metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C23.43D12, C22⋊C8⋊9S3, C8⋊Dic3⋊8C2, (C2×C8).109D6, (C2×C4).33D12, (C2×C12).44D4, C6.8(C2×SD16), C2.D24⋊10C2, C12⋊7D4.2C2, (C2×C6).14SD16, (C22×C6).55D4, C2.14(C8⋊D6), C6.11(C8⋊C22), (C22×C4).101D6, C12.282(C4○D4), (C2×C24).120C22, (C2×C12).745C23, C22.3(C24⋊C2), (C2×D12).10C22, C22.108(C2×D12), C3⋊1(C23.46D4), C4.106(D4⋊2S3), C4⋊Dic3.270C22, (C22×C12).52C22, C6.17(C22.D4), C2.13(C23.21D6), (C2×C4⋊Dic3)⋊5C2, (C3×C22⋊C8)⋊11C2, C2.11(C2×C24⋊C2), (C2×C6).128(C2×D4), (C2×C4).690(C22×S3), SmallGroup(192,294)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C23.43D12
G = < a,b,c,d,e | a2=b2=c2=1, d12=c, e2=cb=bc, dad-1=ab=ba, ac=ca, ae=ea, bd=db, be=eb, cd=dc, ce=ec, ede-1=bd11 >
Subgroups: 384 in 114 conjugacy classes, 43 normal (25 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, C23, C23, Dic3, C12, C12, D6, C2×C6, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C2×D4, C24, D12, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C22×S3, C22×C6, C22⋊C8, D4⋊C4, C4.Q8, C2×C4⋊C4, C4⋊D4, C4⋊Dic3, C4⋊Dic3, C4⋊Dic3, D6⋊C4, C2×C24, C2×D12, C22×Dic3, C2×C3⋊D4, C22×C12, C23.46D4, C8⋊Dic3, C2.D24, C3×C22⋊C8, C2×C4⋊Dic3, C12⋊7D4, C23.43D12
Quotients: C1, C2, C22, S3, D4, C23, D6, SD16, C2×D4, C4○D4, D12, C22×S3, C22.D4, C2×SD16, C8⋊C22, C24⋊C2, C2×D12, D4⋊2S3, C23.46D4, C23.21D6, C2×C24⋊C2, C8⋊D6, C23.43D12
►On 96 points
Generators in S96
(1 59)(2 30)(3 61)(4 32)(5 63)(6 34)(7 65)(8 36)(9 67)(10 38)(11 69)(12 40)(13 71)(14 42)(15 49)(16 44)(17 51)(18 46)(19 53)(20 48)(21 55)(22 26)(23 57)(24 28)(25 74)(27 76)(29 78)(31 80)(33 82)(35 84)(37 86)(39 88)(41 90)(43 92)(45 94)(47 96)(50 93)(52 95)(54 73)(56 75)(58 77)(60 79)(62 81)(64 83)(66 85)(68 87)(70 89)(72 91)
(1 78)(2 79)(3 80)(4 81)(5 82)(6 83)(7 84)(8 85)(9 86)(10 87)(11 88)(12 89)(13 90)(14 91)(15 92)(16 93)(17 94)(18 95)(19 96)(20 73)(21 74)(22 75)(23 76)(24 77)(25 55)(26 56)(27 57)(28 58)(29 59)(30 60)(31 61)(32 62)(33 63)(34 64)(35 65)(36 66)(37 67)(38 68)(39 69)(40 70)(41 71)(42 72)(43 49)(44 50)(45 51)(46 52)(47 53)(48 54)
(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 89 90 24)(2 23 91 88)(3 87 92 22)(4 21 93 86)(5 85 94 20)(6 19 95 84)(7 83 96 18)(8 17 73 82)(9 81 74 16)(10 15 75 80)(11 79 76 14)(12 13 77 78)(25 44 67 62)(26 61 68 43)(27 42 69 60)(28 59 70 41)(29 40 71 58)(30 57 72 39)(31 38 49 56)(32 55 50 37)(33 36 51 54)(34 53 52 35)(45 48 63 66)(46 65 64 47)
G:=sub<Sym(96)| (1,59)(2,30)(3,61)(4,32)(5,63)(6,34)(7,65)(8,36)(9,67)(10,38)(11,69)(12,40)(13,71)(14,42)(15,49)(16,44)(17,51)(18,46)(19,53)(20,48)(21,55)(22,26)(23,57)(24,28)(25,74)(27,76)(29,78)(31,80)(33,82)(35,84)(37,86)(39,88)(41,90)(43,92)(45,94)(47,96)(50,93)(52,95)(54,73)(56,75)(58,77)(60,79)(62,81)(64,83)(66,85)(68,87)(70,89)(72,91), (1,78)(2,79)(3,80)(4,81)(5,82)(6,83)(7,84)(8,85)(9,86)(10,87)(11,88)(12,89)(13,90)(14,91)(15,92)(16,93)(17,94)(18,95)(19,96)(20,73)(21,74)(22,75)(23,76)(24,77)(25,55)(26,56)(27,57)(28,58)(29,59)(30,60)(31,61)(32,62)(33,63)(34,64)(35,65)(36,66)(37,67)(38,68)(39,69)(40,70)(41,71)(42,72)(43,49)(44,50)(45,51)(46,52)(47,53)(48,54), (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), (1,89,90,24)(2,23,91,88)(3,87,92,22)(4,21,93,86)(5,85,94,20)(6,19,95,84)(7,83,96,18)(8,17,73,82)(9,81,74,16)(10,15,75,80)(11,79,76,14)(12,13,77,78)(25,44,67,62)(26,61,68,43)(27,42,69,60)(28,59,70,41)(29,40,71,58)(30,57,72,39)(31,38,49,56)(32,55,50,37)(33,36,51,54)(34,53,52,35)(45,48,63,66)(46,65,64,47)>;
