metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.152D6, C6.962- (1+4), C4⋊C4.208D6, C42.C2⋊8S3, C4.D12⋊38C2, (C4×Dic6)⋊48C2, (C4×D12).25C2, D6.25(C4○D4), (C2×C12).90C23, (C2×C6).238C24, C2.58(Q8○D12), C12.3Q8⋊35C2, D6.D4.3C2, C12.129(C4○D4), (C4×C12).197C22, D6⋊C4.138C22, C4.38(Q8⋊3S3), (C2×D12).225C22, C4⋊Dic3.243C22, C22.259(S3×C23), Dic3⋊C4.123C22, (C22×S3).103C23, (C4×Dic3).144C22, (C2×Dic6).299C22, (C2×Dic3).123C23, C3⋊10(C22.46C24), (S3×C4⋊C4)⋊38C2, C4⋊C4⋊7S3⋊37C2, C4⋊C4⋊S3⋊36C2, C2.89(S3×C4○D4), C6.200(C2×C4○D4), (S3×C2×C4).128C22, (C2×C4).81(C22×S3), C2.23(C2×Q8⋊3S3), (C3×C42.C2)⋊11C2, (C3×C4⋊C4).193C22, SmallGroup(192,1253)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 496 in 214 conjugacy classes, 97 normal (43 characteristic)
C1, C2 [×3], C2 [×3], C3, C4 [×2], C4 [×12], C22, C22 [×7], S3 [×3], C6 [×3], C2×C4 [×3], C2×C4 [×4], C2×C4 [×14], D4 [×2], Q8 [×2], C23 [×2], Dic3 [×6], C12 [×2], C12 [×6], D6 [×2], D6 [×5], C2×C6, C42, C42 [×4], C22⋊C4 [×8], C4⋊C4 [×2], C4⋊C4 [×4], C4⋊C4 [×10], C22×C4 [×4], C2×D4, C2×Q8, Dic6 [×2], C4×S3 [×8], D12 [×2], C2×Dic3 [×2], C2×Dic3 [×4], C2×C12 [×3], C2×C12 [×4], C22×S3 [×2], C2×C4⋊C4, C42⋊C2 [×3], C4×D4, C4×Q8, C22⋊Q8 [×2], C22.D4 [×2], C42.C2, C42.C2 [×2], C42⋊2C2 [×2], C4×Dic3 [×4], Dic3⋊C4 [×4], C4⋊Dic3 [×2], C4⋊Dic3 [×4], D6⋊C4 [×2], D6⋊C4 [×6], C4×C12, C3×C4⋊C4 [×2], C3×C4⋊C4 [×4], C2×Dic6, S3×C2×C4 [×2], S3×C2×C4 [×2], C2×D12, C22.46C24, C4×Dic6, C4×D12, C12.3Q8 [×2], S3×C4⋊C4, C4⋊C4⋊7S3, C4⋊C4⋊7S3 [×2], D6.D4 [×2], C4.D12 [×2], C4⋊C4⋊S3 [×2], C3×C42.C2, C42.152D6
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, C22.46C24, C2×Q8⋊3S3, S3×C4○D4, Q8○D12, C42.152D6
Generators and relations
G = < a,b,c,d | a4=b4=1, c6=d2=b2, ab=ba, cac-1=dad-1=a-1, cbc-1=a2b-1, dbd-1=a2b, dcd-1=c5 >
►On 96 points
(1 75 90 70)(2 71 91 76)(3 77 92 72)(4 61 93 78)(5 79 94 62)(6 63 95 80)(7 81 96 64)(8 65 85 82)(9 83 86 66)(10 67 87 84)(11 73 88 68)(12 69 89 74)(13 53 45 25)(14 26 46 54)(15 55 47 27)(16 28 48 56)(17 57 37 29)(18 30 38 58)(19 59 39 31)(20 32 40 60)(21 49 41 33)(22 34 42 50)(23 51 43 35)(24 36 44 52)
(1 28 7 34)(2 51 8 57)(3 30 9 36)(4 53 10 59)(5 32 11 26)(6 55 12 49)(13 84 19 78)(14 62 20 68)(15 74 21 80)(16 64 22 70)(17 76 23 82)(18 66 24 72)(25 87 31 93)(27 89 33 95)(29 91 35 85)(37 71 43 65)(38 83 44 77)(39 61 45 67)(40 73 46 79)(41 63 47 69)(42 75 48 81)(50 90 56 96)(52 92 58 86)(54 94 60 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 27 7 33)(2 32 8 26)(3 25 9 31)(4 30 10 36)(5 35 11 29)(6 28 12 34)(13 66 19 72)(14 71 20 65)(15 64 21 70)(16 69 22 63)(17 62 23 68)(18 67 24 61)(37 79 43 73)(38 84 44 78)(39 77 45 83)(40 82 46 76)(41 75 47 81)(42 80 48 74)(49 90 55 96)(50 95 56 89)(51 88 57 94)(52 93 58 87)(53 86 59 92)(54 91 60 85)
