metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.144D6, C6.922- 1+4, C6.1282+ 1+4, D6⋊3Q8⋊33C2, C12⋊2Q8⋊31C2, (Q8×Dic3)⋊21C2, (D4×Dic3)⋊32C2, (C2×D4).177D6, C4.4D4⋊15S3, (C2×Q8).164D6, C22⋊C4.37D6, C42⋊2S3⋊21C2, D6⋊3D4.13C2, C2.52(D4○D12), (C2×C6).226C24, (C2×C12).82C23, C2.53(Q8○D12), C12.127(C4○D4), C4.39(D4⋊2S3), (C4×C12).189C22, D6⋊C4.111C22, (C6×D4).159C22, C23.8D6⋊42C2, C23.58(C22×S3), (C22×C6).56C23, (C6×Q8).130C22, Dic3.D4⋊42C2, C23.11D6⋊42C2, C23.21D6⋊27C2, Dic3⋊C4.49C22, (C22×S3).98C23, C4⋊Dic3.236C22, C22.247(S3×C23), (C2×Dic6).38C22, C3⋊9(C22.36C24), (C4×Dic3).136C22, (C2×Dic3).116C23, C6.D4.59C22, (C22×Dic3).146C22, C6.94(C2×C4○D4), (C3×C4.4D4)⋊18C2, C2.58(C2×D4⋊2S3), (S3×C2×C4).122C22, (C2×C4).199(C22×S3), (C2×C3⋊D4).64C22, (C3×C22⋊C4).68C22, SmallGroup(192,1241)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.144D6
G = < a,b,c,d | a4=b4=c6=1, d2=b2, ab=ba, cac-1=dad-1=a-1, cbc-1=a2b-1, dbd-1=b-1, dcd-1=b2c-1 >
Subgroups: 528 in 216 conjugacy classes, 95 normal (43 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, Dic3, C12, C12, D6, C2×C6, C2×C6, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, Dic6, C4×S3, C2×Dic3, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×D4, C3×Q8, C22×S3, C22×C6, C42⋊C2, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C4.4D4, C42⋊2C2, C4⋊Q8, C4×Dic3, C4×Dic3, Dic3⋊C4, Dic3⋊C4, C4⋊Dic3, C4⋊Dic3, D6⋊C4, D6⋊C4, C6.D4, C4×C12, C3×C22⋊C4, C2×Dic6, S3×C2×C4, C22×Dic3, C2×C3⋊D4, C6×D4, C6×Q8, C22.36C24, C12⋊2Q8, C42⋊2S3, Dic3.D4, C23.8D6, C23.11D6, C23.21D6, D4×Dic3, D6⋊3D4, Q8×Dic3, D6⋊3Q8, C3×C4.4D4, C42.144D6
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C24, C22×S3, C2×C4○D4, 2+ 1+4, 2- 1+4, D4⋊2S3, S3×C23, C22.36C24, C2×D4⋊2S3, D4○D12, Q8○D12, C42.144D6
►On 96 points
(1 64 19 67)(2 68 20 65)(3 66 21 69)(4 70 22 61)(5 62 23 71)(6 72 24 63)(7 59 32 83)(8 84 33 60)(9 55 34 79)(10 80 35 56)(11 57 36 81)(12 82 31 58)(13 38 88 54)(14 49 89 39)(15 40 90 50)(16 51 85 41)(17 42 86 52)(18 53 87 37)(25 78 45 93)(26 94 46 73)(27 74 47 95)(28 96 48 75)(29 76 43 91)(30 92 44 77)
(1 43 7 50)(2 41 8 30)(3 45 9 52)(4 37 10 26)(5 47 11 54)(6 39 12 28)(13 62 95 57)(14 82 96 72)(15 64 91 59)(16 84 92 68)(17 66 93 55)(18 80 94 70)(19 29 32 40)(20 51 33 44)(21 25 34 42)(22 53 35 46)(23 27 36 38)(24 49 31 48)(56 73 61 87)(58 75 63 89)(60 77 65 85)(67 76 83 90)(69 78 79 86)(71 74 81 88)
(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 85 95 77)(14 76 96 90)(15 89 91 75)(16 74 92 88)(17 87 93 73)(18 78 94 86)(19 24 32 31)(20 36 33 23)(21 22 34 35)(25 46 42 53)(26 52 37 45)(27 44 38 51)(28 50 39 43)(29 48 40 49)(30 54 41 47)(55 56 66 61)(57 60 62 65)(58 64 63 59)(67 72 83 82)(68 81 84 71)(69 70 79 80)
G:=sub<Sym(96)| (1,64,19,67)(2,68,20,65)(3,66,21,69)(4,70,22,61)(5,62,23,71)(6,72,24,63)(7,59,32,83)(8,84,33,60)(9,55,34,79)(10,80,35,56)(11,57,36,81)(12,82,31,58)(13,38,88,54)(14,49,89,39)(15,40,90,50)(16,51,85,41)(17,42,86,52)(18,53,87,37)(25,78,45,93)(26,94,46,73)(27,74,47,95)(28,96,48,75)(29,76,43,91)(30,92,44,77), (1,43,7,50)(2,41,8,30)(3,45,9,52)(4,37,10,26)(5,47,11,54)(6,39,12,28)(13,62,95,57)(14,82,96,72)(15,64,91,59)(16,84,92,68)(17,66,93,55)(18,80,94,70)(19,29,32,40)(20,51,33,44)(21,25,34,42)(22,53,35,46)(23,27,36,38)(24,49,31,48)(56,73,61,87)(58,75,63,89)(60,77,65,85)(67,76,83,90)(69,78,79,86)(71,74,81,88), (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)(13,85,95,77)(14,76,96,90)(15,89,91,75)(16,74,92,88)(17,87,93,73)(18,78,94,86)(19,24,32,31)(20,36,33,23)(21,22,34,35)(25,46,42,53)(26,52,37,45)(27,44,38,51)(28,50,39,43)(29,48,40,49)(30,54,41,47)(55,56,66,61)(57,60,62,65)(58,64,63,59)(67,72,83,82)(68,81,84,71)(69,70,79,80)>;
