metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C24.75D6, C23.32D12, C23.18Dic6, C12⋊7(C22⋊C4), (C22×C12)⋊14C4, (C2×C12).475D4, (C23×C4).14S3, (C22×C6).26Q8, C2.4(C12⋊7D4), C4⋊2(C6.D4), C6.79(C4⋊D4), (C23×C12).11C2, C22⋊3(C4⋊Dic3), (C22×C4)⋊10Dic3, C22.60(C2×D12), (C22×C4).449D6, (C22×C6).143D4, C6.68(C22⋊Q8), C3⋊4(C23.7Q8), C6.C42⋊24C2, (C23×C6).99C22, C23.36(C2×Dic3), C22.32(C2×Dic6), C6.49(C42⋊C2), C2.5(C12.48D4), C22.63(C4○D12), (C22×C6).363C23, C23.313(C22×S3), (C22×C12).484C22, C22.50(C22×Dic3), C2.12(C23.26D6), (C22×Dic3).66C22, (C2×C6)⋊7(C4⋊C4), C6.56(C2×C4⋊C4), (C2×C6).44(C2×Q8), (C2×C4⋊Dic3)⋊16C2, (C2×C6).549(C2×D4), C6.70(C2×C22⋊C4), C2.16(C2×C4⋊Dic3), (C2×C12).282(C2×C4), (C2×C6).91(C4○D4), (C2×C4).85(C2×Dic3), C2.6(C2×C6.D4), C22.87(C2×C3⋊D4), (C2×C4).260(C3⋊D4), (C22×C6).136(C2×C4), (C2×C6).193(C22×C4), (C2×C6.D4).18C2, SmallGroup(192,771)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C24.75D6
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e6=d, f2=b, ab=ba, faf-1=ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=de5 >
Subgroups: 504 in 234 conjugacy classes, 103 normal (25 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C22, C6, C6, C6, C2×C4, C2×C4, C23, C23, C23, Dic3, C12, C12, C2×C6, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C24, C2×Dic3, C2×C12, C2×C12, C22×C6, C22×C6, C22×C6, C2.C42, C2×C22⋊C4, C2×C4⋊C4, C23×C4, C4⋊Dic3, C6.D4, C22×Dic3, C22×C12, C22×C12, C22×C12, C23×C6, C23.7Q8, C6.C42, C2×C4⋊Dic3, C2×C6.D4, C23×C12, C24.75D6
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, C23, Dic3, D6, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, C4○D4, Dic6, D12, C2×Dic3, C3⋊D4, C22×S3, C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C4⋊D4, C22⋊Q8, C4⋊Dic3, C6.D4, C2×Dic6, C2×D12, C4○D12, C22×Dic3, C2×C3⋊D4, C23.7Q8, C12.48D4, C2×C4⋊Dic3, C23.26D6, C12⋊7D4, C2×C6.D4, C24.75D6
►On 96 points
Generators in S96
(1 59)(2 60)(3 49)(4 50)(5 51)(6 52)(7 53)(8 54)(9 55)(10 56)(11 57)(12 58)(13 78)(14 79)(15 80)(16 81)(17 82)(18 83)(19 84)(20 73)(21 74)(22 75)(23 76)(24 77)(25 67)(26 68)(27 69)(28 70)(29 71)(30 72)(31 61)(32 62)(33 63)(34 64)(35 65)(36 66)(37 89)(38 90)(39 91)(40 92)(41 93)(42 94)(43 95)(44 96)(45 85)(46 86)(47 87)(48 88)
(1 59)(2 60)(3 49)(4 50)(5 51)(6 52)(7 53)(8 54)(9 55)(10 56)(11 57)(12 58)(13 86)(14 87)(15 88)(16 89)(17 90)(18 91)(19 92)(20 93)(21 94)(22 95)(23 96)(24 85)(25 67)(26 68)(27 69)(28 70)(29 71)(30 72)(31 61)(32 62)(33 63)(34 64)(35 65)(36 66)(37 81)(38 82)(39 83)(40 84)(41 73)(42 74)(43 75)(44 76)(45 77)(46 78)(47 79)(48 80)
