metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.122D6, C6.72- (1+4), (C4×Q8)⋊9S3, (Q8×C12)⋊5C2, C4⋊C4.291D6, D6⋊Q8⋊10C2, (C4×Dic6)⋊36C2, (C2×Q8).200D6, C42⋊2S3⋊33C2, C42⋊3S3⋊17C2, Dic3⋊Q8⋊9C2, C4.18(C4○D12), (C2×C6).112C24, D6⋊C4.68C22, Dic6⋊C4⋊17C2, Dic3⋊5D4.10C2, C12.116(C4○D4), C42⋊7S3.10C2, (C2×C12).621C23, (C4×C12).238C22, C12.23D4.7C2, (C6×Q8).212C22, Dic3.21(C4○D4), (C2×D12).140C22, Dic3⋊C4.68C22, (C22×S3).44C23, C4⋊Dic3.303C22, C22.137(S3×C23), (C4×Dic3).81C22, C2.10(Q8.15D6), C3⋊3(C22.50C24), (C2×Dic3).211C23, (C2×Dic6).147C22, C4⋊C4⋊S3⋊10C2, C6.53(C2×C4○D4), C2.27(S3×C4○D4), C2.60(C2×C4○D12), (S3×C2×C4).206C22, (C3×C4⋊C4).340C22, (C2×C4).653(C22×S3), SmallGroup(192,1127)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 488 in 212 conjugacy classes, 97 normal (43 characteristic)
C1, C2 [×3], C2 [×2], C3, C4 [×2], C4 [×13], C22, C22 [×6], S3 [×2], C6 [×3], C2×C4 [×3], C2×C4 [×4], C2×C4 [×10], D4 [×2], Q8 [×6], C23 [×2], Dic3 [×2], Dic3 [×5], C12 [×2], C12 [×6], D6 [×6], C2×C6, C42, C42 [×2], C42 [×4], C22⋊C4 [×10], C4⋊C4, C4⋊C4 [×2], C4⋊C4 [×9], C22×C4 [×2], C2×D4, C2×Q8, C2×Q8 [×2], Dic6 [×4], C4×S3 [×4], D12 [×2], C2×Dic3 [×4], C2×Dic3 [×2], C2×C12 [×3], C2×C12 [×4], C3×Q8 [×2], C22×S3 [×2], C42⋊C2 [×2], C4×D4, C4×Q8, C4×Q8 [×2], C22⋊Q8 [×2], C4.4D4 [×2], C42⋊2C2 [×4], C4⋊Q8, C4×Dic3 [×2], C4×Dic3 [×2], Dic3⋊C4 [×2], Dic3⋊C4 [×6], C4⋊Dic3, D6⋊C4 [×10], C4×C12, C4×C12 [×2], C3×C4⋊C4, C3×C4⋊C4 [×2], C2×Dic6 [×2], S3×C2×C4 [×2], C2×D12, C6×Q8, C22.50C24, C4×Dic6, C42⋊2S3 [×2], C42⋊7S3, C42⋊3S3 [×2], Dic6⋊C4, Dic3⋊5D4, D6⋊Q8 [×2], C4⋊C4⋊S3 [×2], Dic3⋊Q8, C12.23D4, Q8×C12, C42.122D6
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, C22.50C24, C2×C4○D12, Q8.15D6, S3×C4○D4, C42.122D6
Generators and relations
G = < a,b,c,d | a4=b4=d2=1, c6=b2, ab=ba, ac=ca, dad=ab2, cbc-1=dbd=a2b-1, dcd=c5 >
►On 96 points
(1 56 65 23)(2 57 66 24)(3 58 67 13)(4 59 68 14)(5 60 69 15)(6 49 70 16)(7 50 71 17)(8 51 72 18)(9 52 61 19)(10 53 62 20)(11 54 63 21)(12 55 64 22)(25 41 80 91)(26 42 81 92)(27 43 82 93)(28 44 83 94)(29 45 84 95)(30 46 73 96)(31 47 74 85)(32 48 75 86)(33 37 76 87)(34 38 77 88)(35 39 78 89)(36 40 79 90)
(1 42 7 48)(2 87 8 93)(3 44 9 38)(4 89 10 95)(5 46 11 40)(6 91 12 85)(13 28 19 34)(14 78 20 84)(15 30 21 36)(16 80 22 74)(17 32 23 26)(18 82 24 76)(25 55 31 49)(27 57 33 51)(29 59 35 53)(37 72 43 66)(39 62 45 68)(41 64 47 70)(50 75 56 81)(52 77 58 83)(54 79 60 73)(61 88 67 94)(63 90 69 96)(65 92 71 86)
