metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C23.54D12, C4○D12⋊6C4, D12⋊18(C2×C4), (C2×C8).190D6, C4.39(D6⋊C4), Dic6⋊17(C2×C4), C12.419(C2×D4), (C2×C12).175D4, (C2×C4).154D12, C2.D24⋊40C2, C2.5(C8⋊D6), C6.21(C8⋊C22), (C2×M4(2))⋊13S3, (C6×M4(2))⋊21C2, C22.3(D6⋊C4), C2.Dic12⋊40C2, C2.5(C8.D6), (C22×C4).157D6, C22.58(C2×D12), (C22×C6).102D4, C12.28(C22⋊C4), (C2×C12).774C23, C12.116(C22×C4), (C2×C24).320C22, C6.21(C8.C22), C3⋊4(C23.36D4), (C2×D12).201C22, C4⋊Dic3.285C22, (C22×C12).190C22, (C2×Dic6).221C22, C4.74(S3×C2×C4), (C2×C4).54(C4×S3), C2.32(C2×D6⋊C4), (C2×C4⋊Dic3)⋊33C2, (C2×C6).164(C2×D4), C4.112(C2×C3⋊D4), C6.60(C2×C22⋊C4), (C2×C12).110(C2×C4), (C2×C4○D12).13C2, (C2×C4).78(C3⋊D4), (C2×C6).22(C22⋊C4), (C2×C4).723(C22×S3), SmallGroup(192,692)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C23.54D12
G = < a,b,c,d,e | a2=b2=c2=1, d12=c, e2=cb=bc, ab=ba, dad-1=ac=ca, ae=ea, bd=db, be=eb, cd=dc, ce=ec, ede-1=bd11 >
Subgroups: 472 in 162 conjugacy classes, 63 normal (29 characteristic)
C1, C2, C2, C3, C4, C4, C4, C22, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, Dic3, C12, C12, D6, C2×C6, C2×C6, C2×C6, C4⋊C4, C2×C8, M4(2), C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, C24, Dic6, Dic6, C4×S3, D12, D12, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C22×S3, C22×C6, D4⋊C4, Q8⋊C4, C2×C4⋊C4, C2×M4(2), C2×C4○D4, C4⋊Dic3, C4⋊Dic3, C2×C24, C3×M4(2), C2×Dic6, S3×C2×C4, C2×D12, C4○D12, C4○D12, C22×Dic3, C2×C3⋊D4, C22×C12, C23.36D4, C2.Dic12, C2.D24, C2×C4⋊Dic3, C6×M4(2), C2×C4○D12, C23.54D12
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D6, C22⋊C4, C22×C4, C2×D4, C4×S3, D12, C3⋊D4, C22×S3, C2×C22⋊C4, C8⋊C22, C8.C22, D6⋊C4, S3×C2×C4, C2×D12, C2×C3⋊D4, C23.36D4, C8⋊D6, C8.D6, C2×D6⋊C4, C23.54D12
►On 96 points
Generators in S96
(1 59)(2 72)(3 61)(4 50)(5 63)(6 52)(7 65)(8 54)(9 67)(10 56)(11 69)(12 58)(13 71)(14 60)(15 49)(16 62)(17 51)(18 64)(19 53)(20 66)(21 55)(22 68)(23 57)(24 70)(25 80)(26 93)(27 82)(28 95)(29 84)(30 73)(31 86)(32 75)(33 88)(34 77)(35 90)(36 79)(37 92)(38 81)(39 94)(40 83)(41 96)(42 85)(43 74)(44 87)(45 76)(46 89)(47 78)(48 91)
(1 94)(2 95)(3 96)(4 73)(5 74)(6 75)(7 76)(8 77)(9 78)(10 79)(11 80)(12 81)(13 82)(14 83)(15 84)(16 85)(17 86)(18 87)(19 88)(20 89)(21 90)(22 91)(23 92)(24 93)(25 69)(26 70)(27 71)(28 72)(29 49)(30 50)(31 51)(32 52)(33 53)(34 54)(35 55)(36 56)(37 57)(38 58)(39 59)(40 60)(41 61)(42 62)(43 63)(44 64)(45 65)(46 66)(47 67)(48 68)
(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 81 82 24)(2 23 83 80)(3 79 84 22)(4 21 85 78)(5 77 86 20)(6 19 87 76)(7 75 88 18)(8 17 89 74)(9 73 90 16)(10 15 91 96)(11 95 92 14)(12 13 93 94)(25 72 57 40)(26 39 58 71)(27 70 59 38)(28 37 60 69)(29 68 61 36)(30 35 62 67)(31 66 63 34)(32 33 64 65)(41 56 49 48)(42 47 50 55)(43 54 51 46)(44 45 52 53)
