metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.153D6, C6.1332+ (1+4), (C4×D12)⋊48C2, C4⋊C4.209D6, C42.C2⋊9S3, C12⋊D4⋊33C2, Dic3⋊5D4⋊37C2, D6.32(C4○D4), D6.D4⋊35C2, C2.58(D4○D12), (C2×C12).91C23, (C2×C6).239C24, D6⋊C4.41C22, C12.130(C4○D4), (C4×C12).198C22, C4.39(Q8⋊3S3), (C2×D12).268C22, Dic3⋊C4.54C22, C4⋊Dic3.315C22, C22.260(S3×C23), (C22×S3).104C23, (C4×Dic3).145C22, (C2×Dic3).124C23, C3⋊10(C22.47C24), (S3×C4⋊C4)⋊39C2, C4⋊C4⋊7S3⋊38C2, C4⋊C4⋊S3⋊37C2, C2.90(S3×C4○D4), C6.201(C2×C4○D4), (S3×C2×C4).129C22, (C2×C4).82(C22×S3), (C3×C42.C2)⋊12C2, C2.24(C2×Q8⋊3S3), (C3×C4⋊C4).194C22, SmallGroup(192,1254)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 656 in 238 conjugacy classes, 97 normal (43 characteristic)
C1, C2 [×3], C2 [×5], C3, C4 [×2], C4 [×10], C22, C22 [×13], S3 [×5], C6 [×3], C2×C4 [×3], C2×C4 [×4], C2×C4 [×12], D4 [×10], C23 [×4], Dic3 [×4], C12 [×2], C12 [×6], D6 [×2], D6 [×11], C2×C6, C42, C42 [×2], C22⋊C4 [×10], C4⋊C4 [×2], C4⋊C4 [×4], C4⋊C4 [×4], C22×C4 [×6], C2×D4 [×6], C4×S3 [×8], D12 [×10], C2×Dic3 [×2], C2×Dic3 [×2], C2×C12 [×3], C2×C12 [×4], C22×S3 [×2], C22×S3 [×2], C2×C4⋊C4, C42⋊C2, C4×D4 [×4], C4⋊D4 [×4], C22.D4 [×2], C42.C2, C42⋊2C2 [×2], C4×Dic3 [×2], Dic3⋊C4 [×2], C4⋊Dic3 [×2], D6⋊C4 [×2], D6⋊C4 [×8], C4×C12, C3×C4⋊C4 [×2], C3×C4⋊C4 [×4], S3×C2×C4 [×2], S3×C2×C4 [×4], C2×D12 [×2], C2×D12 [×4], C22.47C24, C4×D12 [×2], S3×C4⋊C4, C4⋊C4⋊7S3, Dic3⋊5D4 [×2], D6.D4 [×2], C12⋊D4 [×2], C12⋊D4 [×2], C4⋊C4⋊S3 [×2], C3×C42.C2, C42.153D6
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), Q8⋊3S3 [×2], S3×C23, C22.47C24, C2×Q8⋊3S3, S3×C4○D4, D4○D12, C42.153D6
Generators and relations
G = < a,b,c,d | a4=b4=1, c6=d2=a2b2, ab=ba, cac-1=dad-1=a-1, cbc-1=a2b-1, dbd-1=a2b, dcd-1=c5 >
►On 96 points
(1 75 70 91)(2 92 71 76)(3 77 72 93)(4 94 61 78)(5 79 62 95)(6 96 63 80)(7 81 64 85)(8 86 65 82)(9 83 66 87)(10 88 67 84)(11 73 68 89)(12 90 69 74)(13 41 25 55)(14 56 26 42)(15 43 27 57)(16 58 28 44)(17 45 29 59)(18 60 30 46)(19 47 31 49)(20 50 32 48)(21 37 33 51)(22 52 34 38)(23 39 35 53)(24 54 36 40)
(1 46 64 54)(2 41 65 49)(3 48 66 56)(4 43 67 51)(5 38 68 58)(6 45 69 53)(7 40 70 60)(8 47 71 55)(9 42 72 50)(10 37 61 57)(11 44 62 52)(12 39 63 59)(13 86 31 76)(14 93 32 83)(15 88 33 78)(16 95 34 73)(17 90 35 80)(18 85 36 75)(19 92 25 82)(20 87 26 77)(21 94 27 84)(22 89 28 79)(23 96 29 74)(24 91 30 81)
(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 32 7 26)(2 25 8 31)(3 30 9 36)(4 35 10 29)(5 28 11 34)(6 33 12 27)(13 65 19 71)(14 70 20 64)(15 63 21 69)(16 68 22 62)(17 61 23 67)(18 66 24 72)(37 90 43 96)(38 95 44 89)(39 88 45 94)(40 93 46 87)(41 86 47 92)(42 91 48 85)(49 76 55 82)(50 81 56 75)(51 74 57 80)(52 79 58 73)(53 84 59 78)(54 77 60 83)
