metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.164D6, C6.1022- 1+4, C6.1412+ 1+4, C12⋊Q8⋊41C2, C4⋊C4.119D6, C42⋊2C2⋊7S3, C42⋊7S3⋊9C2, D6⋊Q8⋊45C2, C4.D12⋊42C2, (C4×Dic6)⋊16C2, Dic3⋊D4.5C2, C22⋊C4.82D6, D6.D4⋊43C2, C2.66(D4○D12), (C2×C6).254C24, (C4×C12).36C22, 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), Dic3.D4⋊47C2, C23.11D6⋊47C2, C23.21D6⋊31C2, C4⋊Dic3.319C22, C22.275(S3×C23), Dic3⋊C4.127C22, (C22×S3).113C23, (C2×Dic6).256C22, (C4×Dic3).219C22, (C2×Dic3).267C23, 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
Generators and relations for C42.164D6
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 >
Subgroups: 560 in 216 conjugacy classes, 93 normal (91 characteristic)
C1, C2, C2, C3, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, Q8, C23, C23, Dic3, Dic3, C12, D6, C2×C6, C2×C6, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, Dic6, C4×S3, D12, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C22×S3, C22×C6, C42⋊C2, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C42⋊2C2, C42⋊2C2, C4⋊Q8, C4×Dic3, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C6.D4, C4×C12, C3×C22⋊C4, C3×C4⋊C4, C2×Dic6, S3×C2×C4, C2×D12, C22×Dic3, C2×C3⋊D4, C22.36C24, C4×Dic6, C42⋊7S3, Dic3.D4, Dic3⋊4D4, Dic3⋊D4, C23.11D6, C23.21D6, C12⋊Q8, C4⋊C4⋊7S3, D6.D4, D6⋊Q8, C4.D12, C4⋊C4⋊S3, C3×C42⋊2C2, C42.164D6
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C24, C22×S3, C2×C4○D4, 2+ 1+4, 2- 1+4, S3×C23, C22.36C24, S3×C4○D4, D4○D12, Q8○D12, C42.164D6
►On 96 points
(1 50 23 32)(2 57 24 27)(3 52 13 34)(4 59 14 29)(5 54 15 36)(6 49 16 31)(7 56 17 26)(8 51 18 33)(9 58 19 28)(10 53 20 35)(11 60 21 30)(12 55 22 25)(37 83 71 91)(38 78 72 86)(39 73 61 93)(40 80 62 88)(41 75 63 95)(42 82 64 90)(43 77 65 85)(44 84 66 92)(45 79 67 87)(46 74 68 94)(47 81 69 89)(48 76 70 96)
(1 74 7 80)(2 89 8 95)(3 76 9 82)(4 91 10 85)(5 78 11 84)(6 93 12 87)(13 96 19 90)(14 83 20 77)(15 86 21 92)(16 73 22 79)(17 88 23 94)(18 75 24 81)(25 67 31 61)(26 40 32 46)(27 69 33 63)(28 42 34 48)(29 71 35 65)(30 44 36 38)(37 53 43 59)(39 55 45 49)(41 57 47 51)(50 68 56 62)(52 70 58 64)(54 72 60 66)
(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 14 19 20)(15 24 21 18)(16 17 22 23)(25 50 31 56)(26 55 32 49)(27 60 33 54)(28 53 34 59)(29 58 35 52)(30 51 36 57)(37 64 43 70)(38 69 44 63)(39 62 45 68)(40 67 46 61)(41 72 47 66)(42 65 48 71)(73 80 79 74)(75 78 81 84)(76 83 82 77)(85 96 91 90)(86 89 92 95)(87 94 93 88)