G:=Group( (1,59)(2,30)(3,61)(4,32)(5,63)(6,34)(7,65)(8,36)(9,67)(10,38)(11,69)(12,40)(13,71)(14,42)(15,49)(16,44)(17,51)(18,46)(19,53)(20,48)(21,55)(22,26)(23,57)(24,28)(25,74)(27,76)(29,78)(31,80)(33,82)(35,84)(37,86)(39,88)(41,90)(43,92)(45,94)(47,96)(50,93)(52,95)(54,73)(56,75)(58,77)(60,79)(62,81)(64,83)(66,85)(68,87)(70,89)(72,91), (1,78)(2,79)(3,80)(4,81)(5,82)(6,83)(7,84)(8,85)(9,86)(10,87)(11,88)(12,89)(13,90)(14,91)(15,92)(16,93)(17,94)(18,95)(19,96)(20,73)(21,74)(22,75)(23,76)(24,77)(25,55)(26,56)(27,57)(28,58)(29,59)(30,60)(31,61)(32,62)(33,63)(34,64)(35,65)(36,66)(37,67)(38,68)(39,69)(40,70)(41,71)(42,72)(43,49)(44,50)(45,51)(46,52)(47,53)(48,54), (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), (1,89,90,24)(2,23,91,88)(3,87,92,22)(4,21,93,86)(5,85,94,20)(6,19,95,84)(7,83,96,18)(8,17,73,82)(9,81,74,16)(10,15,75,80)(11,79,76,14)(12,13,77,78)(25,44,67,62)(26,61,68,43)(27,42,69,60)(28,59,70,41)(29,40,71,58)(30,57,72,39)(31,38,49,56)(32,55,50,37)(33,36,51,54)(34,53,52,35)(45,48,63,66)(46,65,64,47) );
G=PermutationGroup([[(1,59),(2,30),(3,61),(4,32),(5,63),(6,34),(7,65),(8,36),(9,67),(10,38),(11,69),(12,40),(13,71),(14,42),(15,49),(16,44),(17,51),(18,46),(19,53),(20,48),(21,55),(22,26),(23,57),(24,28),(25,74),(27,76),(29,78),(31,80),(33,82),(35,84),(37,86),(39,88),(41,90),(43,92),(45,94),(47,96),(50,93),(52,95),(54,73),(56,75),(58,77),(60,79),(62,81),(64,83),(66,85),(68,87),(70,89),(72,91)], [(1,78),(2,79),(3,80),(4,81),(5,82),(6,83),(7,84),(8,85),(9,86),(10,87),(11,88),(12,89),(13,90),(14,91),(15,92),(16,93),(17,94),(18,95),(19,96),(20,73),(21,74),(22,75),(23,76),(24,77),(25,55),(26,56),(27,57),(28,58),(29,59),(30,60),(31,61),(32,62),(33,63),(34,64),(35,65),(36,66),(37,67),(38,68),(39,69),(40,70),(41,71),(42,72),(43,49),(44,50),(45,51),(46,52),(47,53),(48,54)], [(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)], [(1,89,90,24),(2,23,91,88),(3,87,92,22),(4,21,93,86),(5,85,94,20),(6,19,95,84),(7,83,96,18),(8,17,73,82),(9,81,74,16),(10,15,75,80),(11,79,76,14),(12,13,77,78),(25,44,67,62),(26,61,68,43),(27,42,69,60),(28,59,70,41),(29,40,71,58),(30,57,72,39),(31,38,49,56),(32,55,50,37),(33,36,51,54),(34,53,52,35),(45,48,63,66),(46,65,64,47)]])
39 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | 6B | 6C | 6D | 6E | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | 12F | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 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 | 24 | 2 | 2 | 2 | 4 | 12 | 12 | 12 | 12 | 24 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | ··· | 4 |
39 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D6 | D6 | C4○D4 | SD16 | D12 | D12 | C24⋊C2 | C8⋊C22 | D4⋊2S3 | C8⋊D6 |
| kernel | C23.43D12 | C8⋊Dic3 | C2.D24 | C3×C22⋊C8 | C2×C4⋊Dic3 | C12⋊7D4 | C22⋊C8 | C2×C12 | C22×C6 | C2×C8 | C22×C4 | C12 | C2×C6 | C2×C4 | C23 | C22 | C6 | C4 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 4 | 4 | 2 | 2 | 8 | 1 | 2 | 2 |
Matrix representation of C23.43D12 ►in GL4(𝔽73) generated by
| 72 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 46 |
| 0 | 0 | 27 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 72 |
| 72 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 25 | 36 | 0 | 0 |
| 37 | 62 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
| 25 | 36 | 0 | 0 |
| 11 | 48 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 72 | 0 |
G:=sub<GL(4,GF(73))| [72,0,0,0,0,72,0,0,0,0,0,27,0,0,46,0],[1,0,0,0,0,1,0,0,0,0,72,0,0,0,0,72],[72,0,0,0,0,72,0,0,0,0,1,0,0,0,0,1],[25,37,0,0,36,62,0,0,0,0,0,1,0,0,1,0],[25,11,0,0,36,48,0,0,0,0,0,72,0,0,1,0] >;
C23.43D12 in GAP, Magma, Sage, TeX
C_2^3._{43}D_{12} % in TeX
G:=Group("C2^3.43D12"); // GroupNames label
G:=SmallGroup(192,294);
// by ID
G=gap.SmallGroup(192,294);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,254,219,142,1123,136,6278]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=1,d^12=c,e^2=c*b=b*c,d*a*d^-1=a*b=b*a,a*c=c*a,a*e=e*a,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=b*d^11>;
// generators/relations