metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.89D6, C6.472- (1+4), C4⋊C4.267D6, C12⋊2Q8⋊5C2, (C4×Dic6)⋊7C2, Dic3.Q8⋊4C2, (C2×C6).62C24, C2.6(Q8○D12), C22⋊C4.90D6, C12.6Q8⋊3C2, (C4×C12).22C22, (C22×C4).202D6, C4.119(C4○D12), C12.235(C4○D4), (C2×C12).141C23, C42⋊C2.12S3, C4⋊Dic3.31C22, C22.95(S3×C23), C23.93(C22×S3), C23.8D6.1C2, C12.48D4.18C2, Dic3⋊C4.74C22, (C22×C6).132C23, (C2×Dic3).21C23, C6.D4.3C22, (C22×C12).307C22, C3⋊1(C22.35C24), (C4×Dic3).194C22, (C2×Dic6).229C22, C6.27(C2×C4○D4), C2.29(C2×C4○D12), (C3×C4⋊C4).303C22, (C2×C4).269(C22×S3), (C3×C42⋊C2).13C2, (C3×C22⋊C4).111C22, SmallGroup(192,1077)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 392 in 192 conjugacy classes, 95 normal (23 characteristic)
C1, C2, C2 [×2], C2, C3, C4 [×2], C4 [×13], C22, C22 [×3], C6, C6 [×2], C6, C2×C4 [×2], C2×C4 [×4], C2×C4 [×10], Q8 [×4], C23, Dic3 [×8], C12 [×2], C12 [×5], C2×C6, C2×C6 [×3], C42 [×2], C42 [×4], C22⋊C4 [×2], C22⋊C4 [×4], C4⋊C4 [×2], C4⋊C4 [×18], C22×C4, C2×Q8 [×2], Dic6 [×4], C2×Dic3 [×8], C2×C12 [×2], C2×C12 [×4], C2×C12 [×2], C22×C6, C42⋊C2, C4×Q8 [×2], C22⋊Q8 [×2], C42.C2 [×5], C42⋊2C2 [×4], C4⋊Q8, C4×Dic3 [×4], Dic3⋊C4 [×12], C4⋊Dic3 [×6], C6.D4 [×4], C4×C12 [×2], C3×C22⋊C4 [×2], C3×C4⋊C4 [×2], C2×Dic6 [×2], C22×C12, C22.35C24, C4×Dic6 [×2], C12⋊2Q8, C12.6Q8, C23.8D6 [×4], Dic3.Q8 [×4], C12.48D4 [×2], C3×C42⋊C2, C42.89D6
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], C4○D12 [×2], S3×C23, C22.35C24, C2×C4○D12, Q8○D12 [×2], C42.89D6
Generators and relations
G = < a,b,c,d | a4=b4=1, c6=a2, d2=b2, ab=ba, ac=ca, dad-1=a-1, cbc-1=dbd-1=a2b, dcd-1=b2c5 >
►On 96 points
(1 4 7 10)(2 5 8 11)(3 6 9 12)(13 75 19 81)(14 76 20 82)(15 77 21 83)(16 78 22 84)(17 79 23 73)(18 80 24 74)(25 34 31 28)(26 35 32 29)(27 36 33 30)(37 40 43 46)(38 41 44 47)(39 42 45 48)(49 86 55 92)(50 87 56 93)(51 88 57 94)(52 89 58 95)(53 90 59 96)(54 91 60 85)(61 70 67 64)(62 71 68 65)(63 72 69 66)
(1 61 48 30)(2 68 37 25)(3 63 38 32)(4 70 39 27)(5 65 40 34)(6 72 41 29)(7 67 42 36)(8 62 43 31)(9 69 44 26)(10 64 45 33)(11 71 46 28)(12 66 47 35)(13 85 84 57)(14 92 73 52)(15 87 74 59)(16 94 75 54)(17 89 76 49)(18 96 77 56)(19 91 78 51)(20 86 79 58)(21 93 80 53)(22 88 81 60)(23 95 82 55)(24 90 83 50)
(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 59 48 87)(2 92 37 52)(3 57 38 85)(4 90 39 50)(5 55 40 95)(6 88 41 60)(7 53 42 93)(8 86 43 58)(9 51 44 91)(10 96 45 56)(11 49 46 89)(12 94 47 54)(13 26 84 69)(14 62 73 31)(15 36 74 67)(16 72 75 29)(17 34 76 65)(18 70 77 27)(19 32 78 63)(20 68 79 25)(21 30 80 61)(22 66 81 35)(23 28 82 71)(24 64 83 33)