(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 41)(2 46)(3 39)(4 44)(5 37)(6 42)(7 47)(8 40)(9 45)(10 38)(11 43)(12 48)(13 29)(14 34)(15 27)(16 32)(17 25)(18 30)(19 35)(20 28)(21 33)(22 26)(23 31)(24 36)(49 75)(50 80)(51 73)(52 78)(53 83)(54 76)(55 81)(56 74)(57 79)(58 84)(59 77)(60 82)(61 95)(62 88)(63 93)(64 86)(65 91)(66 96)(67 89)(68 94)(69 87)(70 92)(71 85)(72 90)
G:=sub<Sym(96)| (1,56,65,23)(2,57,66,24)(3,58,67,13)(4,59,68,14)(5,60,69,15)(6,49,70,16)(7,50,71,17)(8,51,72,18)(9,52,61,19)(10,53,62,20)(11,54,63,21)(12,55,64,22)(25,41,80,91)(26,42,81,92)(27,43,82,93)(28,44,83,94)(29,45,84,95)(30,46,73,96)(31,47,74,85)(32,48,75,86)(33,37,76,87)(34,38,77,88)(35,39,78,89)(36,40,79,90), (1,42,7,48)(2,87,8,93)(3,44,9,38)(4,89,10,95)(5,46,11,40)(6,91,12,85)(13,28,19,34)(14,78,20,84)(15,30,21,36)(16,80,22,74)(17,32,23,26)(18,82,24,76)(25,55,31,49)(27,57,33,51)(29,59,35,53)(37,72,43,66)(39,62,45,68)(41,64,47,70)(50,75,56,81)(52,77,58,83)(54,79,60,73)(61,88,67,94)(63,90,69,96)(65,92,71,86), (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), (1,41)(2,46)(3,39)(4,44)(5,37)(6,42)(7,47)(8,40)(9,45)(10,38)(11,43)(12,48)(13,29)(14,34)(15,27)(16,32)(17,25)(18,30)(19,35)(20,28)(21,33)(22,26)(23,31)(24,36)(49,75)(50,80)(51,73)(52,78)(53,83)(54,76)(55,81)(56,74)(57,79)(58,84)(59,77)(60,82)(61,95)(62,88)(63,93)(64,86)(65,91)(66,96)(67,89)(68,94)(69,87)(70,92)(71,85)(72,90)>;
G:=Group( (1,56,65,23)(2,57,66,24)(3,58,67,13)(4,59,68,14)(5,60,69,15)(6,49,70,16)(7,50,71,17)(8,51,72,18)(9,52,61,19)(10,53,62,20)(11,54,63,21)(12,55,64,22)(25,41,80,91)(26,42,81,92)(27,43,82,93)(28,44,83,94)(29,45,84,95)(30,46,73,96)(31,47,74,85)(32,48,75,86)(33,37,76,87)(34,38,77,88)(35,39,78,89)(36,40,79,90), (1,42,7,48)(2,87,8,93)(3,44,9,38)(4,89,10,95)(5,46,11,40)(6,91,12,85)(13,28,19,34)(14,78,20,84)(15,30,21,36)(16,80,22,74)(17,32,23,26)(18,82,24,76)(25,55,31,49)(27,57,33,51)(29,59,35,53)(37,72,43,66)(39,62,45,68)(41,64,47,70)(50,75,56,81)(52,77,58,83)(54,79,60,73)(61,88,67,94)(63,90,69,96)(65,92,71,86), (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), (1,41)(2,46)(3,39)(4,44)(5,37)(6,42)(7,47)(8,40)(9,45)(10,38)(11,43)(12,48)(13,29)(14,34)(15,27)(16,32)(17,25)(18,30)(19,35)(20,28)(21,33)(22,26)(23,31)(24,36)(49,75)(50,80)(51,73)(52,78)(53,83)(54,76)(55,81)(56,74)(57,79)(58,84)(59,77)(60,82)(61,95)(62,88)(63,93)(64,86)(65,91)(66,96)(67,89)(68,94)(69,87)(70,92)(71,85)(72,90) );
