×2], Dic3 [×5], C12 [×6], D6 [×6], C2×C6, C2×C6 [×3], C42, C42 [×3], C22⋊C4 [×3], C22⋊C4 [×9], C4⋊C4 [×3], C4⋊C4 [×7], C22×C4 [×3], C2×D4 [×3], C2×Q8 [×3], Dic6 [×4], C4×S3 [×3], D12, C2×Dic3 [×6], C2×Dic3, C3⋊D4 [×3], C2×C12 [×6], C22×S3 [×2], C22×C6, C42⋊C2, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8 [×3], C22.D4 [×2], C4.4D4 [×3], C42⋊2C2, C42⋊2C2, C4⋊Q8, C4×Dic3 [×3], Dic3⋊C4 [×4], C4⋊Dic3 [×3], D6⋊C4 [×8], C6.D4, C4×C12, C3×C22⋊C4 [×3], C3×C4⋊C4 [×3], C2×Dic6 [×3], S3×C2×C4 [×2], C2×D12, C22×Dic3, C2×C3⋊D4 [×2], C22.36C24, C4×Dic6, C42⋊7S3, Dic3.D4, Dic3⋊4D4, Dic3⋊D4, C23.11D6 [×2], C22.D12, C12⋊Q8, C4⋊C4⋊7S3, D6.D4, D6⋊Q8, C4.D12, C4⋊C4⋊S3, C3×C42⋊2C2, C42.164D6
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), S3×C23, C22.36C24, S3×C4○D4, D4○D12, Q8○D12, C42.164D6
Generators and relations
G = < a,b,c,d | a4=b4=1, c6=d2=b2, ab=ba, cac-1=ab2, dad-1=a-1, cbc-1=a2b-1, dbd-1=a2b, dcd-1=c5 >
►On 96 points
(1 48 86 17)(2 43 87 24)(3 38 88 19)(4 45 89 14)(5 40 90 21)(6 47 91 16)(7 42 92 23)(8 37 93 18)(9 44 94 13)(10 39 95 20)(11 46 96 15)(12 41 85 22)(25 77 62 54)(26 84 63 49)(27 79 64 56)(28 74 65 51)(29 81 66 58)(30 76 67 53)(31 83 68 60)(32 78 69 55)(33 73 70 50)(34 80 71 57)(35 75 72 52)(36 82 61 59)
(1 36 7 30)(2 68 8 62)(3 26 9 32)(4 70 10 64)(5 28 11 34)(6 72 12 66)(13 55 19 49)(14 73 20 79)(15 57 21 51)(16 75 22 81)(17 59 23 53)(18 77 24 83)(25 87 31 93)(27 89 33 95)(29 91 35 85)(37 54 43 60)(38 84 44 78)(39 56 45 50)(40 74 46 80)(41 58 47 52)(42 76 48 82)(61 92 67 86)(63 94 69 88)(65 96 71 90)
(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 6 7 12)(2 11 8 5)(3 4 9 10)(13 39 19 45)(14 44 20 38)(15 37 21 43)(16 42 22 48)(17 47 23 41)(18 40 24 46)(25 28 31 34)(26 33 32 27)(29 36 35 30)(49 73 55 79)(50 78 56 84)(51 83 57 77)(52 76 58 82)(53 81 59 75)(54 74 60 80)(61 72 67 66)(62 65 68 71)(63 70 69 64)(85 86 91 92)(87 96 93 90)(88 89 94 95)
G:=sub<Sym(96)| (1,48,86,17)(2,43,87,24)(3,38,88,19)(4,45,89,14)(5,40,90,21)(6,47,91,16)(7,42,92,23)(8,37,93,18)(9,44,94,13)(10,39,95,20)(11,46,96,15)(12,41,85,22)(25,77,62,54)(26,84,63,49)(27,79,64,56)(28,74,65,51)(29,81,66,58)(30,76,67,53)(31,83,68,60)(32,78,69,55)(33,73,70,50)(34,80,71,57)(35,75,72,52)(36,82,61,59), (1,36,7,30)(2,68,8,62)(3,26,9,32)(4,70,10,64)(5,28,11,34)(6,72,12,66)(13,55,19,49)(14,73,20,79)(15,57,21,51)(16,75,22,81)(17,59,23,53)(18,77,24,83)(25,87