G:=sub<Sym(96)| (1,4,7,10)(2,5,8,11)(3,6,9,12)(13,75,19,81)(14,76,20,82)(15,77,21,83)(16,78,22,84)(17,79,23,73)(18,80,24,74)(25,34,31,28)(26,35,32,29)(27,36,33,30)(37,40,43,46)(38,41,44,47)(39,42,45,48)(49,86,55,92)(50,87,56,93)(51,88,57,94)(52,89,58,95)(53,90,59,96)(54,91,60,85)(61,70,67,64)(62,71,68,65)(63,72,69,66), (1,61,48,30)(2,68,37,25)(3,63,38,32)(4,70,39,27)(5,65,40,34)(6,72,41,29)(7,67,42,36)(8,62,43,31)(9,69,44,26)(10,64,45,33)(11,71,46,28)(12,66,47,35)(13,85,84,57)(14,92,73,52)(15,87,74,59)(16,94,75,54)(17,89,76,49)(18,96,77,56)(19,91,78,51)(20,86,79,58)(21,93,80,53)(22,88,81,60)(23,95,82,55)(24,90,83,50), (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), (1,59,48,87)(2,92,37,52)(3,57,38,85)(4,90,39,50)(5,55,40,95)(6,88,41,60)(7,53,42,93)(8,86,43,58)(9,51,44,91)(10,96,45,56)(11,49,46,89)(12,94,47,54)(13,26,84,69)(14,62,73,31)(15,36,74,67)(16,72,75,29)(17,34,76,65)(18,70,77,27)(19,32,78,63)(20,68,79,25)(21,30,80,61)(22,66,81,35)(23,28,82,71)(24,64,83,33)>;
G:=Group( (1,4,7,10)(2,5,8,11)(3,6,9,12)(13,75,19,81)(14,76,20,82)(15,77,21,83)(16,78,22,84)(17,79,23,73)(18,80,24,74)(25,34,31,28)(26,35,32,29)(27,36,33,30)(37,40,43,46)(38,41,44,47)(39,42,45,48)(49,86,55,92)(50,87,56,93)(51,88,57,94)(52,89,58,95)(53,90,59,96)(54,91,60,85)(61,70,67,64)(62,71,68,65)(63,72,69,66), (1,61,48,30)(2,68,37,25)(3,63,38,32)(4,70,39,27)(5,65,40,34)(6,72,41,29)(7,67,42,36)(8,62,43,31)(9,69,44,26)(10,64,45,33)(11,71,46,28)(12,66,47,35)(13,85,84,57)(14,92,73,52)(15,87,74,59)(16,94,75,54)(17,89,76,49)(18,96,77,56)(19,91,78,51)(20,86,79,58)(21,93,80,53)(22,88,81,60)(23,95,82,55)(24,90,83,50), (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), (1,59,48,87)(2,92,37,52)(3,57,38,85)(4,90,39,50)(5,55,40,95)(6,88,41,60)(7,53,42,93)(8,86,43,58)(9,51,44,91)(10,96,45,56)(11,49,46,89)(12,94,47,54)(13,26,84,69)(14,62,73,31)(15,36,74,67)(16,72,75,29)(17,34,76,65)(18,70,77,27)(19,32,78,63)(20,68,79,25)(21,30,80,61)(22,66,81,35)(23,28,82,71)(24,64,83,33) );
G=PermutationGroup([(1,4,7,10),(2,5,8,11),(3,6,9,12),(13,75,19,81),(14,76,20,82),(15,77,21,83),(16,78,22,84),(17,79,23,73),(18,80,24,74),(25,34,31,28),(26,35,32,29),(27,36,33,30),(37,40,43,46),(38,41,44,47),(39,42,45,48),(49,86,55,92),(50,87,56,93),(51,88,57,94),(52,89,58,95),(53,90,59,96),(54,91,60,85),(61,70,67,64),(62,71,68,65),(63,72,69,66)], [(1,61,48,30),(2,68,37,25),(3,63,38,32),(4,70,39,27),(5,65,40,34),(6,72,41,29),(7,67,42,36),(8,62,43,31),(9,69,44,26),(10,64,45,33),(11,71,46,28),(12,66,47,35),(13,85,84,57),(14,92,73,52),(15,87,74,59),(16,94,75,54),(17,89,76,49),(18,96,77,56),(19,91,78,51),(20,86,79,58),(21,93,80,53),(22,88,81,60),(23,95,82,55),(24,90,83,50)], [(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)], [(1,59,48,87),(2,92,37,52),(3,57,38,85),(4,90,39,50),(5,55,40,95),(6,88,41,60),(7,53,42,93),(8,86,43,58),(9,51,44,91),(10,96,45,56),(11,49,46,89),(12,94,47,54),(13,26,84,69),(14,62,73,31),(15,36,74,67),(16,72,75,29),(17,34,76,65),(18,70,77,27),(19,32,78,63),(20,68,79,25),(21,30,80,61),(22,66,81,35),(23,28,82,71),(24,64,83,33)])
