metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.155D6, C6.1342+ (1+4), C4⋊C4.113D6, C12⋊D4⋊34C2, C4⋊D12⋊15C2, C42.C2⋊11S3, Dic3⋊5D4⋊38C2, C42⋊2S3⋊22C2, D6.D4⋊36C2, C2.59(D4○D12), (C2×C6).241C24, C12.131(C4○D4), (C4×C12).200C22, (C2×C12).189C23, D6⋊C4.112C22, C4.20(Q8⋊3S3), (C2×D12).166C22, C22.262(S3×C23), Dic3⋊C4.124C22, (C22×S3).106C23, C3⋊6(C22.34C24), (C2×Dic3).261C23, (C4×Dic3).146C22, C6.118(C2×C4○D4), (S3×C2×C4).131C22, (C3×C42.C2)⋊14C2, C2.25(C2×Q8⋊3S3), (C3×C4⋊C4).196C22, (C2×C4).594(C22×S3), SmallGroup(192,1256)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 736 in 240 conjugacy classes, 95 normal (19 characteristic)
C1, C2, C2 [×2], C2 [×5], C3, C4 [×2], C4 [×9], C22, C22 [×15], S3 [×5], C6, C6 [×2], C2×C4, C2×C4 [×6], C2×C4 [×9], D4 [×12], C23 [×5], Dic3 [×3], C12 [×2], C12 [×6], D6 [×15], C2×C6, C42, C42, C22⋊C4 [×10], C4⋊C4 [×6], C4⋊C4 [×2], C22×C4 [×5], C2×D4 [×10], C4×S3 [×6], D12 [×12], C2×Dic3, C2×Dic3 [×2], C2×C12, C2×C12 [×6], C22×S3, C22×S3 [×4], C42⋊C2, C4×D4 [×2], C4⋊D4 [×6], C22.D4 [×4], C42.C2, C4⋊1D4, C4×Dic3, Dic3⋊C4 [×2], D6⋊C4 [×10], C4×C12, C3×C4⋊C4 [×6], S3×C2×C4, S3×C2×C4 [×4], C2×D12 [×10], C22.34C24, C42⋊2S3, C4⋊D12, Dic3⋊5D4 [×2], D6.D4 [×4], C12⋊D4 [×6], C3×C42.C2, C42.155D6
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], Q8⋊3S3 [×2], S3×C23, C22.34C24, C2×Q8⋊3S3, D4○D12 [×2], C42.155D6
Generators and relations
G = < a,b,c,d | a4=b4=1, c6=d2=a2, ab=ba, cac-1=ab2, dad-1=a-1b2, cbc-1=a2b, dbd-1=a2b-1, dcd-1=c5 >
►On 96 points
(1 84 7 78)(2 59 8 53)(3 74 9 80)(4 49 10 55)(5 76 11 82)(6 51 12 57)(13 96 19 90)(14 48 20 42)(15 86 21 92)(16 38 22 44)(17 88 23 94)(18 40 24 46)(25 95 31 89)(26 47 32 41)(27 85 33 91)(28 37 34 43)(29 87 35 93)(30 39 36 45)(50 72 56 66)(52 62 58 68)(54 64 60 70)(61 83 67 77)(63 73 69 79)(65 75 71 81)
(1 18 62 31)(2 13 63 26)(3 20 64 33)(4 15 65 28)(5 22 66 35)(6 17 67 30)(7 24 68 25)(8 19 69 32)(9 14 70 27)(10 21 71 34)(11 16 72 29)(12 23 61 36)(37 49 86 75)(38 56 87 82)(39 51 88 77)(40 58 89 84)(41 53 90 79)(42 60 91 74)(43 55 92 81)(44 50 93 76)(45 57 94 83)(46 52 95 78)(47 59 96 73)(48 54 85 80)
(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 83 7 77)(2 76 8 82)(3 81 9 75)(4 74 10 80)(5 79 11 73)(6 84 12 78)(13 87 19 93)(14 92 20 86)(15 85 21 91)(16 90 22 96)(17 95 23 89)(18 88 24 94)(25 45 31 39)(26 38 32 44)(27 43 33 37)(28 48 34 42)(29 41 35 47)(30 46 36 40)(49 64 55 70)(50 69 56 63)(51 62 57 68)(52 67 58 61)(53 72 59 66)(54 65 60 71)
