metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.95D6, C6.972+ (1+4), (C4×D12)⋊10C2, Dic3⋊D4⋊3C2, C4⋊C4.272D6, C12⋊D4⋊12C2, C12⋊7D4⋊29C2, C42⋊3S3⋊4C2, C23.9D6⋊4C2, C2.9(D4○D12), (C2×C6).74C24, D6⋊C4.3C22, C22⋊C4.98D6, Dic3⋊5D4⋊12C2, C42⋊C2⋊14S3, D6.28(C4○D4), C4.96(C4○D12), (C4×C12).25C22, C12.3Q8⋊13C2, (C22×C4).211D6, C12.198(C4○D4), (C2×C12).149C23, C23.96(C22×S3), (C2×D12).137C22, Dic3⋊C4.98C22, (C22×S3).22C23, C4⋊Dic3.195C22, C22.103(S3×C23), (C22×C6).144C23, (C2×Dic3).27C23, (C4×Dic3).70C22, (C22×C12).232C22, C3⋊2(C22.47C24), C6.D4.97C22, (S3×C4⋊C4)⋊13C2, (C4×C3⋊D4)⋊12C2, C2.13(S3×C4○D4), C2.33(C2×C4○D12), C6.134(C2×C4○D4), (S3×C2×C4).61C22, (C3×C42⋊C2)⋊16C2, (C3×C4⋊C4).310C22, (C2×C4).276(C22×S3), (C2×C3⋊D4).104C22, (C3×C22⋊C4).114C22, SmallGroup(192,1089)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 632 in 238 conjugacy classes, 97 normal (43 characteristic)
C1, C2 [×3], C2 [×5], C3, C4 [×2], C4 [×10], C22, C22 [×13], S3 [×4], C6 [×3], C6, C2×C4 [×2], C2×C4 [×4], C2×C4 [×13], D4 [×10], C23, C23 [×3], Dic3 [×5], C12 [×2], C12 [×5], D6 [×2], D6 [×8], C2×C6, C2×C6 [×3], C42 [×2], C42, C22⋊C4 [×2], C22⋊C4 [×8], C4⋊C4 [×2], C4⋊C4 [×8], C22×C4, C22×C4 [×5], C2×D4 [×6], C4×S3 [×6], D12 [×6], C2×Dic3 [×3], C2×Dic3 [×2], C3⋊D4 [×4], C2×C12 [×2], C2×C12 [×4], C2×C12 [×2], C22×S3, C22×S3 [×2], C22×C6, C2×C4⋊C4, C42⋊C2, C4×D4 [×4], C4⋊D4 [×4], C22.D4 [×2], C42.C2, C42⋊2C2 [×2], C4×Dic3, Dic3⋊C4, Dic3⋊C4 [×4], C4⋊Dic3, C4⋊Dic3 [×2], D6⋊C4, D6⋊C4 [×6], C6.D4, C4×C12 [×2], C3×C22⋊C4 [×2], C3×C4⋊C4 [×2], S3×C2×C4, S3×C2×C4 [×4], C2×D12, C2×D12 [×2], C2×C3⋊D4, C2×C3⋊D4 [×2], C22×C12, C22.47C24, C4×D12 [×2], C42⋊3S3 [×2], C23.9D6 [×2], Dic3⋊D4 [×2], C12.3Q8, S3×C4⋊C4, Dic3⋊5D4, C12⋊D4, C4×C3⋊D4, C12⋊7D4, C3×C42⋊C2, C42.95D6
Quotients:
C1, C2 [×15], C22 [×35], S3, C23 [×15], D6 [×7], C4○D4 [×4], C24, C22×S3 [×7], C2×C4○D4 [×2], 2+ (1+4), C4○D12 [×2], S3×C23, C22.47C24, C2×C4○D12, S3×C4○D4, D4○D12, C42.95D6
Generators and relations
G = < a,b,c,d | a4=b4=1, c6=d2=b2, ab=ba, cac-1=dad-1=ab2, bc=cb, dbd-1=a2b-1, dcd-1=c5 >
►On 96 points
(1 41 81 26)(2 48 82 33)(3 43 83 28)(4 38 84 35)(5 45 73 30)(6 40 74 25)(7 47 75 32)(8 42 76 27)(9 37 77 34)(10 44 78 29)(11 39 79 36)(12 46 80 31)(13 91 54 69)(14 86 55 64)(15 93 56 71)(16 88 57 66)(17 95 58 61)(18 90 59 68)(19 85 60 63)(20 92 49 70)(21 87 50 65)(22 94 51 72)(23 89 52 67)(24 96 53 62)
(1 95 7 89)(2 96 8 90)(3 85 9 91)(4 86 10 92)(5 87 11 93)(6 88 12 94)(13 28 19 34)(14 29 20 35)(15 30 21 36)(16 31 22 25)(17 32 23 26)(18 33 24 27)(37 54 43 60)(38 55 44 49)(39 56 45 50)(40 57 46 51)(41 58 47 52)(42 59 48 53)(61 75 67 81)(62 76 68 82)(63 77 69 83)(64 78 70 84)(65 79 71 73)(66 80 72 74)