G:=sub<Sym(96)| (1,59)(2,72)(3,61)(4,50)(5,63)(6,52)(7,65)(8,54)(9,67)(10,56)(11,69)(12,58)(13,71)(14,60)(15,49)(16,62)(17,51)(18,64)(19,53)(20,66)(21,55)(22,68)(23,57)(24,70)(25,80)(26,93)(27,82)(28,95)(29,84)(30,73)(31,86)(32,75)(33,88)(34,77)(35,90)(36,79)(37,92)(38,81)(39,94)(40,83)(41,96)(42,85)(43,74)(44,87)(45,76)(46,89)(47,78)(48,91), (1,94)(2,95)(3,96)(4,73)(5,74)(6,75)(7,76)(8,77)(9,78)(10,79)(11,80)(12,81)(13,82)(14,83)(15,84)(16,85)(17,86)(18,87)(19,88)(20,89)(21,90)(22,91)(23,92)(24,93)(25,69)(26,70)(27,71)(28,72)(29,49)(30,50)(31,51)(32,52)(33,53)(34,54)(35,55)(36,56)(37,57)(38,58)(39,59)(40,60)(41,61)(42,62)(43,63)(44,64)(45,65)(46,66)(47,67)(48,68), (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), (1,81,82,24)(2,23,83,80)(3,79,84,22)(4,21,85,78)(5,77,86,20)(6,19,87,76)(7,75,88,18)(8,17,89,74)(9,73,90,16)(10,15,91,96)(11,95,92,14)(12,13,93,94)(25,72,57,40)(26,39,58,71)(27,70,59,38)(28,37,60,69)(29,68,61,36)(30,35,62,67)(31,66,63,34)(32,33,64,65)(41,56,49,48)(42,47,50,55)(43,54,51,46)(44,45,52,53)>;
G:=Group( (1,59)(2,72)(3,61)(4,50)(5,63)(6,52)(7,65)(8,54)(9,67)(10,56)(11,69)(12,58)(13,71)(14,60)(15,49)(16,62)(17,51)(18,64)(19,53)(20,66)(21,55)(22,68)(23,57)(24,70)(25,80)(26,93)(27,82)(28,95)(29,84)(30,73)(31,86)(32,75)(33,88)(34,77)(35,90)(36,79)(37,92)(38,81)(39,94)(40,83)(41,96)(42,85)(43,74)(44,87)(45,76)(46,89)(47,78)(48,91), (1,94)(2,95)(3,96)(4,73)(5,74)(6,75)(7,76)(8,77)(9,78)(10,79)(11,80)(12,81)(13,82)(14,83)(15,84)(16,85)(17,86)(18,87)(19,88)(20,89)(21,90)(22,91)(23,92)(24,93)(25,69)(26,70)(27,71)(28,72)(29,49)(30,50)(31,51)(32,52)(33,53)(34,54)(35,55)(36,56)(37,57)(38,58)(39,59)(40,60)(41,61)(42,62)(43,63)(44,64)(45,65)(46,66)(47,67)(48,68), (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), (1,81,82,24)(2,23,83,80)(3,79,84,22)(4,21,85,78)(5,77,86,20)(6,19,87,76)(7,75,88,18)(8,17,89,74)(9,73,90,16)(10,15,91,96)(11,95,92,14)(12,13,93,94)(25,72,57,40)(26,39,58,71)(27,70,59,38)(28,37,60,69)(29,68,61,36)(30,35,62,67)(31,66,63,34)(32,33,64,65)(41,56,49,48)(42,47,50,55)(43,54,51,46)(44,45,52,53) );
G=PermutationGroup([[(1,59),(2,72),(3,61),(4,50),(5,63),(6,52),(7,65),(8,54),(9,67),(10,56),(11,69),(12,58),(13,71),(14,60),(15,49),(16,62),(17,51),(18,64),(19,53),(20,66),(21,55),(22,68),(23,57),(24,70),(25,80),(26,93),(27,82),(28,95),(29,84),(30,73),(31,86),(32,75),(33,88),(34,77),(35,90),(36,79),(37,92),(38,81),(39,94),(40,83),(41,96),(42,85),(43,74),(44,87),(45,76),(46,89),(47,78),(48,91)], [(1,94),(2,95),(3,96),(4,73),(5,74),(6,75),(7,76),(8,77),(9,78),(10,79),(11,80),(12,81),(13,82),(14,83),(15,84),(16,85),(17,86),(18,87),(19,88),(20,89),(21,90),(22,91),(23,92),(24,93),(25,69),(26,70),(27,71),(28,72),(29,49