,31,93)(27,89,33,95)(29,91,35,85)(37,54,43,60)(38,84,44,78)(39,56,45,50)(40,74,46,80)(41,58,47,52)(42,76,48,82)(61,92,67,86)(63,94,69,88)(65,96,71,90), (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), (1,6,7,12)(2,11,8,5)(3,4,9,10)(13,39,19,45)(14,44,20,38)(15,37,21,43)(16,42,22,48)(17,47,23,41)(18,40,24,46)(25,28,31,34)(26,33,32,27)(29,36,35,30)(49,73,55,79)(50,78,56,84)(51,83,57,77)(52,76,58,82)(53,81,59,75)(54,74,60,80)(61,72,67,66)(62,65,68,71)(63,70,69,64)(85,86,91,92)(87,96,93,90)(88,89,94,95)>;
G:=Group( (1,48,86,17)(2,43,87,24)(3,38,88,19)(4,45,89,14)(5,40,90,21)(6,47,91,16)(7,42,92,23)(8,37,93,18)(9,44,94,13)(10,39,95,20)(11,46,96,15)(12,41,85,22)(25,77,62,54)(26,84,63,49)(27,79,64,56)(28,74,65,51)(29,81,66,58)(30,76,67,53)(31,83,68,60)(32,78,69,55)(33,73,70,50)(34,80,71,57)(35,75,72,52)(36,82,61,59), (1,36,7,30)(2,68,8,62)(3,26,9,32)(4,70,10,64)(5,28,11,34)(6,72,12,66)(13,55,19,49)(14,73,20,79)(15,57,21,51)(16,75,22,81)(17,59,23,53)(18,77,24,83)(25,87,31,93)(27,89,33,95)(29,91,35,85)(37,54,43,60)(38,84,44,78)(39,56,45,50)(40,74,46,80)(41,58,47,52)(42,76,48,82)(61,92,67,86)(63,94,69,88)(65,96,71,90), (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), (1,6,7,12)(2,11,8,5)(3,4,9,10)(13,39,19,45)(14,44,20,38)(15,37,21,43)(16,42,22,48)(17,47,23,41)(18,40,24,46)(25,28,31,34)(26,33,32,27)(29,36,35,30)(49,73,55,79)(50,78,56,84)(51,83,57,77)(52,76,58,82)(53,81,59,75)(54,74,60,80)(61,72,67,66)(62,65,68,71)(63,70,69,64)(85,86,91,92)(87,96,93,90)(88,89,94,95) );
G=PermutationGroup([(1,48,86,17),(2,43,87,24),(3,38,88,19),(4,45,89,14),(5,40,90,21),(6,47,91,16),(7,42,92,23),(8,37,93,18),(9,44,94,13),(10,39,95,20),(11,46,96,15),(12,41,85,22),(25,77,62,54),(26,84,63,49),(27,79,64,56),(28,74,65,51),(29,81,66,58),(30,76,67,53),(31,83,68,60),(32,78,69,55),(33,73,70,50),(34,80,71,57),(35,75,72,52),(36,82,61,59)], [(1,36,7,30),(2,68,8,62),(3,26,9,32),(4,70,10,64),(5,28,11,34),(6,72,12,66),(13,55,19,49),(14,73,20,79),(15,57,21,51),(16,75,22,81),(17,59,23,53),(18,77,24,83),(25,87,31,93),(27,89,33,95),(29,91,35,85),(37,54,43,60),(38,84,44,78),(39,56,45,50),(40,74,46,80),(41,58,47,52),(42,76,48,82),(61,92,67,86),(63,94,69,88),(65,96,71,90)], [(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)], [(1,6,7,12),(2,11,8,5),(3,4,9,10),(13,39,19,45),(14,44,20,38),(15,37,21,43),(16,42,22,48),(17,47,23,41),(18,40,24,46),(25,28,31,34),(26,33,32,27),(29,36,35,30),(49,73,55,79),(50,78,56,84),(51,83,57,77),(52,76,58,82),(53,81,59,75),(54,74,60,80),(61,72,67,66),(62,65,68,71),(63,70,69,64),(85,86,91,92),(87,96,93,90),(88,89,94,95)])
