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×C6).238C24, (C2×C12).90C23, C2.58(Q8○D12), C4.Dic6⋊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, (C2×Dic6).299C22, (C4×Dic3).144C22, (C2×Dic3).123C23, C3⋊10(C22.46C24), (S3×C4⋊C4)⋊38C2, C4⋊C4⋊S3⋊36C2, C4⋊C4⋊7S3⋊37C2, 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
Generators and relations for C42.152D6
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 >
Subgroups: 496 in 214 conjugacy classes, 97 normal (43 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C2×C4, C2×C4, C2×C4, D4, Q8, C23, Dic3, C12, C12, D6, D6, C2×C6, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, Dic6, C4×S3, D12, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×S3, C2×C4⋊C4, C42⋊C2, C4×D4, C4×Q8, C22⋊Q8, C22.D4, C42.C2, C42.C2, C42⋊2C2, C4×Dic3, Dic3⋊C4, C4⋊Dic3, C4⋊Dic3, D6⋊C4, D6⋊C4, C4×C12, C3×C4⋊C4, C3×C4⋊C4, C2×Dic6, S3×C2×C4, S3×C2×C4, C2×D12, C22.46C24, C4×Dic6, C4×D12, C4.Dic6, S3×C4⋊C4, C4⋊C4⋊7S3, C4⋊C4⋊7S3, D6.D4, C4.D12, C4⋊C4⋊S3, C3×C42.C2, C42.152D6
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C24, C22×S3, C2×C4○D4, 2- 1+4, Q8⋊3S3, S3×C23, C22.46C24, C2×Q8⋊3S3, S3×C4○D4, Q8○D12, C42.152D6
►On 96 points
(1 88 68 49)(2 50 69 89)(3 90 70 51)(4 52 71 91)(5 92 72 53)(6 54 61 93)(7 94 62 55)(8 56 63 95)(9 96 64 57)(10 58 65 85)(11 86 66 59)(12 60 67 87)(13 28 75 48)(14 37 76 29)(15 30 77 38)(16 39 78 31)(17 32 79 40)(18 41 80 33)(19 34 81 42)(20 43 82 35)(21 36 83 44)(22 45 84 25)(23 26 73 46)(24 47 74 27)
(1 23 7 17)(2 80 8 74)(3 13 9 19)(4 82 10 76)(5 15 11 21)(6 84 12 78)(14 71 20 65)(16 61 22 67)(18 63 24 69)(25 60 31 54)(26 94 32 88)(27 50 33 56)(28 96 34 90)(29 52 35 58)(30 86 36 92)(37 91 43 85)(38 59 44 53)(39 93 45 87)(40 49 46 55)(41 95 47 89)(42 51 48 57)(62 79 68 73)(64 81 70 75)(66 83 72 77)
(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 22 7 16)(2 15 8 21)(3 20 9 14)(4 13 10 19)(5 18 11 24)(6 23 12 17)(25 94 31 88)(26 87 32 93)(27 92 33 86)(28 85 34 91)(29 90 35 96)(30 95 36 89)(37 51 43 57)(38 56 44 50)(39 49 45 55)(40 54 46 60)(41 59 47 53)(42 52 48 58)(61 73 67 79)(62 78 68 84)(63 83 69 77)(64 76 70 82)(65 81 71 75)(66 74 72 80)
G:=sub<Sym(96)| (1,88,68,49)(2,50,69,89)(3,90,70,51)(4,52,71,91)(5,92,72,53)(6,54,61,93)(7,94,62,55)(8,56,63,95)(9,96,64,57)(10,58,65,85)(11,86,66,59)(12,60,67,87)(13,28,75,48)(14,37,76,29)(15,30,77,38)(16,39,78,31)(17,32,79,40)(18,41,80,33)(19,34,81,42)(20,43,82,35)(21,36,83,44)(22,45,84,25)(23,26,73,46)(24,47,74,27), (1,23,7,17)(2,80,8,74)(3,13,9,19)(4,82,10,76)(5,15,11,21)(6,84,12,78)(14,71,20,65)(16,61,22,67)(18,63,24,69)(25,60,31,54)(26,94,32,88)(27,50,33,56)(28,96,34,90)(29,52,35,58)(30,86,36,92)(37,91,43,85)(38,59,44,53)(39,93,45,87)(40,49,46,55)(41,95,47,89)(42,51,48,57)(62,79,68,73)(64,81,70,75)(66,83,72,77), (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), (1,22,7,16)(2,15,8,21)(3,20,9,14)(4,13,10,19)(5,18,11,24)(6,23,12,17)(25,94,31,88)(26,87,32,93)(27,92,33,86)(28,85,34,91)(29,90,35,96)(30,95,36,89)(37,51,43,57)(38,56,44,50)(39,49,45,55)(40,54,46,60)(41,59,47,53)(42,52,48,58)(61,73,67,79)(62,78,68,84)(63,83,69,77)(64,76,70,82)(65,81,71,75)(66,74,72,80)>;
