metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C23.52D12, M4(2)⋊1Dic3, C24⋊3(C2×C4), C8⋊Dic3⋊3C2, C8⋊1(C2×Dic3), (C2×C8).75D6, C24⋊1C4⋊17C2, C12.21(C4⋊C4), (C2×C12).26Q8, C12.76(C2×Q8), (C2×C12).167D4, (C2×C4).149D12, C2.3(C8⋊D6), (C3×M4(2))⋊1C4, C4.6(C4⋊Dic3), C4.42(C2×Dic6), (C2×C4).15Dic6, C6.19(C8⋊C22), (C2×C24).61C22, C2.4(C8.D6), C22.56(C2×D12), (C22×C4).149D6, (C22×C6).100D4, (C6×M4(2)).1C2, (C2×M4(2)).1S3, C3⋊5(M4(2)⋊C4), C12.173(C22×C4), (C2×C12).772C23, C6.20(C8.C22), C22.6(C4⋊Dic3), C4.27(C22×Dic3), C4⋊Dic3.284C22, (C22×C12).180C22, C23.26D6.17C2, C6.50(C2×C4⋊C4), (C2×C6).16(C4⋊C4), (C2×C6).162(C2×D4), C2.14(C2×C4⋊Dic3), (C2×C12).100(C2×C4), (C2×C4⋊Dic3).39C2, (C2×C4).21(C2×Dic3), (C2×C4).720(C22×S3), SmallGroup(192,680)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C23.52D12
G = < a,b,c,d,e | a2=b2=c2=1, d12=c, e2=cb=bc, ab=ba, dad-1=eae-1=ac=ca, bd=db, be=eb, cd=dc, ce=ec, ede-1=d11 >
Subgroups: 280 in 118 conjugacy classes, 71 normal (29 characteristic)
C1, C2, C2, C3, C4, C4, C4, C22, C22, C22, C6, C6, C8, C2×C4, C2×C4, C2×C4, C23, Dic3, C12, C12, C2×C6, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, C22×C4, C24, C2×Dic3, C2×C12, C2×C12, C22×C6, C4.Q8, C2.D8, C2×C4⋊C4, C42⋊C2, C2×M4(2), C4×Dic3, C4⋊Dic3, C4⋊Dic3, C4⋊Dic3, C6.D4, C2×C24, C3×M4(2), C22×Dic3, C22×C12, M4(2)⋊C4, C8⋊Dic3, C24⋊1C4, C2×C4⋊Dic3, C23.26D6, C6×M4(2), C23.52D12
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, C23, Dic3, D6, C4⋊C4, C22×C4, C2×D4, C2×Q8, Dic6, D12, C2×Dic3, C22×S3, C2×C4⋊C4, C8⋊C22, C8.C22, C4⋊Dic3, C2×Dic6, C2×D12, C22×Dic3, M4(2)⋊C4, C8⋊D6, C8.D6, C2×C4⋊Dic3, C23.52D12
►On 96 points
Generators in S96
(1 73)(2 86)(3 75)(4 88)(5 77)(6 90)(7 79)(8 92)(9 81)(10 94)(11 83)(12 96)(13 85)(14 74)(15 87)(16 76)(17 89)(18 78)(19 91)(20 80)(21 93)(22 82)(23 95)(24 84)(25 68)(26 57)(27 70)(28 59)(29 72)(30 61)(31 50)(32 63)(33 52)(34 65)(35 54)(36 67)(37 56)(38 69)(39 58)(40 71)(41 60)(42 49)(43 62)(44 51)(45 64)(46 53)(47 66)(48 55)
(1 73)(2 74)(3 75)(4 76)(5 77)(6 78)(7 79)(8 80)(9 81)(10 82)(11 83)(12 84)(13 85)(14 86)(15 87)(16 88)(17 89)(18 90)(19 91)(20 92)(21 93)(22 94)(23 95)(24 96)(25 68)(26 69)(27 70)(28 71)(29 72)(30 49)(31 50)(32 51)(33 52)(34 53)(35 54)(36 55)(37 56)(38 57)(39 58)(40 59)(41 60)(42 61)(43 62)(44 63)(45 64)(46 65)(47 66)(48 67)
(1 13)(2 14)(3 15)(4 16)(5 17)(6 18)(7 19)(8 20)(9 21)(10 22)(11 23)(12 24)(25 37)(26 38)(27 39)(28 40)(29 41)(30 42)(31 43)(32 44)(33 45)(34 46)(35 47)(36 48)(49 61)(50 62)(51 63)(52 64)(53 65)(54 66)(55 67)(56 68)(57 69)(58 70)(59 71)(60 72)(73 85)(74 86)(75 87)(76 88)(77 89)(78 90)(79 91)(80 92)(81 93)(82 94)(83 95)(84 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 48 85 55)(2 35 86 66)(3 46 87 53)(4 33 88 64)(5 44 89 51)(6 31 90 62)(7 42 91 49)(8 29 92 60)(9 40 93 71)(10 27 94 58)(11 38 95 69)(12 25 96 56)(13 36 73 67)(14 47 74 54)(15 34 75 65)(16 45 76 52)(17 32 77 63)(18 43 78 50)(19 30 79 61)(20 41 80 72)(21 28 81 59)(22 39 82 70)(23 26 83 57)(24 37 84 68)