(1 66)(2 67)(3 68)(4 69)(5 70)(6 71)(7 72)(8 61)(9 62)(10 63)(11 64)(12 65)(13 46)(14 47)(15 48)(16 37)(17 38)(18 39)(19 40)(20 41)(21 42)(22 43)(23 44)(24 45)(25 60)(26 49)(27 50)(28 51)(29 52)(30 53)(31 54)(32 55)(33 56)(34 57)(35 58)(36 59)(73 93)(74 94)(75 95)(76 96)(77 85)(78 86)(79 87)(80 88)(81 89)(82 90)(83 91)(84 92)
(1 7)(2 8)(3 9)(4 10)(5 11)(6 12)(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)(25 31)(26 32)(27 33)(28 34)(29 35)(30 36)(37 43)(38 44)(39 45)(40 46)(41 47)(42 48)(49 55)(50 56)(51 57)(52 58)(53 59)(54 60)(61 67)(62 68)(63 69)(64 70)(65 71)(66 72)(73 79)(74 80)(75 81)(76 82)(77 83)(78 84)(85 91)(86 92)(87 93)(88 94)(89 95)(90 96)
(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 14 59 87)(2 13 60 86)(3 24 49 85)(4 23 50 96)(5 22 51 95)(6 21 52 94)(7 20 53 93)(8 19 54 92)(9 18 55 91)(10 17 56 90)(11 16 57 89)(12 15 58 88)(25 78 67 46)(26 77 68 45)(27 76 69 44)(28 75 70 43)(29 74 71 42)(30 73 72 41)(31 84 61 40)(32 83 62 39)(33 82 63 38)(34 81 64 37)(35 80 65 48)(36 79 66 47)
G:=sub<Sym(96)| (1,59)(2,60)(3,49)(4,50)(5,51)(6,52)(7,53)(8,54)(9,55)(10,56)(11,57)(12,58)(13,78)(14,79)(15,80)(16,81)(17,82)(18,83)(19,84)(20,73)(21,74)(22,75)(23,76)(24,77)(25,67)(26,68)(27,69)(28,70)(29,71)(30,72)(31,61)(32,62)(33,63)(34,64)(35,65)(36,66)(37,89)(38,90)(39,91)(40,92)(41,93)(42,94)(43,95)(44,96)(45,85)(46,86)(47,87)(48,88), (1,59)(2,60)(3,49)(4,50)(5,51)(6,52)(7,53)(8,54)(9,55)(10,56)(11,57)(12,58)(13,86)(14,87)(15,88)(16,89)(17,90)(18,91)(19,92)(20,93)(21,94)(22,95)(23,96)(24,85)(25,67)(26,68)(27,69)(28,70)(29,71)(30,72)(31,61)(32,62)(33,63)(34,64)(35,65)(36,66)(37,81)(38,82)(39,83)(40,84)(41,73)(42,74)(43,75)(44,76)(45,77)(46,78)(47,79)(48,80), (1,66)(2,67)(3,68)(4,69)(5,70)(6,71)(7,72)(8,61)(9,62)(10,63)(11,64)(12,65)(13,46)(14,47)(15,48)(16,37)(17,38)(18,39)(19,40)(20,41)(21,42)(22,43)(23,44)(24,45)(25,60)(26,49)(27,50)(28,51)(29,52)(30,53)(31,54)(32,55)(33,56)(34,57)(35,58)(36,59)(73,93)(74,94)(75,95)(76,96)(77,85)(78,86)(79,87)(80,88)(81,89)(82,90)(83,91)(84,92), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24)(25,31)(26,32)(27,33)(28,34)(29,35)(30,36)(37,43)(38,44)(39,45)(40,46)(41,47)(42,48)(49,55)(50,56)(51,57)(52,58)(53,59)(54,60)(61,67)(62,68)(63,69)(64,70)(65,71)(66,72)(73,79)(74,80)(75,81)(76,82)(77,83)(78,84)(85,91)(86,92)(87,93)(88,94)(89,95)(90,96), (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), (1,14,59,87)(2,13,60,86)(3,24,49,85)(4,23,50,96)(5,22,51,95)(6,21,52,94)(7,20,53,93)(8,19,54,92)(9,18,55,91)(10,17,56,90)(11,16,57,89)(12,15,58,88)(25,78,67,46)(26,77,68,45)(27,76,69,44)(28,75,70,43)(29,74,71,42)(30,73,72,41)(31,84,61,40)(32,83,62,39)(33,82,63,38)(34,81,64,37)(35,80,65,48)(36,79,66,47)>;
