G = C6.452+ 1+4 order 192 = 26·3
45th non-split extension by C6 of 2+ 1+4 acting via 2+ 1+4/C2×D4=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C6.452+ 1+4, C12⋊Q8⋊21C2, C4⋊D4⋊19S3, C4⋊C4.183D6, (C2×D4).94D6, D6⋊3D4⋊25C2, (D4×Dic3)⋊23C2, C22⋊C4.51D6, (C2×C6).160C24, (C2×C12).43C23, D6⋊C4.16C22, (C22×C4).243D6, C23.14D6⋊15C2, C12.204(C4○D4), C4.97(D4⋊2S3), C2.47(D4⋊6D6), C23.12D6⋊19C2, (C6×D4).126C22, C23.16D6⋊7C2, C23.30(C22×S3), Dic3.23(C4○D4), C23.11D6⋊21C2, C23.26D6⋊27C2, Dic3⋊C4.19C22, (C22×S3).67C23, C4⋊Dic3.373C22, C22.181(S3×C23), (C22×C6).189C23, (C4×Dic3).97C22, (C2×Dic3).79C23, (C22×C12).244C22, C3⋊3(C22.49C24), (C2×Dic6).156C22, C6.D4.112C22, (C22×Dic3).113C22, (C4×C3⋊D4)⋊20C2, C4⋊C4⋊7S3⋊21C2, C2.44(S3×C4○D4), (C3×C4⋊D4)⋊22C2, C6.157(C2×C4○D4), (S3×C2×C4).87C22, C2.39(C2×D4⋊2S3), (C3×C4⋊C4).148C22, (C2×C4).588(C22×S3), (C2×C3⋊D4).33C22, (C3×C22⋊C4).17C22, SmallGroup(192,1175)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C6.452+ 1+4
G = < a,b,c,d,e | a6=b4=c2=e2=1, d2=a3b2, ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ece=a3c, ede=a3b2d >
Subgroups: 576 in 236 conjugacy classes, 97 normal (43 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C23, Dic3, Dic3, C12, C12, D6, C2×C6, C2×C6, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×D4, C2×Q8, Dic6, C4×S3, C2×Dic3, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C2×C12, C3×D4, C22×S3, C22×C6, C22×C6, C42⋊C2, C4×D4, C4⋊D4, C4⋊D4, C4.4D4, C4⋊Q8, C4×Dic3, C4×Dic3, Dic3⋊C4, Dic3⋊C4, C4⋊Dic3, D6⋊C4, D6⋊C4, C6.D4, C6.D4, C3×C22⋊C4, C3×C4⋊C4, C2×Dic6, S3×C2×C4, C22×Dic3, C2×C3⋊D4, C2×C3⋊D4, C22×C12, C6×D4, C6×D4, C22.49C24, C23.16D6, C23.11D6, C12⋊Q8, C4⋊C4⋊7S3, C23.26D6, C4×C3⋊D4, D4×Dic3, C23.12D6, D6⋊3D4, C23.14D6, C3×C4⋊D4, C6.452+ 1+4
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C24, C22×S3, C2×C4○D4, 2+ 1+4, D4⋊2S3, S3×C23, C22.49C24, C2×D4⋊2S3, D4⋊6D6, S3×C4○D4, C6.452+ 1+4
►On 96 points
(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 82 7 76)(2 83 8 77)(3 84 9 78)(4 79 10 73)(5 80 11 74)(6 81 12 75)(13 94 19 88)(14 95 20 89)(15 96 21 90)(16 91 22 85)(17 92 23 86)(18 93 24 87)(25 58 31 52)(26 59 32 53)(27 60 33 54)(28 55 34 49)(29 56 35 50)(30 57 36 51)(37 70 43 64)(38 71 44 65)(39 72 45 66)(40 67 46 61)(41 68 47 62)(42 69 48 63)
(1 49)(2 50)(3 51)(4 52)(5 53)(6 54)(7 55)(8 56)(9 57)(10 58)(11 59)(12 60)(13 61)(14 62)(15 63)(16 64)(17 65)(18 66)(19 67)(20 68)(21 69)(22 70)(23 71)(24 72)(25 73)(26 74)(27 75)(28 76)(29 77)(30 78)(31 79)(32 80)(33 81)(34 82)(35 83)(36 84)(37 85)(38 86)(39 87)(40 88)(41 89)(42 90)(43 91)(44 92)(45 93)(46 94)(47 95)(48 96)
(1 22 10 13)(2 21 11 18)(3 20 12 17)(4 19 7 16)(5 24 8 15)(6 23 9 14)(25 46 34 37)(26 45 35 42)(27 44 36 41)(28 43 31 40)(29 48 32 39)(30 47 33 38)(49 70 58 61)(50 69 59 66)(51 68 60 65)(52 67 55 64)(53 72 56 63)(54 71 57 62)(73 94 82 85)(74 93 83 90)(75 92 84 89)(76 91 79 88)(77 96 80 87)(78 95 81 86)
(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 40)(26 41)(27 42)(28 37)(29 38)(30 39)(31 46)(32 47)(33 48)(34 43)(35 44)(36 45)(49 64)(50 65)(51 66)(52 61)(53 62)(54 63)(55 70)(56 71)(57 72)(58 67)(59 68)(60 69)(73 85)(74 86)(75 87)(76 88)(77 89)(78 90)(79 91)(80 92)(81 93)(82 94)(83 95)(84 96)
