metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.165D6, C6.1032- (1+4), C6.1432+ (1+4), C12⋊Q8⋊42C2, C4⋊C4.120D6, C12⋊2Q8⋊9C2, C42⋊3S3⋊2C2, C42⋊2C2⋊9S3, C4.D12⋊43C2, D6⋊Q8⋊46C2, (C4×C12).9C22, C22⋊C4.42D6, C2.68(D4○D12), Dic3.Q8⋊40C2, (C2×C6).256C24, (C2×C12).97C23, D6⋊C4.48C22, C2.67(Q8○D12), C12.3Q8⋊41C2, C23.9D6.5C2, C23.8D6⋊47C2, C4⋊Dic3.55C22, C23.72(C22×S3), (C22×C6).70C23, Dic3.D4⋊48C2, Dic3⋊C4.11C22, C22.277(S3×C23), C23.11D6.5C2, (C2×Dic6).43C22, C22.D12.3C2, (C22×S3).115C23, C3⋊5(C22.57C24), (C4×Dic3).153C22, (C2×Dic3).132C23, C6.D4.70C22, (C22×Dic3).155C22, C4⋊C4⋊S3⋊45C2, (S3×C2×C4).137C22, (C3×C42⋊2C2)⋊11C2, (C3×C4⋊C4).207C22, (C2×C4).212(C22×S3), (C2×C3⋊D4).76C22, (C3×C22⋊C4).81C22, SmallGroup(192,1271)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 480 in 196 conjugacy classes, 91 normal (all characteristic)
C1, C2 [×3], C2 [×2], C3, C4 [×13], C22, C22 [×6], S3, C6 [×3], C6, C2×C4 [×6], C2×C4 [×9], D4, Q8 [×3], C23, C23, Dic3 [×7], C12 [×6], D6 [×3], C2×C6, C2×C6 [×3], C42, C42 [×2], C22⋊C4 [×3], C22⋊C4 [×7], C4⋊C4 [×3], C4⋊C4 [×13], C22×C4 [×2], C2×D4, C2×Q8 [×3], Dic6 [×3], C4×S3, C2×Dic3 [×7], C2×Dic3, C3⋊D4, C2×C12 [×6], C22×S3, C22×C6, C22⋊Q8 [×4], C22.D4 [×2], C4.4D4, C42.C2 [×2], C42⋊2C2, C42⋊2C2 [×3], C4⋊Q8 [×2], C4×Dic3 [×2], Dic3⋊C4 [×7], C4⋊Dic3 [×6], D6⋊C4 [×5], C6.D4 [×2], C4×C12, C3×C22⋊C4 [×3], C3×C4⋊C4 [×3], C2×Dic6 [×3], S3×C2×C4, C22×Dic3, C2×C3⋊D4, C22.57C24, C12⋊2Q8, C42⋊3S3, Dic3.D4 [×2], C23.8D6, C23.9D6, C23.11D6, C22.D12, C12⋊Q8, Dic3.Q8, C12.3Q8, D6⋊Q8, C4.D12, C4⋊C4⋊S3, C3×C42⋊2C2, C42.165D6
Quotients:
C1, C2 [×15], C22 [×35], S3, C23 [×15], D6 [×7], C24, C22×S3 [×7], 2+ (1+4), 2- (1+4) [×2], S3×C23, C22.57C24, D4○D12, Q8○D12 [×2], C42.165D6
Generators and relations
G = < a,b,c,d | a4=b4=1, c6=d2=a2b2, ab=ba, cac-1=ab2, dad-1=a-1, cbc-1=a2b-1, dbd-1=b-1, dcd-1=c5 >
►On 96 points
(1 69 39 91)(2 86 40 64)(3 71 41 93)(4 88 42 66)(5 61 43 95)(6 90 44 68)(7 63 45 85)(8 92 46 70)(9 65 47 87)(10 94 48 72)(11 67 37 89)(12 96 38 62)(13 77 56 25)(14 32 57 84)(15 79 58 27)(16 34 59 74)(17 81 60 29)(18 36 49 76)(19 83 50 31)(20 26 51 78)(21 73 52 33)(22 28 53 80)(23 75 54 35)(24 30 55 82)
(1 80 45 34)(2 75 46 29)(3 82 47 36)(4 77 48 31)(5 84 37 26)(6 79 38 33)(7 74 39 28)(8 81 40 35)(9 76 41 30)(10 83 42 25)(11 78 43 32)(12 73 44 27)(13 94 50 66)(14 89 51 61)(15 96 52 68)(16 91 53 63)(17 86 54 70)(18 93 55 65)(19 88 56 72)(20 95 57 67)(21 90 58 62)(22 85 59 69)(23 92 60 64)(24 87 49 71)
(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 55 19 49)(14 60 20 54)(15 53 21 59)(16 58 22 52)(17 51 23 57)(18 56 24 50)(25 82 31 76)(26 75 32 81)(27 80 33 74)(28 73 34 79)(29 78 35 84)(30 83 36 77)(37 46 43 40)(38 39 44 45)(41 42 47 48)(61 64 67 70)(62 69 68 63)(65 72 71 66)(85 96 91 90)(86 89 92 95)(87 94 93 88)