Matrix representation ►G ⊆ GL8(𝔽13)
| 1 | 0 | 0 | 0 | 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 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 4 | 12 | 3 |
| 0 | 0 | 0 | 0 | 6 | 0 | 8 | 1 |
| 8 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 | 5 | 0 |
| 0 | 0 | 0 | 0 | 7 | 5 | 5 | 11 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 | 12 | 0 | 8 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 3 | 12 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 7 | 12 | 3 |
| 0 | 0 | 0 | 0 | 1 | 8 | 8 | 1 |
| 1 | 8 | 0 | 0 | 0 | 0 | 0 | 0 |
| 3 | 12 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 4 | 1 | 1 | 10 |
| 0 | 0 | 0 | 0 | 11 | 0 | 5 | 0 |
| 0 | 0 | 0 | 0 | 7 | 0 | 5 | 12 |
G:=sub<GL(8,GF(13))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,1,9,6,0,0,0,0,12,0,4,0,0,0,0,0,0,0,12,8,0,0,0,0,0,0,3,1],[8,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,7,0,1,0,0,0,0,0,5,0,12,0,0,0,0,5,5,1,0,0,0,0,0,0,11,0,8],[1,3,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,12,9,1,0,0,0,0,1,0,7,8,0,0,0,0,0,0,12,8,0,0,0,0,0,0,3,1],[1,3,0,0,0,0,0,0,8,12,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,8,4,11,7,0,0,0,0,0,1,0,0,0,0,0,0,12,1,5,5,0,0,0,0,0,10,0,12] >;
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | ··· | 4F | 4G | 4H | 4I | 4J | ··· | 4Q | 6A | 6B | 6C | 6D | 6E | 12A | 12B | 12C | 12D | 12E | ··· | 12N |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 6 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 4 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 12 | ··· | 12 | 2 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D6 | D6 | D6 | D6 | C4○D4 | C4○D12 | 2- (1+4) | Q8○D12 |
| kernel | C42.89D6 | C4×Dic6 | C12⋊2Q8 | C12.6Q8 | C23.8D6 | Dic3.Q8 | C12.48D4 | C3×C42⋊C2 | C42⋊C2 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | C12 | C4 | C6 | C2 |
| # reps | 1 | 2 | 1 | 1 | 4 | 4 | 2 | 1 | 1 | 2 | 2 | 2 | 1 | 4 | 8 | 2 | 4 |
In GAP, Magma, Sage, TeX
C_4^2._{89}D_6 % in TeX
G:=Group("C4^2.89D6"); // GroupNames label
G:=SmallGroup(192,1077);
// by ID
G=gap.SmallGroup(192,1077);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,477,232,100,675,297,136,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^6=a^2,d^2=b^2,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,c*b*c^-1=d*b*d^-1=a^2*b,d*c*d^-1=b^2*c^5>;
// generators/relations