(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 16 7 22)(2 21 8 15)(3 14 9 20)(4 19 10 13)(5 24 11 18)(6 17 12 23)(25 67 31 61)(26 72 32 66)(27 65 33 71)(28 70 34 64)(29 63 35 69)(30 68 36 62)(37 86 43 92)(38 91 44 85)(39 96 45 90)(40 89 46 95)(41 94 47 88)(42 87 48 93)(49 83 55 77)(50 76 56 82)(51 81 57 75)(52 74 58 80)(53 79 59 73)(54 84 60 78)
G:=sub<Sym(96)| (1,41,81,26)(2,48,82,33)(3,43,83,28)(4,38,84,35)(5,45,73,30)(6,40,74,25)(7,47,75,32)(8,42,76,27)(9,37,77,34)(10,44,78,29)(11,39,79,36)(12,46,80,31)(13,91,54,69)(14,86,55,64)(15,93,56,71)(16,88,57,66)(17,95,58,61)(18,90,59,68)(19,85,60,63)(20,92,49,70)(21,87,50,65)(22,94,51,72)(23,89,52,67)(24,96,53,62), (1,95,7,89)(2,96,8,90)(3,85,9,91)(4,86,10,92)(5,87,11,93)(6,88,12,94)(13,28,19,34)(14,29,20,35)(15,30,21,36)(16,31,22,25)(17,32,23,26)(18,33,24,27)(37,54,43,60)(38,55,44,49)(39,56,45,50)(40,57,46,51)(41,58,47,52)(42,59,48,53)(61,75,67,81)(62,76,68,82)(63,77,69,83)(64,78,70,84)(65,79,71,73)(66,80,72,74), (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), (1,16,7,22)(2,21,8,15)(3,14,9,20)(4,19,10,13)(5,24,11,18)(6,17,12,23)(25,67,31,61)(26,72,32,66)(27,65,33,71)(28,70,34,64)(29,63,35,69)(30,68,36,62)(37,86,43,92)(38,91,44,85)(39,96,45,90)(40,89,46,95)(41,94,47,88)(42,87,48,93)(49,83,55,77)(50,76,56,82)(51,81,57,75)(52,74,58,80)(53,79,59,73)(54,84,60,78)>;
G:=Group( (1,41,81,26)(2,48,82,33)(3,43,83,28)(4,38,84,35)(5,45,73,30)(6,40,74,25)(7,47,75,32)(8,42,76,27)(9,37,77,34)(10,44,78,29)(11,39,79,36)(12,46,80,31)(13,91,54,69)(14,86,55,64)(15,93,56,71)(16,88,57,66)(17,95,58,61)(18,90,59,68)(19,85,60,63)(20,92,49,70)(21,87,50,65)(22,94,51,72)(23,89,52,67)(24,96,53,62), (1,95,7,89)(2,96,8,90)(3,85,9,91)(4,86,10,92)(5,87,11,93)(6,88,12,94)(13,28,19,34)(14,29,20,35)(15,30,21,36)(16,31,22,25)(17,32,23,26)(18,33,24,27)(37,54,43,60)(38,55,44,49)(39,56,45,50)(40,57,46,51)(41,58,47,52)(42,59,48,53)(61,75,67,81)(62,76,68,82)(63,77,69,83)(64,78,70,84)(65,79,71,73)(66,80,72,74), (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), (1,16,7,22)(2,21,8,15)(3,14,9,20)(4,19,10,13)(5,24,11,18)(6,17,12,23)(25,67,31,61)(26,72,32,66)(27,65,33,71)(28,70,34,64)(29,63,35,69)(30,68,36,62)(37,86,43,92)(38,91,44,85)(39,96,45,90)(40,89,46,95)(41,94,47,88)(42,87,48,93)(49,83,55,77)(50,76,56,82)(51,81,57,75)(52,74,58,80)(53,79,59,73)(54,84,60,78) );