G:=sub<Sym(96)| (1,84,7,78)(2,59,8,53)(3,74,9,80)(4,49,10,55)(5,76,11,82)(6,51,12,57)(13,96,19,90)(14,48,20,42)(15,86,21,92)(16,38,22,44)(17,88,23,94)(18,40,24,46)(25,95,31,89)(26,47,32,41)(27,85,33,91)(28,37,34,43)(29,87,35,93)(30,39,36,45)(50,72,56,66)(52,62,58,68)(54,64,60,70)(61,83,67,77)(63,73,69,79)(65,75,71,81), (1,18,62,31)(2,13,63,26)(3,20,64,33)(4,15,65,28)(5,22,66,35)(6,17,67,30)(7,24,68,25)(8,19,69,32)(9,14,70,27)(10,21,71,34)(11,16,72,29)(12,23,61,36)(37,49,86,75)(38,56,87,82)(39,51,88,77)(40,58,89,84)(41,53,90,79)(42,60,91,74)(43,55,92,81)(44,50,93,76)(45,57,94,83)(46,52,95,78)(47,59,96,73)(48,54,85,80), (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), (1,83,7,77)(2,76,8,82)(3,81,9,75)(4,74,10,80)(5,79,11,73)(6,84,12,78)(13,87,19,93)(14,92,20,86)(15,85,21,91)(16,90,22,96)(17,95,23,89)(18,88,24,94)(25,45,31,39)(26,38,32,44)(27,43,33,37)(28,48,34,42)(29,41,35,47)(30,46,36,40)(49,64,55,70)(50,69,56,63)(51,62,57,68)(52,67,58,61)(53,72,59,66)(54,65,60,71)>;
G:=Group( (1,84,7,78)(2,59,8,53)(3,74,9,80)(4,49,10,55)(5,76,11,82)(6,51,12,57)(13,96,19,90)(14,48,20,42)(15,86,21,92)(16,38,22,44)(17,88,23,94)(18,40,24,46)(25,95,31,89)(26,47,32,41)(27,85,33,91)(28,37,34,43)(29,87,35,93)(30,39,36,45)(50,72,56,66)(52,62,58,68)(54,64,60,70)(61,83,67,77)(63,73,69,79)(65,75,71,81), (1,18,62,31)(2,13,63,26)(3,20,64,33)(4,15,65,28)(5,22,66,35)(6,17,67,30)(7,24,68,25)(8,19,69,32)(9,14,70,27)(10,21,71,34)(11,16,72,29)(12,23,61,36)(37,49,86,75)(38,56,87,82)(39,51,88,77)(40,58,89,84)(41,53,90,79)(42,60,91,74)(43,55,92,81)(44,50,93,76)(45,57,94,83)(46,52,95,78)(47,59,96,73)(48,54,85,80), (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), (1,83,7,77)(2,76,8,82)(3,81,9,75)(4,74,10,80)(5,79,11,73)(6,84,12,78)(13,87,19,93)(14,92,20,86)(15,85,21,91)(16,90,22,96)(17,95,23,89)(18,88,24,94)(25,45,31,39)(26,38,32,44)(27,43,33,37)(28,48,34,42)(29,41,35,47)(30,46,36,40)(49,64,55,70)(50,69,56,63)(51,62,57,68)(52,67,58,61)(53,72,59,66)(54,65,60,71) );
G=PermutationGroup([(1,84,7,78),(2,59,8,53),(3,74,9,80),(4,49,10,55),(5,76,11,82),(6,51,12,57),(13,96,19,90),(14,48,20,42),(15,86,21,92),(16,38,22,44),(17,88,23,94),(18,40,24,46),(25,95,31,89),(26,47,32,41),(27,85,33,91),(28,37,34,43),(29,87,35,93),(30,39,36,45),(50,72,56,66),(52,62,58,68),(54,64,60,70),(61,83,67,77),(63,73,69,79),(65,75,71,81)], [(1,18,62,31),(2,13,63,26),(3,20,64,33),(4,15,65,28),(5,22,66,35),(6,17,67,30),(7,24,68,25),(8,19,69,32),(9,14,70,27),(10,21,71,34),(11,16,72,29),(12,23,61,36),(37,49,86,75),(38,56,87,82),(39,51,88,77),(40,58,89,84),(41,53,90,79),(42,60,91,74),(43,55,92,81),(44,50,93,76),(45,57,94,83),(46,52,95,78),(47,59,96,73),(48,54,85,80)], [(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)], [(1,83,7,77),(2,76,8,82),(3,81,9,75),(4,74,10,80),(5,79,11,73),(6,84,12,78),(13,87,19,93),(14,92,20,86),(15,85,21,91),(16,90,22,96),(17,95,23,89),(18,88,24,94),(25,45,31,39),(26,38,32,44),(27,43,33,37),(28,48,34,42),(29,41,35,47),(30,46,36,40),(49,64,55,70),(50,69,56,63),(51,62,57,68),(52,67,58,61),(53,72,59,66),(54,65,60,71)])