G:=sub<Sym(96)| (1,69,39,91)(2,86,40,64)(3,71,41,93)(4,88,42,66)(5,61,43,95)(6,90,44,68)(7,63,45,85)(8,92,46,70)(9,65,47,87)(10,94,48,72)(11,67,37,89)(12,96,38,62)(13,77,56,25)(14,32,57,84)(15,79,58,27)(16,34,59,74)(17,81,60,29)(18,36,49,76)(19,83,50,31)(20,26,51,78)(21,73,52,33)(22,28,53,80)(23,75,54,35)(24,30,55,82), (1,80,45,34)(2,75,46,29)(3,82,47,36)(4,77,48,31)(5,84,37,26)(6,79,38,33)(7,74,39,28)(8,81,40,35)(9,76,41,30)(10,83,42,25)(11,78,43,32)(12,73,44,27)(13,94,50,66)(14,89,51,61)(15,96,52,68)(16,91,53,63)(17,86,54,70)(18,93,55,65)(19,88,56,72)(20,95,57,67)(21,90,58,62)(22,85,59,69)(23,92,60,64)(24,87,49,71), (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)(13,55,19,49)(14,60,20,54)(15,53,21,59)(16,58,22,52)(17,51,23,57)(18,56,24,50)(25,82,31,76)(26,75,32,81)(27,80,33,74)(28,73,34,79)(29,78,35,84)(30,83,36,77)(37,46,43,40)(38,39,44,45)(41,42,47,48)(61,64,67,70)(62,69,68,63)(65,72,71,66)(85,96,91,90)(86,89,92,95)(87,94,93,88)>;
G:=Group( (1,69,39,91)(2,86,40,64)(3,71,41,93)(4,88,42,66)(5,61,43,95)(6,90,44,68)(7,63,45,85)(8,92,46,70)(9,65,47,87)(10,94,48,72)(11,67,37,89)(12,96,38,62)(13,77,56,25)(14,32,57,84)(15,79,58,27)(16,34,59,74)(17,81,60,29)(18,36,49,76)(19,83,50,31)(20,26,51,78)(21,73,52,33)(22,28,53,80)(23,75,54,35)(24,30,55,82), (1,80,45,34)(2,75,46,29)(3,82,47,36)(4,77,48,31)(5,84,37,26)(6,79,38,33)(7,74,39,28)(8,81,40,35)(9,76,41,30)(10,83,42,25)(11,78,43,32)(12,73,44,27)(13,94,50,66)(14,89,51,61)(15,96,52,68)(16,91,53,63)(17,86,54,70)(18,93,55,65)(19,88,56,72)(20,95,57,67)(21,90,58,62)(22,85,59,69)(23,92,60,64)(24,87,49,71), (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)(13,55,19,49)(14,60,20,54)(15,53,21,59)(16,58,22,52)(17,51,23,57)(18,56,24,50)(25,82,31,76)(26,75,32,81)(27,80,33,74)(28,73,34,79)(29,78,35,84)(30,83,36,77)(37,46,43,40)(38,39,44,45)(41,42,47,48)(61,64,67,70)(62,69,68,63)(65,72,71,66)(85,96,91,90)(86,89,92,95)(87,94,93,88) );
G=PermutationGroup([(1,69,39,91),(2,86,40,64),(3,71,41,93),(4,88,42,66),(5,61,43,95),(6,90,44,68),(7,63,45,85),(8,92,46,70),(9,65,47,87),(10,94,48,72),(11,67,37,89),(12,96,38,62),(13,77,56,25),(14,32,57,84),(15,79,58,27),(16,34,59,74),(17,81,60,29),(18,36,49,76),(19,83,50,31),(20,26,51,78),(21,73,52,33),(22,28,53,80),(23,75,54,35),(24,30,55,82)], [(1,80,45,34),(2,75,46,29),(3,82,47,36),(4,77,48,31),(5,84,37,26),(6,79,38,33),(7,74,39,28),(8,81,40,35),(9,76,41,30),(10,83,42,25),(11,78,43,32),(12,73,44,27),(13,94,50,66),(14,89,51,61),(15,96,52,68),(16,91,53,63),(17,86,54,70),(18,93,55,65),(19,88,56,72),(20,95,57,67),(21,90,58,62),(22,85,59,69),(23,92,60,64),(24,87,49,71)], [(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),(13,55,19,49),(14,60,20,54),(15,53,21,59),(16,58,22,52),(17,51,23,57),(18,56,24,50),(25,82,31,76),(26,75,32,81),(27,80,33,74),(28,73,34,79),(29,78,35,84),(30,83,36,77),(37,46,43,40),(38,39,44,45),(41,42,47,48),(61,64,67,70),(62,69,68,63),(65,72,71,66),(85,96,91,90),(86,89,92,95),(87,94,93,88)])