G:=Group( (1,59)(2,60)(3,49)(4,50)(5,51)(6,52)(7,53)(8,54)(9,55)(10,56)(11,57)(12,58)(13,78)(14,79)(15,80)(16,81)(17,82)(18,83)(19,84)(20,73)(21,74)(22,75)(23,76)(24,77)(25,67)(26,68)(27,69)(28,70)(29,71)(30,72)(31,61)(32,62)(33,63)(34,64)(35,65)(36,66)(37,89)(38,90)(39,91)(40,92)(41,93)(42,94)(43,95)(44,96)(45,85)(46,86)(47,87)(48,88), (1,59)(2,60)(3,49)(4,50)(5,51)(6,52)(7,53)(8,54)(9,55)(10,56)(11,57)(12,58)(13,86)(14,87)(15,88)(16,89)(17,90)(18,91)(19,92)(20,93)(21,94)(22,95)(23,96)(24,85)(25,67)(26,68)(27,69)(28,70)(29,71)(30,72)(31,61)(32,62)(33,63)(34,64)(35,65)(36,66)(37,81)(38,82)(39,83)(40,84)(41,73)(42,74)(43,75)(44,76)(45,77)(46,78)(47,79)(48,80), (1,66)(2,67)(3,68)(4,69)(5,70)(6,71)(7,72)(8,61)(9,62)(10,63)(11,64)(12,65)(13,46)(14,47)(15,48)(16,37)(17,38)(18,39)(19,40)(20,41)(21,42)(22,43)(23,44)(24,45)(25,60)(26,49)(27,50)(28,51)(29,52)(30,53)(31,54)(32,55)(33,56)(34,57)(35,58)(36,59)(73,93)(74,94)(75,95)(76,96)(77,85)(78,86)(79,87)(80,88)(81,89)(82,90)(83,91)(84,92), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24)(25,31)(26,32)(27,33)(28,34)(29,35)(30,36)(37,43)(38,44)(39,45)(40,46)(41,47)(42,48)(49,55)(50,56)(51,57)(52,58)(53,59)(54,60)(61,67)(62,68)(63,69)(64,70)(65,71)(66,72)(73,79)(74,80)(75,81)(76,82)(77,83)(78,84)(85,91)(86,92)(87,93)(88,94)(89,95)(90,96), (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), (1,14,59,87)(2,13,60,86)(3,24,49,85)(4,23,50,96)(5,22,51,95)(6,21,52,94)(7,20,53,93)(8,19,54,92)(9,18,55,91)(10,17,56,90)(11,16,57,89)(12,15,58,88)(25,78,67,46)(26,77,68,45)(27,76,69,44)(28,75,70,43)(29,74,71,42)(30,73,72,41)(31,84,61,40)(32,83,62,39)(33,82,63,38)(34,81,64,37)(35,80,65,48)(36,79,66,47) );
G=PermutationGroup([[(1,59),(2,60),(3,49),(4,50),(5,51),(6,52),(7,53),(8,54),(9,55),(10,56),(11,57),(12,58),(13,78),(14,79),(15,80),(16,81),(17,82),(18,83),(19,84),(20,73),(21,74),(22,75),(23,76),(24,77),(25,67),(26,68),(27,69),(28,70),(29,71),(30,72),(31,61),(32,62),(33,63),(34,64),(35,65),(36,66),(37,89),(38,90),(39,91),(40,92),(41,93),(42,94),(43,95),(44,96),(45,85),(46,86),(47,87),(48,88)], [(1,59),(2,60),(3,49),(4,50),(5,51),(6,52),(7,53),(8,54),(9,55),(10,56),(11,57),(12,58),(13,86),(14,87),(15,88),(16,89),(17,90),(18,91),(19,92),(20,93),(21,94),(22,95),(23,96),(24,85),(25,67),(26,68),(27,69),(28,70),(29,71),(30,72),(31,61),(32,62),(33,63),(34,64),(35,65),(36,66),(37,81),(38,82),(39,83),(40,84),(41,73),(42,74),(43,75),(44,76),(45,77),(46,78),(47,79),(48,80)], [(1,66),(2,67),(3,68),(4,69),(5,70),(6,71),(7,72),(8,61),(9,62),(10,63),(11,64),(12,65),(13,46),(14,47),(15,48),(16,37),(17,38),(18,39),(19,40),(20,41),(21,42),(22,43),(23,44),(24,45),(25,60),(26,49),(27,50),(28,51),(29,52),(30,53),(31,54),(32,55),(33,56),(34,57),(35,58),(36,59),(73,93),(74,94),(75,95),(76,96),(77,85),(78,86),(79,87),(80,88),(81,89),(82,90),(83,91),(84,92)], [(1,7),(2,8),(3,9),(4,10),(5,11