G:=Group( (1,88,68,49)(2,50,69,89)(3,90,70,51)(4,52,71,91)(5,92,72,53)(6,54,61,93)(7,94,62,55)(8,56,63,95)(9,96,64,57)(10,58,65,85)(11,86,66,59)(12,60,67,87)(13,28,75,48)(14,37,76,29)(15,30,77,38)(16,39,78,31)(17,32,79,40)(18,41,80,33)(19,34,81,42)(20,43,82,35)(21,36,83,44)(22,45,84,25)(23,26,73,46)(24,47,74,27), (1,23,7,17)(2,80,8,74)(3,13,9,19)(4,82,10,76)(5,15,11,21)(6,84,12,78)(14,71,20,65)(16,61,22,67)(18,63,24,69)(25,60,31,54)(26,94,32,88)(27,50,33,56)(28,96,34,90)(29,52,35,58)(30,86,36,92)(37,91,43,85)(38,59,44,53)(39,93,45,87)(40,49,46,55)(41,95,47,89)(42,51,48,57)(62,79,68,73)(64,81,70,75)(66,83,72,77), (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), (1,22,7,16)(2,15,8,21)(3,20,9,14)(4,13,10,19)(5,18,11,24)(6,23,12,17)(25,94,31,88)(26,87,32,93)(27,92,33,86)(28,85,34,91)(29,90,35,96)(30,95,36,89)(37,51,43,57)(38,56,44,50)(39,49,45,55)(40,54,46,60)(41,59,47,53)(42,52,48,58)(61,73,67,79)(62,78,68,84)(63,83,69,77)(64,76,70,82)(65,81,71,75)(66,74,72,80) );
G=PermutationGroup([[(1,88,68,49),(2,50,69,89),(3,90,70,51),(4,52,71,91),(5,92,72,53),(6,54,61,93),(7,94,62,55),(8,56,63,95),(9,96,64,57),(10,58,65,85),(11,86,66,59),(12,60,67,87),(13,28,75,48),(14,37,76,29),(15,30,77,38),(16,39,78,31),(17,32,79,40),(18,41,80,33),(19,34,81,42),(20,43,82,35),(21,36,83,44),(22,45,84,25),(23,26,73,46),(24,47,74,27)], [(1,23,7,17),(2,80,8,74),(3,13,9,19),(4,82,10,76),(5,15,11,21),(6,84,12,78),(14,71,20,65),(16,61,22,67),(18,63,24,69),(25,60,31,54),(26,94,32,88),(27,50,33,56),(28,96,34,90),(29,52,35,58),(30,86,36,92),(37,91,43,85),(38,59,44,53),(39,93,45,87),(40,49,46,55),(41,95,47,89),(42,51,48,57),(62,79,68,73),(64,81,70,75),(66,83,72,77)], [(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)], [(1,22,7,16),(2,15,8,21),(3,20,9,14),(4,13,10,19),(5,18,11,24),(6,23,12,17),(25,94,31,88),(26,87,32,93),(27,92,33,86),(28,85,34,91),(29,90,35,96),(30,95,36,89),(37,51,43,57),(38,56,44,50),(39,49,45,55),(40,54,46,60),(41,59,47,53),(42,52,48,58),(61,73,67,79),(62,78,68,84),(63,83,69,77),(64,76,70,82),(65,81,71,75),(66,74,72,80)]])
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 | C4.Dic6 | 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 |
Matrix representation of C42.152D6 ►in 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] >;
C42.152D6 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