G:=sub<Sym(96)| (1,75,70,91)(2,92,71,76)(3,77,72,93)(4,94,61,78)(5,79,62,95)(6,96,63,80)(7,81,64,85)(8,86,65,82)(9,83,66,87)(10,88,67,84)(11,73,68,89)(12,90,69,74)(13,41,25,55)(14,56,26,42)(15,43,27,57)(16,58,28,44)(17,45,29,59)(18,60,30,46)(19,47,31,49)(20,50,32,48)(21,37,33,51)(22,52,34,38)(23,39,35,53)(24,54,36,40), (1,46,64,54)(2,41,65,49)(3,48,66,56)(4,43,67,51)(5,38,68,58)(6,45,69,53)(7,40,70,60)(8,47,71,55)(9,42,72,50)(10,37,61,57)(11,44,62,52)(12,39,63,59)(13,86,31,76)(14,93,32,83)(15,88,33,78)(16,95,34,73)(17,90,35,80)(18,85,36,75)(19,92,25,82)(20,87,26,77)(21,94,27,84)(22,89,28,79)(23,96,29,74)(24,91,30,81), (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), (1,32,7,26)(2,25,8,31)(3,30,9,36)(4,35,10,29)(5,28,11,34)(6,33,12,27)(13,65,19,71)(14,70,20,64)(15,63,21,69)(16,68,22,62)(17,61,23,67)(18,66,24,72)(37,90,43,96)(38,95,44,89)(39,88,45,94)(40,93,46,87)(41,86,47,92)(42,91,48,85)(49,76,55,82)(50,81,56,75)(51,74,57,80)(52,79,58,73)(53,84,59,78)(54,77,60,83)>;
G:=Group( (1,75,70,91)(2,92,71,76)(3,77,72,93)(4,94,61,78)(5,79,62,95)(6,96,63,80)(7,81,64,85)(8,86,65,82)(9,83,66,87)(10,88,67,84)(11,73,68,89)(12,90,69,74)(13,41,25,55)(14,56,26,42)(15,43,27,57)(16,58,28,44)(17,45,29,59)(18,60,30,46)(19,47,31,49)(20,50,32,48)(21,37,33,51)(22,52,34,38)(23,39,35,53)(24,54,36,40), (1,46,64,54)(2,41,65,49)(3,48,66,56)(4,43,67,51)(5,38,68,58)(6,45,69,53)(7,40,70,60)(8,47,71,55)(9,42,72,50)(10,37,61,57)(11,44,62,52)(12,39,63,59)(13,86,31,76)(14,93,32,83)(15,88,33,78)(16,95,34,73)(17,90,35,80)(18,85,36,75)(19,92,25,82)(20,87,26,77)(21,94,27,84)(22,89,28,79)(23,96,29,74)(24,91,30,81), (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), (1,32,7,26)(2,25,8,31)(3,30,9,36)(4,35,10,29)(5,28,11,34)(6,33,12,27)(13,65,19,71)(14,70,20,64)(15,63,21,69)(16,68,22,62)(17,61,23,67)(18,66,24,72)(37,90,43,96)(38,95,44,89)(39,88,45,94)(40,93,46,87)(41,86,47,92)(42,91,48,85)(49,76,55,82)(50,81,56,75)(51,74,57,80)(52,79,58,73)(53,84,59,78)(54,77,60,83) );
G=PermutationGroup([(1,75,70,91),(2,92,71,76),(3,77,72,93),(4,94,61,78),(5,79,62,95),(6,96,63,80),(7,81,64,85),(8,86,65,82),(9,83,66,87),(10,88,67,84),(11,73,68,89),(12,90,69,74),(13,41,25,55),(14,56,26,42),(15,43,27,57),(16,58,28,44),(17,45,29,59),(18,60,30,46),(19,47,31,49),(20,50,32,48),(21,37,33,51),(22,52,34,38),(23,39,35,53),(24,54,36,40)], [(1,46,64,54),(2,41,65,49),(3,48,66,56),(4,43,67,51),(5,38,68,58),(6,45,69,53),(7,40,70,60),(8,47,71,55