G:=sub<Sym(96)| (1,75,90,70)(2,71,91,76)(3,77,92,72)(4,61,93,78)(5,79,94,62)(6,63,95,80)(7,81,96,64)(8,65,85,82)(9,83,86,66)(10,67,87,84)(11,73,88,68)(12,69,89,74)(13,53,45,25)(14,26,46,54)(15,55,47,27)(16,28,48,56)(17,57,37,29)(18,30,38,58)(19,59,39,31)(20,32,40,60)(21,49,41,33)(22,34,42,50)(23,51,43,35)(24,36,44,52), (1,28,7,34)(2,51,8,57)(3,30,9,36)(4,53,10,59)(5,32,11,26)(6,55,12,49)(13,84,19,78)(14,62,20,68)(15,74,21,80)(16,64,22,70)(17,76,23,82)(18,66,24,72)(25,87,31,93)(27,89,33,95)(29,91,35,85)(37,71,43,65)(38,83,44,77)(39,61,45,67)(40,73,46,79)(41,63,47,69)(42,75,48,81)(50,90,56,96)(52,92,58,86)(54,94,60,88), (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), (1,27,7,33)(2,32,8,26)(3,25,9,31)(4,30,10,36)(5,35,11,29)(6,28,12,34)(13,66,19,72)(14,71,20,65)(15,64,21,70)(16,69,22,63)(17,62,23,68)(18,67,24,61)(37,79,43,73)(38,84,44,78)(39,77,45,83)(40,82,46,76)(41,75,47,81)(42,80,48,74)(49,90,55,96)(50,95,56,89)(51,88,57,94)(52,93,58,87)(53,86,59,92)(54,91,60,85)>;
G:=Group( (1,75,90,70)(2,71,91,76)(3,77,92,72)(4,61,93,78)(5,79,94,62)(6,63,95,80)(7,81,96,64)(8,65,85,82)(9,83,86,66)(10,67,87,84)(11,73,88,68)(12,69,89,74)(13,53,45,25)(14,26,46,54)(15,55,47,27)(16,28,48,56)(17,57,37,29)(18,30,38,58)(19,59,39,31)(20,32,40,60)(21,49,41,33)(22,34,42,50)(23,51,43,35)(24,36,44,52), (1,28,7,34)(2,51,8,57)(3,30,9,36)(4,53,10,59)(5,32,11,26)(6,55,12,49)(13,84,19,78)(14,62,20,68)(15,74,21,80)(16,64,22,70)(17,76,23,82)(18,66,24,72)(25,87,31,93)(27,89,33,95)(29,91,35,85)(37,71,43,65)(38,83,44,77)(39,61,45,67)(40,73,46,79)(41,63,47,69)(42,75,48,81)(50,90,56,96)(52,92,58,86)(54,94,60,88), (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), (1,27,7,33)(2,32,8,26)(3,25,9,31)(4,30,10,36)(5,35,11,29)(6,28,12,34)(13,66,19,72)(14,71,20,65)(15,64,21,70)(16,69,22,63)(17,62,23,68)(18,67,24,61)(37,79,43,73)(38,84,44,78)(39,77,45,83)(40,82,46,76)(41,75,47,81)(42,80,48,74)(49,90,55,96)(50,95,56,89)(51,88,57,94)(52,93,58,87)(53,86,59,92)(54,91,60,85) );
G=PermutationGroup([(1,75,90,70),(2,71,91,76),(3,77,92,72),(4,61,93,78),(5,79,94,62),(6,63,95,80),(7,81,96,64),(8,65,85,82),(9,83,86,66),(10,67,87,84),(11,73,88,68),(12,69,89,74),(13,53,45,25),(14,26,46,54),(15,55,47,27),(16,28,48,56),(17,57,37,29),(18,30,38,58),(19,59,39,31),(20,32,40,60),(21,49,41,33),(22,34,42,50),(23,51,43,35),(24,36,44,52)], [(1,28,7,34),(2,51,8,57),(3,30,9,36),(4,53,10,59),(5,32,11,26),(