G=PermutationGroup([(1,56,65,23),(2,57,66,24),(3,58,67,13),(4,59,68,14),(5,60,69,15),(6,49,70,16),(7,50,71,17),(8,51,72,18),(9,52,61,19),(10,53,62,20),(11,54,63,21),(12,55,64,22),(25,41,80,91),(26,42,81,92),(27,43,82,93),(28,44,83,94),(29,45,84,95),(30,46,73,96),(31,47,74,85),(32,48,75,86),(33,37,76,87),(34,38,77,88),(35,39,78,89),(36,40,79,90)], [(1,42,7,48),(2,87,8,93),(3,44,9,38),(4,89,10,95),(5,46,11,40),(6,91,12,85),(13,28,19,34),(14,78,20,84),(15,30,21,36),(16,80,22,74),(17,32,23,26),(18,82,24,76),(25,55,31,49),(27,57,33,51),(29,59,35,53),(37,72,43,66),(39,62,45,68),(41,64,47,70),(50,75,56,81),(52,77,58,83),(54,79,60,73),(61,88,67,94),(63,90,69,96),(65,92,71,86)], [(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)], [(1,41),(2,46),(3,39),(4,44),(5,37),(6,42),(7,47),(8,40),(9,45),(10,38),(11,43),(12,48),(13,29),(14,34),(15,27),(16,32),(17,25),(18,30),(19,35),(20,28),(21,33),(22,26),(23,31),(24,36),(49,75),(50,80),(51,73),(52,78),(53,83),(54,76),(55,81),(56,74),(57,79),(58,84),(59,77),(60,82),(61,95),(62,88),(63,93),(64,86),(65,91),(66,96),(67,89),(68,94),(69,87),(70,92),(71,85),(72,90)])
Matrix representation ►G ⊆ GL4(𝔽13) generated by
| 3 | 6 | 0 | 0 |
| 7 | 10 | 0 | 0 |
| 0 | 0 | 5 | 0 |
| 0 | 0 | 0 | 5 |
| 8 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 8 | 1 |
| 8 | 8 | 0 | 0 |
| 5 | 0 | 0 | 0 |
| 0 | 0 | 12 | 3 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 12 | 12 | 0 | 0 |
| 0 | 0 | 1 | 10 |
| 0 | 0 | 0 | 12 |
G:=sub<GL(4,GF(13))| [3,7,0,0,6,10,0,0,0,0,5,0,0,0,0,5],[8,0,0,0,0,8,0,0,0,0,12,8,0,0,0,1],[8,5,0,0,8,0,0,0,0,0,12,0,0,0,3,1],[1,12,0,0,0,12,0,0,0,0,1,0,0,0,10,12] >;
45 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | ··· | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | 4Q | 4R | 4S | 6A | 6B | 6C | 12A | 12B | 12C | 12D | 12E | ··· | 12P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 12 | 12 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 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 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | |||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D6 | D6 | D6 | C4○D4 | C4○D4 | C4○D12 | 2- (1+4) | Q8.15D6 | S3×C4○D4 |
| kernel | C42.122D6 | C4×Dic6 | C42⋊2S3 | C42⋊7S3 | C42⋊3S3 | Dic6⋊C4 | Dic3⋊5D4 | D6⋊Q8 | C4⋊C4⋊S3 | Dic3⋊Q8 | C12.23D4 | Q8×C12 | C4×Q8 | C42 | C4⋊C4 | C2×Q8 | Dic3 | C12 | C4 | C6 | C2 | C2 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 3 | 3 | 1 | 4 | 4 | 8 | 1 | 2 | 2 |
In GAP, Magma, Sage, TeX
C_4^2._{122}D_6 % in TeX
G:=Group("C4^2.122D6"); // GroupNames label
G:=SmallGroup(192,1127);
// by ID
G=gap.SmallGroup(192,1127);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,232,758,100,794,136,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=d^2=1,c^6=b^2,a*b=b*a,a*c=c*a,d*a*d=a*b^2,c*b*c^-1=d*b*d=a^2*b^-1,d*c*d=c^5>;
// generators/relations