G:=sub<Sym(96)| (1,73)(2,86)(3,75)(4,88)(5,77)(6,90)(7,79)(8,92)(9,81)(10,94)(11,83)(12,96)(13,85)(14,74)(15,87)(16,76)(17,89)(18,78)(19,91)(20,80)(21,93)(22,82)(23,95)(24,84)(25,68)(26,57)(27,70)(28,59)(29,72)(30,61)(31,50)(32,63)(33,52)(34,65)(35,54)(36,67)(37,56)(38,69)(39,58)(40,71)(41,60)(42,49)(43,62)(44,51)(45,64)(46,53)(47,66)(48,55), (1,73)(2,74)(3,75)(4,76)(5,77)(6,78)(7,79)(8,80)(9,81)(10,82)(11,83)(12,84)(13,85)(14,86)(15,87)(16,88)(17,89)(18,90)(19,91)(20,92)(21,93)(22,94)(23,95)(24,96)(25,68)(26,69)(27,70)(28,71)(29,72)(30,49)(31,50)(32,51)(33,52)(34,53)(35,54)(36,55)(37,56)(38,57)(39,58)(40,59)(41,60)(42,61)(43,62)(44,63)(45,64)(46,65)(47,66)(48,67), (1,13)(2,14)(3,15)(4,16)(5,17)(6,18)(7,19)(8,20)(9,21)(10,22)(11,23)(12,24)(25,37)(26,38)(27,39)(28,40)(29,41)(30,42)(31,43)(32,44)(33,45)(34,46)(35,47)(36,48)(49,61)(50,62)(51,63)(52,64)(53,65)(54,66)(55,67)(56,68)(57,69)(58,70)(59,71)(60,72)(73,85)(74,86)(75,87)(76,88)(77,89)(78,90)(79,91)(80,92)(81,93)(82,94)(83,95)(84,96), (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), (1,48,85,55)(2,35,86,66)(3,46,87,53)(4,33,88,64)(5,44,89,51)(6,31,90,62)(7,42,91,49)(8,29,92,60)(9,40,93,71)(10,27,94,58)(11,38,95,69)(12,25,96,56)(13,36,73,67)(14,47,74,54)(15,34,75,65)(16,45,76,52)(17,32,77,63)(18,43,78,50)(19,30,79,61)(20,41,80,72)(21,28,81,59)(22,39,82,70)(23,26,83,57)(24,37,84,68)>;
G:=Group( (1,73)(2,86)(3,75)(4,88)(5,77)(6,90)(7,79)(8,92)(9,81)(10,94)(11,83)(12,96)(13,85)(14,74)(15,87)(16,76)(17,89)(18,78)(19,91)(20,80)(21,93)(22,82)(23,95)(24,84)(25,68)(26,57)(27,70)(28,59)(29,72)(30,61)(31,50)(32,63)(33,52)(34,65)(35,54)(36,67)(37,56)(38,69)(39,58)(40,71)(41,60)(42,49)(43,62)(44,51)(45,64)(46,53)(47,66)(48,55), (1,73)(2,74)(3,75)(4,76)(5,77)(6,78)(7,79)(8,80)(9,81)(10,82)(11,83)(12,84)(13,85)(14,86)(15,87)(16,88)(17,89)(18,90)(19,91)(20,92)(21,93)(22,94)(23,95)(24,96)(25,68)(26,69)(27,70)(28,71)(29,72)(30,49)(31,50)(32,51)(33,52)(34,53)(35,54)(36,55)(37,56)(38,57)(39,58)(40,59)(41,60)(42,61)(43,62)(44,63)(45,64)(46,65)(47,66)(48,67), (1,13)(2,14)(3,15)(4,16)(5,17)(6,18)(7,19)(8,20)(9,21)(10,22)(11,23)(12,24)(25,37)(26,38)(27,39)(28,40)(29,41)(30,42)(31,43)(32,44)(33,45)(34,46)(35,47)(36,48)(49,61)(50,62)(51,63)(52,64)(53,65)(54,66)(55,67)(56,68)(57,69)(58,70)(59,71)(60,72)(73,85)(74,86)(75,87)(76,88)(77,89)(78,90)(79,91)(80,92)(81,93)(82,94)(83,95)(84,96), (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), (1,48,85,55)(2,35,86,66)(3,46,87,53)(4,33,88,64)(5,44,89,51)(6,31,90,62)(7,42,91,49)(8,29,92,60)(9,40,93,71)(10,27,94,58)(11,38,95,69)(12,25,96,56)(13,36,73,67)(14,47,74,54)(15,34,75,65)(16,45,76,52)(17,32,77,63)(18,43,78,50)(19,30,79,61)(20,41,80,72)(21,28,81,59)(22,39,82,70)(23,26,83,57)(24,37,84,68) );