6,55,12,49),(13,84,19,78),(14,62,20,68),(15,74,21,80),(16,64,22,70),(17,76,23,82),(18,66,24,72),(25,87,31,93),(27,89,33,95),(29,91,35,85),(37,71,43,65),(38,83,44,77),(39,61,45,67),(40,73,46,79),(41,63,47,69),(42,75,48,81),(50,90,56,96),(52,92,58,86),(54,94,60,88)], [(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)], [(1,27,7,33),(2,32,8,26),(3,25,9,31),(4,30,10,36),(5,35,11,29),(6,28,12,34),(13,66,19,72),(14,71,20,65),(15,64,21,70),(16,69,22,63),(17,62,23,68),(18,67,24,61),(37,79,43,73),(38,84,44,78),(39,77,45,83),(40,82,46,76),(41,75,47,81),(42,80,48,74),(49,90,55,96),(50,95,56,89),(51,88,57,94),(52,93,58,87),(53,86,59,92),(54,91,60,85)])
Matrix representation ►G ⊆ GL6(𝔽13)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 0 | 0 | 0 |
| 0 | 0 | 0 | 8 | 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 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 3 |
| 0 | 0 | 0 | 0 | 8 | 12 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 | 11 |
| 0 | 0 | 0 | 0 | 0 | 5 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 0 | 0 | 8 |
G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,5,0,0,0,0,0,0,8,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,8,0,0,0,0,3,12],[1,12,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,8,0,0,0,0,0,11,5],[1,0,0,0,0,0,1,12,0,0,0,0,0,0,0,12,0,0,0,0,12,0,0,0,0,0,0,0,8,0,0,0,0,0,0,8] >;
39 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3 | 4A | 4B | 4C | 4D | 4E | ··· | 4I | 4J | ··· | 4O | 4P | 4Q | 4R | 6A | 6B | 6C | 12A | ··· | 12F | 12G | 12H | 12I | 12J |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 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 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 6 | ··· | 6 | 12 | 12 | 12 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
39 irreducible representations
| dim | 1 | 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 | C2 | S3 | D6 | D6 | C4○D4 | C4○D4 | 2- (1+4) | Q8⋊3S3 | S3×C4○D4 | Q8○D12 |
| kernel | C42.152D6 | C4×Dic6 | C4×D12 | C12.3Q8 | S3×C4⋊C4 | C4⋊C4⋊7S3 | D6.D4 | C4.D12 | C4⋊C4⋊S3 | C3×C42.C2 | C42.C2 | C42 | C4⋊C4 | C12 | D6 | C6 | C4 | C2 | C2 |
| # reps | 1 | 1 | 1 | 2 | 1 | 3 | 2 | 2 | 2 | 1 | 1 | 1 | 6 | 4 | 4 | 1 | 2 | 2 | 2 |
In GAP, Magma, Sage, TeX
C_4^2._{152}D_6 % in TeX
G:=Group("C4^2.152D6"); // GroupNames label
G:=SmallGroup(192,1253);
// by ID
G=gap.SmallGroup(192,1253);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,120,219,268,1571,297,192,6278]);
// 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=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