G:=sub<Sym(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), (1,82,7,76)(2,83,8,77)(3,84,9,78)(4,79,10,73)(5,80,11,74)(6,81,12,75)(13,94,19,88)(14,95,20,89)(15,96,21,90)(16,91,22,85)(17,92,23,86)(18,93,24,87)(25,58,31,52)(26,59,32,53)(27,60,33,54)(28,55,34,49)(29,56,35,50)(30,57,36,51)(37,70,43,64)(38,71,44,65)(39,72,45,66)(40,67,46,61)(41,68,47,62)(42,69,48,63), (1,49)(2,50)(3,51)(4,52)(5,53)(6,54)(7,55)(8,56)(9,57)(10,58)(11,59)(12,60)(13,61)(14,62)(15,63)(16,64)(17,65)(18,66)(19,67)(20,68)(21,69)(22,70)(23,71)(24,72)(25,73)(26,74)(27,75)(28,76)(29,77)(30,78)(31,79)(32,80)(33,81)(34,82)(35,83)(36,84)(37,85)(38,86)(39,87)(40,88)(41,89)(42,90)(43,91)(44,92)(45,93)(46,94)(47,95)(48,96), (1,22,10,13)(2,21,11,18)(3,20,12,17)(4,19,7,16)(5,24,8,15)(6,23,9,14)(25,46,34,37)(26,45,35,42)(27,44,36,41)(28,43,31,40)(29,48,32,39)(30,47,33,38)(49,70,58,61)(50,69,59,66)(51,68,60,65)(52,67,55,64)(53,72,56,63)(54,71,57,62)(73,94,82,85)(74,93,83,90)(75,92,84,89)(76,91,79,88)(77,96,80,87)(78,95,81,86), (1,13)(2,14)(3,15)(4,16)(5,17)(6,18)(7,19)(8,20)(9,21)(10,22)(11,23)(12,24)(25,40)(26,41)(27,42)(28,37)(29,38)(30,39)(31,46)(32,47)(33,48)(34,43)(35,44)(36,45)(49,64)(50,65)(51,66)(52,61)(53,62)(54,63)(55,70)(56,71)(57,72)(58,67)(59,68)(60,69)(73,85)(74,86)(75,87)(76,88)(77,89)(78,90)(79,91)(80,92)(81,93)(82,94)(83,95)(84,96)>;
G:=Group( (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), (1,82,7,76)(2,83,8,77)(3,84,9,78)(4,79,10,73)(5,80,11,74)(6,81,12,75)(13,94,19,88)(14,95,20,89)(15,96,21,90)(16,91,22,85)(17,92,23,86)(18,93,24,87)(25,58,31,52)(26,59,32,53)(27,60,33,54)(28,55,34,49)(29,56,35,50)(30,57,36,51)(37,70,43,64)(38,71,44,65)(39,72,45,66)(40,67,46,61)(41,68,47,62)(42,69,48,63), (1,49)(2,50)(3,51)(4,52)(5,53)(6,54)(7,55)(8,56)(9,57)(10,58)(11,59)(12,60)(13,61)(14,62)(15,63)(16,64)(17,65)(18,66)(19,67)(20,68)(21,69)(22,70)(23,71)(24,72)(25,73)(26,74)(27,75)(28,76)(29,77)(30,78)(31,79)(32,80)(33,81)(34,82)(35,83)(36,84)(37,85)(38,86)(39,87)(40,88)(41,89)(42,90)(43,91)(44,92)(45,93)(46,94)(47,95)(48,96), (1,22,10,13)(2,21,11,18)(3,20,12,17)(4,19,7,16)(5,24,8,15)(6,23,9,14)(25,46,34,37)(26,45,35,42)(27,44,36,41)(28,43,31,40)(29,48,32,39)(30,47,33,38)(49,70,58,61)(50,69,59,66)(51,68,60,65)(52,67,55,64)(53,72,56,63)(54,71,57,62)(73,94,82,85)(74,93,83,90)(75,92,84,89)(76,91,79,88)(77,96,80,87)(78,95,81,86), (1,13)(2,14)(3,15)(4,16)(5,17)(6,18)(7,19)(8,20)(9,21)(10,22)(11,23)(12,24)(25,40)(26,41)(27,42)(28,37)(29,38)(30,39)(31,46)(32,47)(33,48)(34,43)(35,44)(36,45)(49,64)(50,65)(51,66)(52,61)(53,62)(54,63)(55,70)(56,71)(57,72)(58,67)(59,68)(60,69)(73,85)(74,86)(75,87)(76,88)(77,89)(78,90)(79,91)(80,92)(81,93)(82,94)(83,95)(84,96) );