G:=Group( (1,64,19,67)(2,68,20,65)(3,66,21,69)(4,70,22,61)(5,62,23,71)(6,72,24,63)(7,59,32,83)(8,84,33,60)(9,55,34,79)(10,80,35,56)(11,57,36,81)(12,82,31,58)(13,38,88,54)(14,49,89,39)(15,40,90,50)(16,51,85,41)(17,42,86,52)(18,53,87,37)(25,78,45,93)(26,94,46,73)(27,74,47,95)(28,96,48,75)(29,76,43,91)(30,92,44,77), (1,43,7,50)(2,41,8,30)(3,45,9,52)(4,37,10,26)(5,47,11,54)(6,39,12,28)(13,62,95,57)(14,82,96,72)(15,64,91,59)(16,84,92,68)(17,66,93,55)(18,80,94,70)(19,29,32,40)(20,51,33,44)(21,25,34,42)(22,53,35,46)(23,27,36,38)(24,49,31,48)(56,73,61,87)(58,75,63,89)(60,77,65,85)(67,76,83,90)(69,78,79,86)(71,74,81,88), (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)(13,85,95,77)(14,76,96,90)(15,89,91,75)(16,74,92,88)(17,87,93,73)(18,78,94,86)(19,24,32,31)(20,36,33,23)(21,22,34,35)(25,46,42,53)(26,52,37,45)(27,44,38,51)(28,50,39,43)(29,48,40,49)(30,54,41,47)(55,56,66,61)(57,60,62,65)(58,64,63,59)(67,72,83,82)(68,81,84,71)(69,70,79,80) );
G=PermutationGroup([[(1,64,19,67),(2,68,20,65),(3,66,21,69),(4,70,22,61),(5,62,23,71),(6,72,24,63),(7,59,32,83),(8,84,33,60),(9,55,34,79),(10,80,35,56),(11,57,36,81),(12,82,31,58),(13,38,88,54),(14,49,89,39),(15,40,90,50),(16,51,85,41),(17,42,86,52),(18,53,87,37),(25,78,45,93),(26,94,46,73),(27,74,47,95),(28,96,48,75),(29,76,43,91),(30,92,44,77)], [(1,43,7,50),(2,41,8,30),(3,45,9,52),(4,37,10,26),(5,47,11,54),(6,39,12,28),(13,62,95,57),(14,82,96,72),(15,64,91,59),(16,84,92,68),(17,66,93,55),(18,80,94,70),(19,29,32,40),(20,51,33,44),(21,25,34,42),(22,53,35,46),(23,27,36,38),(24,49,31,48),(56,73,61,87),(58,75,63,89),(60,77,65,85),(67,76,83,90),(69,78,79,86),(71,74,81,88)], [(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),(13,85,95,77),(14,76,96,90),(15,89,91,75),(16,74,92,88),(17,87,93,73),(18,78,94,86),(19,24,32,31),(20,36,33,23),(21,22,34,35),(25,46,42,53),(26,52,37,45),(27,44,38,51),(28,50,39,43),(29,48,40,49),(30,54,41,47),(55,56,66,61),(57,60,62,65),(58,64,63,59),(67,72,83,82),(68,81,84,71),(69,70,79,80)]])
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | ··· | 4O | 6A | 6B | 6C | 6D | 6E | 12A | ··· | 12F | 12G | 12H |
| 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 | 6 | 12 | ··· | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 12 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 12 | ··· | 12 | 2 | 2 | 2 | 8 | 8 | 4 | ··· | 4 | 8 | 8 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D6 | D6 | D6 | D6 | C4○D4 | 2+ 1+4 | 2- 1+4 | D4⋊2S3 | D4○D12 | Q8○D12 |
| kernel | C42.144D6 | C12⋊2Q8 | C42⋊2S3 | Dic3.D4 | C23.8D6 | C23.11D6 | C23.21D6 | D4×Dic3 | D6⋊3D4 | Q8×Dic3 | D6⋊3Q8 | C3×C4.4D4 | C4.4D4 | C42 | C22⋊C4 | C2×D4 | C2×Q8 | C12 | C6 | C6 | C4 | C2 | C2 |
| # reps | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 4 | 1 | 1 | 2 | 2 | 2 |
Matrix representation of C42.144D6 ►in GL8(𝔽13)
| 12 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 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 | 3 | 7 | 0 | 0 |
| 0 | 0 | 0 | 0 | 6 | 10 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 3 | 7 |
| 0 | 0 | 0 | 0 | 0 | 0 | 6 | 10 |
| 8 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 5 | 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 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 12 |
| 0 | 0 | 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(8,GF(13))| [12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,3,6,0,0,0,0,0,0,7,10,0,0,0,0,0,0,0,0,3,6,0,0,0,0,0,0,7,10],[8,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0],[0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0] >;
C42.144D6 in GAP, Magma, Sage, TeX
C_4^2._{144}D_6 % in TeX
G:=Group("C4^2.144D6"); // GroupNames label
G:=SmallGroup(192,1241);
// by ID
G=gap.SmallGroup(192,1241);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,758,100,675,570,409,6278]);
// 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=d*a*d^-1=a^-1,c*b*c^-1=a^2*b^-1,d*b*d^-1=b^-1,d*c*d^-1=b^2*c^-1>;
// generators/relations