),(6,12),(13,19),(14,20),(15,21),(16,22),(17,23),(18,24),(25,31),(26,32),(27,33),(28,34),(29,35),(30,36),(37,43),(38,44),(39,45),(40,46),(41,47),(42,48),(49,55),(50,56),(51,57),(52,58),(53,59),(54,60),(61,67),(62,68),(63,69),(64,70),(65,71),(66,72),(73,79),(74,80),(75,81),(76,82),(77,83),(78,84),(85,91),(86,92),(87,93),(88,94),(89,95),(90,96)], [(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)], [(1,14,59,87),(2,13,60,86),(3,24,49,85),(4,23,50,96),(5,22,51,95),(6,21,52,94),(7,20,53,93),(8,19,54,92),(9,18,55,91),(10,17,56,90),(11,16,57,89),(12,15,58,88),(25,78,67,46),(26,77,68,45),(27,76,69,44),(28,75,70,43),(29,74,71,42),(30,73,72,41),(31,84,61,40),(32,83,62,39),(33,82,63,38),(34,81,64,37),(35,80,65,48),(36,79,66,47)]])
60 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 3 | 4A | ··· | 4H | 4I | ··· | 4P | 6A | ··· | 6O | 12A | ··· | 12P |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 3 | 4 | ··· | 4 | 4 | ··· | 4 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 12 | ··· | 12 | 2 | ··· | 2 | 2 | ··· | 2 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | - | - | + | + | - | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | S3 | D4 | D4 | Q8 | Dic3 | D6 | D6 | C4○D4 | C3⋊D4 | Dic6 | D12 | C4○D12 |
| kernel | C24.75D6 | C6.C42 | C2×C4⋊Dic3 | C2×C6.D4 | C23×C12 | C22×C12 | C23×C4 | C2×C12 | C22×C6 | C22×C6 | C22×C4 | C22×C4 | C24 | C2×C6 | C2×C4 | C23 | C23 | C22 |
| # reps | 1 | 2 | 2 | 2 | 1 | 8 | 1 | 4 | 2 | 2 | 4 | 2 | 1 | 4 | 8 | 4 | 4 | 8 |
Matrix representation of C24.75D6 ►in GL6(𝔽13)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 2 | 12 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 3 | 10 | 0 | 0 | 0 | 0 |
| 3 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 0 | 0 | 0 |
| 0 | 0 | 6 | 7 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 0 | 8 | 7 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 12 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 9 | 0 | 0 |
| 0 | 0 | 9 | 10 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 1 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,2,0,0,0,0,0,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[3,3,0,0,0,0,10,6,0,0,0,0,0,0,2,6,0,0,0,0,0,7,0,0,0,0,0,0,2,8,0,0,0,0,0,7],[1,12,0,0,0,0,0,12,0,0,0,0,0,0,3,9,0,0,0,0,9,10,0,0,0,0,0,0,12,0,0,0,0,0,1,1] >;
C24.75D6 in GAP, Magma, Sage, TeX
C_2^4._{75}D_6 % in TeX
G:=Group("C2^4.75D6"); // GroupNames label
G:=SmallGroup(192,771);
// by ID
G=gap.SmallGroup(192,771);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,56,477,232,422,6278]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^6=d,f^2=b,a*b=b*a,f*a*f^-1=a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=d*e^5>;
// generators/relations