Matrix representation ►G ⊆ GL8(𝔽13)
| 0 | 12 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 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 | 1 | 2 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 | 0 | 2 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 12 |
| 0 | 0 | 0 | 0 | 0 | 1 | 1 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 12 | 0 | 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 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 8 | 0 | 0 | 0 | 0 | 0 | 0 |
| 8 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 5 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 | 0 | 0 | 8 |
| 0 | 0 | 0 | 0 | 0 | 5 | 5 | 0 |
| 8 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 | 0 | 0 | 3 |
| 0 | 0 | 0 | 0 | 0 | 8 | 3 | 0 |
| 0 | 0 | 0 | 0 | 0 | 5 | 5 | 0 |
| 0 | 0 | 0 | 0 | 5 | 0 | 0 | 8 |
G:=sub<GL(8,GF(13))| [0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,12,1,0,0,0,0,0,1,0,0,1,0,0,0,0,2,0,0,1,0,0,0,0,0,2,12,0],[0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0],[0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,8,5,0,0,0,0,0,5,0,0,5,0,0,0,0,0,0,0,5,0,0,0,0,0,0,8,0],[8,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,0,5,0,0,5,0,0,0,0,0,8,5,0,0,0,0,0,0,3,5,0,0,0,0,0,3,0,0,8] >;
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | ··· | 2H | 3 | 4A | 4B | 4C | ··· | 4H | 4I | 4J | 4K | 4L | 4M | 6A | 6B | 6C | 12A | ··· | 12F | 12G | 12H | 12I | 12J |
| order | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 3 | 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 | 6 | 6 | 6 | 6 | 12 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D6 | D6 | C4○D4 | 2+ (1+4) | Q8⋊3S3 | D4○D12 |
| kernel | C42.155D6 | C42⋊2S3 | C4⋊D12 | Dic3⋊5D4 | D6.D4 | C12⋊D4 | C3×C42.C2 | C42.C2 | C42 | C4⋊C4 | C12 | C6 | C4 | C2 |
| # reps | 1 | 1 | 1 | 2 | 4 | 6 | 1 | 1 | 1 | 6 | 4 | 2 | 2 | 4 |
In GAP, Magma, Sage, TeX
C_4^2._{155}D_6 % in TeX
G:=Group("C4^2.155D6"); // GroupNames label
G:=SmallGroup(192,1256);
// by ID
G=gap.SmallGroup(192,1256);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,758,219,184,675,570,80,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^6=d^2=a^2,a*b=b*a,c*a*c^-1=a*b^2,d*a*d^-1=a^-1*b^2,c*b*c^-1=a^2*b,d*b*d^-1=a^2*b^-1,d*c*d^-1=c^5>;
// generators/relations