G=PermutationGroup([[(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)], [(1,82,7,76),(2,83,8,77),(3,84,9,78),(4,79,10,73),(5,80,11,74),(6,81,12,75),(13,94,19,88),(14,95,20,89),(15,96,21,90),(16,91,22,85),(17,92,23,86),(18,93,24,87),(25,58,31,52),(26,59,32,53),(27,60,33,54),(28,55,34,49),(29,56,35,50),(30,57,36,51),(37,70,43,64),(38,71,44,65),(39,72,45,66),(40,67,46,61),(41,68,47,62),(42,69,48,63)], [(1,49),(2,50),(3,51),(4,52),(5,53),(6,54),(7,55),(8,56),(9,57),(10,58),(11,59),(12,60),(13,61),(14,62),(15,63),(16,64),(17,65),(18,66),(19,67),(20,68),(21,69),(22,70),(23,71),(24,72),(25,73),(26,74),(27,75),(28,76),(29,77),(30,78),(31,79),(32,80),(33,81),(34,82),(35,83),(36,84),(37,85),(38,86),(39,87),(40,88),(41,89),(42,90),(43,91),(44,92),(45,93),(46,94),(47,95),(48,96)], [(1,22,10,13),(2,21,11,18),(3,20,12,17),(4,19,7,16),(5,24,8,15),(6,23,9,14),(25,46,34,37),(26,45,35,42),(27,44,36,41),(28,43,31,40),(29,48,32,39),(30,47,33,38),(49,70,58,61),(50,69,59,66),(51,68,60,65),(52,67,55,64),(53,72,56,63),(54,71,57,62),(73,94,82,85),(74,93,83,90),(75,92,84,89),(76,91,79,88),(77,96,80,87),(78,95,81,86)], [(1,13),(2,14),(3,15),(4,16),(5,17),(6,18),(7,19),(8,20),(9,21),(10,22),(11,23),(12,24),(25,40),(26,41),(27,42),(28,37),(29,38),(30,39),(31,46),(32,47),(33,48),(34,43),(35,44),(36,45),(49,64),(50,65),(51,66),(52,61),(53,62),(54,63),(55,70),(56,71),(57,72),(58,67),(59,68),(60,69),(73,85),(74,86),(75,87),(76,88),(77,89),(78,90),(79,91),(80,92),(81,93),(82,94),(83,95),(84,96)]])
39 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | ··· | 4N | 4O | 4P | 4Q | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 12A | 12B | 12C | 12D | 12E | 12F |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 12 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 6 | ··· | 6 | 12 | 12 | 12 | 2 | 2 | 2 | 4 | 4 | 8 | 8 | 4 | 4 | 4 | 4 | 8 | 8 |
39 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D6 | D6 | D6 | D6 | C4○D4 | C4○D4 | 2+ 1+4 | D4⋊2S3 | D4⋊6D6 | S3×C4○D4 |
| kernel | C6.452+ 1+4 | C23.16D6 | C23.11D6 | C12⋊Q8 | C4⋊C4⋊7S3 | C23.26D6 | C4×C3⋊D4 | D4×Dic3 | C23.12D6 | D6⋊3D4 | C23.14D6 | C3×C4⋊D4 | C4⋊D4 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | Dic3 | C12 | C6 | C4 | C2 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 2 | 1 | 1 | 2 | 1 | 1 | 3 | 4 | 4 | 1 | 2 | 2 | 2 |
Matrix representation of C6.452+ 1+4 ►in GL6(𝔽13)
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 12 | 0 | 0 |
| 0 | 0 | 1 | 0 | 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 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 0 | 1 | 5 |
| 0 | 8 | 0 | 0 | 0 | 0 |
| 5 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 10 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 12 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 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 | 8 | 1 |
G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,1,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,8,1,0,0,0,0,0,5],[0,5,0,0,0,0,8,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,10,12],[0,12,0,0,0,0,1,0,0,0,0,0,0,0,1,12,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,8,0,0,0,0,0,1] >;
C6.452+ 1+4 in GAP, Magma, Sage, TeX
C_6._{45}2_+^{1+4} % in TeX
G:=Group("C6.45ES+(2,2)"); // GroupNames label
G:=SmallGroup(192,1175);
// by ID
G=gap.SmallGroup(192,1175);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,477,758,219,1571,570,297,6278]);
// Polycyclic
G:=Group<a,b,c,d,e|a^6=b^4=c^2=e^2=1,d^2=a^3*b^2,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e=a^3*c,e*d*e=a^3*b^2*d>;
// generators/relations