),(30,50),(31,51),(32,52),(33,53),(34,54),(35,55),(36,56),(37,57),(38,58),(39,59),(40,60),(41,61),(42,62),(43,63),(44,64),(45,65),(46,66),(47,67),(48,68)], [(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)], [(1,81,82,24),(2,23,83,80),(3,79,84,22),(4,21,85,78),(5,77,86,20),(6,19,87,76),(7,75,88,18),(8,17,89,74),(9,73,90,16),(10,15,91,96),(11,95,92,14),(12,13,93,94),(25,72,57,40),(26,39,58,71),(27,70,59,38),(28,37,60,69),(29,68,61,36),(30,35,62,67),(31,66,63,34),(32,33,64,65),(41,56,49,48),(42,47,50,55),(43,54,51,46),(44,45,52,53)]])
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | ··· | 4J | 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 | 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 | 12 | 12 | 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 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | S3 | D4 | D4 | D6 | D6 | C4×S3 | D12 | C3⋊D4 | D12 | C8⋊C22 | C8.C22 | C8⋊D6 | C8.D6 |
| kernel | C23.54D12 | C2.Dic12 | C2.D24 | C2×C4⋊Dic3 | C6×M4(2) | C2×C4○D12 | C4○D12 | C2×M4(2) | C2×C12 | C22×C6 | C2×C8 | C22×C4 | C2×C4 | C2×C4 | C2×C4 | C23 | C6 | C6 | C2 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 8 | 1 | 3 | 1 | 2 | 1 | 4 | 2 | 4 | 2 | 1 | 1 | 2 | 2 |
Matrix representation of C23.54D12 ►in GL6(𝔽73)
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 39 | 41 | 57 | 13 |
| 0 | 0 | 32 | 7 | 60 | 70 |
| 0 | 0 | 54 | 11 | 34 | 32 |
| 0 | 0 | 62 | 65 | 41 | 66 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 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 | 1 |
| 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 |
| 43 | 30 | 0 | 0 | 0 | 0 |
| 43 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 2 |
| 0 | 0 | 1 | 72 | 71 | 2 |
| 0 | 0 | 7 | 69 | 0 | 1 |
| 0 | 0 | 4 | 3 | 72 | 1 |
| 43 | 30 | 0 | 0 | 0 | 0 |
| 60 | 30 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 72 | 71 | 2 |
| 0 | 0 | 0 | 72 | 0 | 2 |
| 0 | 0 | 1 | 72 | 72 | 1 |
| 0 | 0 | 0 | 72 | 0 | 1 |
G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,39,32,54,62,0,0,41,7,11,65,0,0,57,60,34,41,0,0,13,70,32,66],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[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],[43,43,0,0,0,0,30,13,0,0,0,0,0,0,0,1,7,4,0,0,72,72,69,3,0,0,0,71,0,72,0,0,2,2,1,1],[43,60,0,0,0,0,30,30,0,0,0,0,0,0,1,0,1,0,0,0,72,72,72,72,0,0,71,0,72,0,0,0,2,2,1,1] >;
C23.54D12 in GAP, Magma, Sage, TeX
C_2^3._{54}D_{12} % in TeX
G:=Group("C2^3.54D12"); // GroupNames label
G:=SmallGroup(192,692);
// by ID
G=gap.SmallGroup(192,692);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,254,387,142,1123,136,6278]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=1,d^12=c,e^2=c*b=b*c,a*b=b*a,d*a*d^-1=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