G:=sub<Sym(96)| (1,50,23,32)(2,57,24,27)(3,52,13,34)(4,59,14,29)(5,54,15,36)(6,49,16,31)(7,56,17,26)(8,51,18,33)(9,58,19,28)(10,53,20,35)(11,60,21,30)(12,55,22,25)(37,83,71,91)(38,78,72,86)(39,73,61,93)(40,80,62,88)(41,75,63,95)(42,82,64,90)(43,77,65,85)(44,84,66,92)(45,79,67,87)(46,74,68,94)(47,81,69,89)(48,76,70,96), (1,74,7,80)(2,89,8,95)(3,76,9,82)(4,91,10,85)(5,78,11,84)(6,93,12,87)(13,96,19,90)(14,83,20,77)(15,86,21,92)(16,73,22,79)(17,88,23,94)(18,75,24,81)(25,67,31,61)(26,40,32,46)(27,69,33,63)(28,42,34,48)(29,71,35,65)(30,44,36,38)(37,53,43,59)(39,55,45,49)(41,57,47,51)(50,68,56,62)(52,70,58,64)(54,72,60,66), (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)(13,14,19,20)(15,24,21,18)(16,17,22,23)(25,50,31,56)(26,55,32,49)(27,60,33,54)(28,53,34,59)(29,58,35,52)(30,51,36,57)(37,64,43,70)(38,69,44,63)(39,62,45,68)(40,67,46,61)(41,72,47,66)(42,65,48,71)(73,80,79,74)(75,78,81,84)(76,83,82,77)(85,96,91,90)(86,89,92,95)(87,94,93,88)>;
G:=Group( (1,50,23,32)(2,57,24,27)(3,52,13,34)(4,59,14,29)(5,54,15,36)(6,49,16,31)(7,56,17,26)(8,51,18,33)(9,58,19,28)(10,53,20,35)(11,60,21,30)(12,55,22,25)(37,83,71,91)(38,78,72,86)(39,73,61,93)(40,80,62,88)(41,75,63,95)(42,82,64,90)(43,77,65,85)(44,84,66,92)(45,79,67,87)(46,74,68,94)(47,81,69,89)(48,76,70,96), (1,74,7,80)(2,89,8,95)(3,76,9,82)(4,91,10,85)(5,78,11,84)(6,93,12,87)(13,96,19,90)(14,83,20,77)(15,86,21,92)(16,73,22,79)(17,88,23,94)(18,75,24,81)(25,67,31,61)(26,40,32,46)(27,69,33,63)(28,42,34,48)(29,71,35,65)(30,44,36,38)(37,53,43,59)(39,55,45,49)(41,57,47,51)(50,68,56,62)(52,70,58,64)(54,72,60,66), (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)(13,14,19,20)(15,24,21,18)(16,17,22,23)(25,50,31,56)(26,55,32,49)(27,60,33,54)(28,53,34,59)(29,58,35,52)(30,51,36,57)(37,64,43,70)(38,69,44,63)(39,62,45,68)(40,67,46,61)(41,72,47,66)(42,65,48,71)(73,80,79,74)(75,78,81,84)(76,83,82,77)(85,96,91,90)(86,89,92,95)(87,94,93,88) );
G=PermutationGroup([[(1,50,23,32),(2,57,24,27),(3,52,13,34),(4,59,14,29),(5,54,15,36),(6,49,16,31),(7,56,17,26),(8,51,18,33),(9,58,19,28),(10,53,20,35),(11,60,21,30),(12,55,22,25),(37,83,71,91),(38,78,72,86),(39,73,61,93),(40,80,62,88),(41,75,63,95),(42,82,64,90),(43,77,65,85),(44,84,66,92),(45,79,67,87),(46,74,68,94),(47,81,69,89),(48,76,70,96)], [(1,74,7,80),(2,89,8,95),(3,76,9,82),(4,91,10,85),(5,78,11,84),(6,93,12,87),(13,96,19,90),(14,83,20,77),(15,86,21,92),(16,73,22,79),(17,88,23,94),(18,75,24,81),(25,67,31,61),(26,40,32,46),(27,69,33,63),(28,42,34,48),(29,71,35,65),(30,44,36,38),(37,53,43,59),(39,55,45,49),(41,57,47,51),(50,68,56,62),(52,70,58,64),(54,72,60,66)], [(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),(13,14,19,20),(15,24,21,18),(16,17,22,23),(25,50,31,56),(26,55,32,49),(27,60,33,54),(28,53,34,59),(29,58,35,52),(30,51,36,57),(37,64,43,70),(38,69,44,63),(39,62,45,68),(40,67,46,61),(41,72,47,66),(42,65,48,71),(73,80,79,74),(75,78,81,84),(76,83,82,77),(85,96,91,90),(86,89,92,95),(87,94,93,88)]])
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 | C23.21D6 | 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 |
Matrix representation of C42.164D6 ►in 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] >;
C42.164D6 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