G=PermutationGroup([(1,41,81,26),(2,48,82,33),(3,43,83,28),(4,38,84,35),(5,45,73,30),(6,40,74,25),(7,47,75,32),(8,42,76,27),(9,37,77,34),(10,44,78,29),(11,39,79,36),(12,46,80,31),(13,91,54,69),(14,86,55,64),(15,93,56,71),(16,88,57,66),(17,95,58,61),(18,90,59,68),(19,85,60,63),(20,92,49,70),(21,87,50,65),(22,94,51,72),(23,89,52,67),(24,96,53,62)], [(1,95,7,89),(2,96,8,90),(3,85,9,91),(4,86,10,92),(5,87,11,93),(6,88,12,94),(13,28,19,34),(14,29,20,35),(15,30,21,36),(16,31,22,25),(17,32,23,26),(18,33,24,27),(37,54,43,60),(38,55,44,49),(39,56,45,50),(40,57,46,51),(41,58,47,52),(42,59,48,53),(61,75,67,81),(62,76,68,82),(63,77,69,83),(64,78,70,84),(65,79,71,73),(66,80,72,74)], [(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)], [(1,16,7,22),(2,21,8,15),(3,14,9,20),(4,19,10,13),(5,24,11,18),(6,17,12,23),(25,67,31,61),(26,72,32,66),(27,65,33,71),(28,70,34,64),(29,63,35,69),(30,68,36,62),(37,86,43,92),(38,91,44,85),(39,96,45,90),(40,89,46,95),(41,94,47,88),(42,87,48,93),(49,83,55,77),(50,76,56,82),(51,81,57,75),(52,74,58,80),(53,79,59,73),(54,84,60,78)])
Matrix representation ►G ⊆ GL4(𝔽13) generated by
| 5 | 0 | 0 | 0 |
| 0 | 5 | 0 | 0 |
| 0 | 0 | 5 | 0 |
| 0 | 0 | 0 | 8 |
| 2 | 4 | 0 | 0 |
| 9 | 11 | 0 | 0 |
| 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 8 |
| 12 | 12 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 12 | 0 |
| 3 | 10 | 0 | 0 |
| 7 | 10 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 12 | 0 |
G:=sub<GL(4,GF(13))| [5,0,0,0,0,5,0,0,0,0,5,0,0,0,0,8],[2,9,0,0,4,11,0,0,0,0,8,0,0,0,0,8],[12,1,0,0,12,0,0,0,0,0,0,12,0,0,1,0],[3,7,0,0,10,10,0,0,0,0,0,12,0,0,1,0] >;
45 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 3 | 4A | ··· | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | 6A | 6B | 6C | 6D | 6E | 12A | 12B | 12C | 12D | 12E | ··· | 12N |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 4 | 6 | 6 | 12 | 12 | 2 | 2 | ··· | 2 | 4 | 4 | 6 | 6 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
45 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D6 | D6 | D6 | D6 | C4○D4 | C4○D4 | C4○D12 | 2+ (1+4) | S3×C4○D4 | D4○D12 |
| kernel | C42.95D6 | C4×D12 | C42⋊3S3 | C23.9D6 | Dic3⋊D4 | C12.3Q8 | S3×C4⋊C4 | Dic3⋊5D4 | C12⋊D4 | C4×C3⋊D4 | C12⋊7D4 | C3×C42⋊C2 | C42⋊C2 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | C12 | D6 | C4 | C6 | C2 | C2 |
| # reps | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 4 | 4 | 8 | 1 | 2 | 2 |
In GAP, Magma, Sage, TeX
C_4^2._{95}D_6 % in TeX
G:=Group("C4^2.95D6"); // GroupNames label
G:=SmallGroup(192,1089);
// by ID
G=gap.SmallGroup(192,1089);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,387,100,1571,136,6278]);
// 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=d*a*d^-1=a*b^2,b*c=c*b,d*b*d^-1=a^2*b^-1,d*c*d^-1=c^5>;
// generators/relations