G=PermutationGroup([[(1,73),(2,86),(3,75),(4,88),(5,77),(6,90),(7,79),(8,92),(9,81),(10,94),(11,83),(12,96),(13,85),(14,74),(15,87),(16,76),(17,89),(18,78),(19,91),(20,80),(21,93),(22,82),(23,95),(24,84),(25,68),(26,57),(27,70),(28,59),(29,72),(30,61),(31,50),(32,63),(33,52),(34,65),(35,54),(36,67),(37,56),(38,69),(39,58),(40,71),(41,60),(42,49),(43,62),(44,51),(45,64),(46,53),(47,66),(48,55)], [(1,73),(2,74),(3,75),(4,76),(5,77),(6,78),(7,79),(8,80),(9,81),(10,82),(11,83),(12,84),(13,85),(14,86),(15,87),(16,88),(17,89),(18,90),(19,91),(20,92),(21,93),(22,94),(23,95),(24,96),(25,68),(26,69),(27,70),(28,71),(29,72),(30,49),(31,50),(32,51),(33,52),(34,53),(35,54),(36,55),(37,56),(38,57),(39,58),(40,59),(41,60),(42,61),(43,62),(44,63),(45,64),(46,65),(47,66),(48,67)], [(1,13),(2,14),(3,15),(4,16),(5,17),(6,18),(7,19),(8,20),(9,21),(10,22),(11,23),(12,24),(25,37),(26,38),(27,39),(28,40),(29,41),(30,42),(31,43),(32,44),(33,45),(34,46),(35,47),(36,48),(49,61),(50,62),(51,63),(52,64),(53,65),(54,66),(55,67),(56,68),(57,69),(58,70),(59,71),(60,72),(73,85),(74,86),(75,87),(76,88),(77,89),(78,90),(79,91),(80,92),(81,93),(82,94),(83,95),(84,96)], [(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)], [(1,48,85,55),(2,35,86,66),(3,46,87,53),(4,33,88,64),(5,44,89,51),(6,31,90,62),(7,42,91,49),(8,29,92,60),(9,40,93,71),(10,27,94,58),(11,38,95,69),(12,25,96,56),(13,36,73,67),(14,47,74,54),(15,34,75,65),(16,45,76,52),(17,32,77,63),(18,43,78,50),(19,30,79,61),(20,41,80,72),(21,28,81,59),(22,39,82,70),(23,26,83,57),(24,37,84,68)]])
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 6A | 6B | 6C | 6D | 6E | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | 12F | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 12 | ··· | 12 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | ··· | 4 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | + | + | - | + | - | + | + | + | - | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | S3 | D4 | Q8 | D4 | D6 | Dic3 | D6 | Dic6 | D12 | D12 | C8⋊C22 | C8.C22 | C8⋊D6 | C8.D6 |
| kernel | C23.52D12 | C8⋊Dic3 | C24⋊1C4 | C2×C4⋊Dic3 | C23.26D6 | C6×M4(2) | C3×M4(2) | C2×M4(2) | C2×C12 | C2×C12 | C22×C6 | C2×C8 | M4(2) | C22×C4 | C2×C4 | C2×C4 | C23 | C6 | C6 | C2 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 8 | 1 | 1 | 2 | 1 | 2 | 4 | 1 | 4 | 2 | 2 | 1 | 1 | 2 | 2 |
Matrix representation of C23.52D12 ►in GL6(𝔽73)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 72 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 |
| 0 | 0 | 0 | 0 | 1 | 72 |
| 0 | 0 | 14 | 66 | 0 | 0 |
| 0 | 0 | 7 | 7 | 0 | 0 |
| 63 | 22 | 0 | 0 | 0 | 0 |
| 12 | 10 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 20 | 55 |
| 0 | 0 | 0 | 0 | 2 | 53 |
| 0 | 0 | 53 | 18 | 0 | 0 |
| 0 | 0 | 71 | 20 | 0 | 0 |
G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[0,72,0,0,0,0,1,1,0,0,0,0,0,0,0,0,14,7,0,0,0,0,66,7,0,0,0,1,0,0,0,0,72,72,0,0],[63,12,0,0,0,0,22,10,0,0,0,0,0,0,0,0,53,71,0,0,0,0,18,20,0,0,20,2,0,0,0,0,55,53,0,0] >;
C23.52D12 in GAP, Magma, Sage, TeX
C_2^3._{52}D_{12} % in TeX
G:=Group("C2^3.52D12"); // GroupNames label
G:=SmallGroup(192,680);
// by ID
G=gap.SmallGroup(192,680);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,56,422,387,100,1684,102,6278]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=1,d^12=c,e^2=c*b=b*c,a*b=b*a,d*a*d^-1=e*a*e^-1=a*c=c*a,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^11>;
// generators/relations