Matrix representation ►G ⊆ GL8(𝔽13)
| 5 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 5 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 9 | 10 |
| 0 | 0 | 0 | 0 | 0 | 0 | 10 | 4 |
| 0 | 0 | 0 | 0 | 4 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 9 | 0 | 0 |
| 10 | 7 | 0 | 0 | 0 | 0 | 0 | 0 |
| 6 | 3 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 10 | 7 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 12 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 5 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 0 | 0 | 0 | 0 | 0 |
| 8 | 8 | 0 | 0 | 0 | 0 | 0 | 0 |
| 5 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 9 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 9 | 10 |
| 0 | 0 | 0 | 0 | 0 | 0 | 10 | 4 |
| 0 | 0 | 5 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 8 | 0 | 0 | 0 | 0 |
| 8 | 8 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 5 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 9 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 4 | 3 |
| 0 | 0 | 0 | 0 | 0 | 0 | 3 | 9 |
G:=sub<GL(8,GF(13))| [5,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,0,4,3,0,0,0,0,0,0,3,9,0,0,0,0,9,10,0,0,0,0,0,0,10,4,0,0],[10,6,0,0,0,0,0,0,7,3,0,0,0,0,0,0,0,0,10,6,0,0,0,0,0,0,7,3,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0],[0,0,8,5,0,0,0,0,0,0,8,0,0,0,0,0,5,8,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,0,0,4,3,0,0,0,0,0,0,3,9,0,0,0,0,0,0,0,0,9,10,0,0,0,0,0,0,10,4],[0,0,8,0,0,0,0,0,0,0,8,5,0,0,0,0,5,0,0,0,0,0,0,0,5,8,0,0,0,0,0,0,0,0,0,0,4,3,0,0,0,0,0,0,3,9,0,0,0,0,0,0,0,0,4,3,0,0,0,0,0,0,3,9] >;
33 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | ··· | 4F | 4G | ··· | 4M | 6A | 6B | 6C | 6D | 12A | ··· | 12F | 12G | 12H | 12I |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | ··· | 4 | 4 | ··· | 4 | 6 | 6 | 6 | 6 | 12 | ··· | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 4 | 12 | 2 | 4 | ··· | 4 | 12 | ··· | 12 | 2 | 2 | 2 | 8 | 4 | ··· | 4 | 8 | 8 | 8 |
33 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 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 | 2+ (1+4) | 2- (1+4) | D4○D12 | Q8○D12 |
| kernel | C42.165D6 | C12⋊2Q8 | C42⋊3S3 | Dic3.D4 | C23.8D6 | C23.9D6 | C23.11D6 | C22.D12 | C12⋊Q8 | Dic3.Q8 | C12.3Q8 | D6⋊Q8 | C4.D12 | C4⋊C4⋊S3 | C3×C42⋊2C2 | C42⋊2C2 | C42 | C22⋊C4 | C4⋊C4 | C6 | C6 | C2 | C2 |
| # reps | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 1 | 2 | 2 | 4 |
In GAP, Magma, Sage, TeX
C_4^2._{165}D_6 % in TeX
G:=Group("C4^2.165D6"); // GroupNames label
G:=SmallGroup(192,1271);
// by ID
G=gap.SmallGroup(192,1271);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,758,219,1571,570,297,80,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^6=d^2=a^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=b^-1,d*c*d^-1=c^5>;
// generators/relations