Matrix representation ►G ⊆ GL6(𝔽13)
| 6 | 11 | 0 | 0 | 0 | 0 |
| 11 | 7 | 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 |
| 9 | 10 | 0 | 0 | 0 | 0 |
| 10 | 4 | 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 |
| 5 | 0 | 0 | 0 | 0 | 0 |
| 4 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 9 | 3 | 3 |
| 0 | 0 | 4 | 0 | 10 | 0 |
| 0 | 0 | 3 | 3 | 4 | 4 |
| 0 | 0 | 10 | 0 | 9 | 0 |
| 5 | 0 | 0 | 0 | 0 | 0 |
| 0 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 9 | 3 | 3 |
| 0 | 0 | 0 | 4 | 0 | 10 |
| 0 | 0 | 3 | 3 | 4 | 4 |
| 0 | 0 | 0 | 10 | 0 | 9 |
G:=sub<GL(6,GF(13))| [6,11,0,0,0,0,11,7,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,1,0,0,0,0,0,0,1,0,0],[9,10,0,0,0,0,10,4,0,0,0,0,0,0,10,6,0,0,0,0,7,3,0,0,0,0,0,0,10,6,0,0,0,0,7,3],[5,4,0,0,0,0,0,8,0,0,0,0,0,0,9,4,3,10,0,0,9,0,3,0,0,0,3,10,4,9,0,0,3,0,4,0],[5,0,0,0,0,0,0,5,0,0,0,0,0,0,9,0,3,0,0,0,9,4,3,10,0,0,3,0,4,0,0,0,3,10,4,9] >;
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3 | 4A | 4B | 4C | ··· | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 6A | 6B | 6C | 6D | 12A | ··· | 12F | 12G | 12H | 12I |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 12 | ··· | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 4 | 12 | 12 | 2 | 2 | 2 | 4 | ··· | 4 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 8 | 4 | ··· | 4 | 8 | 8 | 8 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D6 | D6 | D6 | C4○D4 | 2+ (1+4) | 2- (1+4) | S3×C4○D4 | D4○D12 | Q8○D12 |
| kernel | C42.164D6 | C4×Dic6 | C42⋊7S3 | Dic3.D4 | Dic3⋊4D4 | Dic3⋊D4 | C23.11D6 | C22.D12 | C12⋊Q8 | C4⋊C4⋊7S3 | D6.D4 | D6⋊Q8 | C4.D12 | C4⋊C4⋊S3 | C3×C42⋊2C2 | C42⋊2C2 | C42 | C22⋊C4 | C4⋊C4 | Dic3 | C6 | C6 | C2 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 4 | 1 | 1 | 2 | 2 | 2 |
In GAP, Magma, Sage, TeX
C_4^2._{164}D_6 % in TeX
G:=Group("C4^2.164D6"); // GroupNames label
G:=SmallGroup(192,1269);
// by ID
G=gap.SmallGroup(192,1269);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,120,758,219,268,675,570,80,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^6=d^2=b^2,a*b=b*a,c*a*c^-1=a*b^2,d*a*d^-1=a^-1,c*b*c^-1=a^2*b^-1,d*b*d^-1=a^2*b,d*c*d^-1=c^5>;
// generators/relations