),(9,42,72,50),(10,37,61,57),(11,44,62,52),(12,39,63,59),(13,86,31,76),(14,93,32,83),(15,88,33,78),(16,95,34,73),(17,90,35,80),(18,85,36,75),(19,92,25,82),(20,87,26,77),(21,94,27,84),(22,89,28,79),(23,96,29,74),(24,91,30,81)], [(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)], [(1,32,7,26),(2,25,8,31),(3,30,9,36),(4,35,10,29),(5,28,11,34),(6,33,12,27),(13,65,19,71),(14,70,20,64),(15,63,21,69),(16,68,22,62),(17,61,23,67),(18,66,24,72),(37,90,43,96),(38,95,44,89),(39,88,45,94),(40,93,46,87),(41,86,47,92),(42,91,48,85),(49,76,55,82),(50,81,56,75),(51,74,57,80),(52,79,58,73),(53,84,59,78),(54,77,60,83)])
Matrix representation ►G ⊆ GL6(𝔽13)
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 5 | 0 | 0 |
| 0 | 0 | 5 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 | 0 |
| 0 | 0 | 0 | 0 | 11 | 8 |
| 12 | 12 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 0 | 0 | 0 |
| 0 | 0 | 0 | 8 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 8 |
| 0 | 0 | 0 | 0 | 3 | 1 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 0 | 0 | 0 |
| 0 | 0 | 0 | 5 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 | 0 |
| 0 | 0 | 0 | 0 | 0 | 5 |
G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,5,0,0,0,0,5,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,5,11,0,0,0,0,0,8],[12,1,0,0,0,0,12,0,0,0,0,0,0,0,5,0,0,0,0,0,0,8,0,0,0,0,0,0,12,3,0,0,0,0,8,1],[1,0,0,0,0,0,1,12,0,0,0,0,0,0,8,0,0,0,0,0,0,5,0,0,0,0,0,0,5,0,0,0,0,0,0,5] >;
39 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 3 | 4A | 4B | 4C | 4D | 4E | ··· | 4I | 4J | ··· | 4O | 4P | 6A | 6B | 6C | 12A | ··· | 12F | 12G | 12H | 12I | 12J |
| order | 1 | 2 | 2 | 2 | 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 | 6 | 6 | 12 | 12 | 12 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 6 | ··· | 6 | 12 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
39 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D6 | D6 | C4○D4 | C4○D4 | 2+ (1+4) | Q8⋊3S3 | S3×C4○D4 | D4○D12 |
| kernel | C42.153D6 | C4×D12 | S3×C4⋊C4 | C4⋊C4⋊7S3 | Dic3⋊5D4 | D6.D4 | C12⋊D4 | C4⋊C4⋊S3 | C3×C42.C2 | C42.C2 | C42 | C4⋊C4 | C12 | D6 | C6 | C4 | C2 | C2 |
| # reps | 1 | 2 | 1 | 1 | 2 | 2 | 4 | 2 | 1 | 1 | 1 | 6 | 4 | 4 | 1 | 2 | 2 | 2 |
In GAP, Magma, Sage, TeX
C_4^2._{153}D_6 % in TeX
G:=Group("C4^2.153D6"); // GroupNames label
G:=SmallGroup(192,1254);
// by ID
G=gap.SmallGroup(192,1254);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,387,100,1571,185,192,6278]);
// 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=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