direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×S3×Q16, C24.33C23, C12.10C24, Dic6.6C23, Dic12⋊15C22, C6⋊2(C2×Q16), (C6×Q16)⋊7C2, C4.46(S3×D4), C3⋊2(C22×Q16), (C4×S3).30D4, D6.65(C2×D4), C12.85(C2×D4), (C2×C8).246D6, C3⋊C8.22C23, C8.39(C22×S3), C4.10(S3×C23), (C2×Q8).176D6, (C3×Q16)⋊8C22, C3⋊Q16⋊8C22, (C2×Dic12)⋊20C2, (C3×Q8).4C23, (S3×Q8).4C22, (C4×S3).27C23, (S3×C8).15C22, (C2×C24).98C22, Dic3.13(C2×D4), C6.111(C22×D4), C22.142(S3×D4), Q8.14(C22×S3), (C2×C12).527C23, (C2×Dic3).123D4, (C22×S3).112D4, (C6×Q8).149C22, (C2×Dic6).198C22, (S3×C2×C8).6C2, C2.84(C2×S3×D4), (C2×S3×Q8).8C2, (C2×C3⋊Q16)⋊27C2, (C2×C6).400(C2×D4), (C2×C3⋊C8).285C22, (S3×C2×C4).259C22, (C2×C4).615(C22×S3), SmallGroup(192,1322)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 600 in 258 conjugacy classes, 111 normal (23 characteristic)
C1, C2, C2 [×2], C2 [×4], C3, C4 [×2], C4 [×10], C22, C22 [×6], S3 [×4], C6, C6 [×2], C8 [×2], C8 [×2], C2×C4, C2×C4 [×17], Q8 [×4], Q8 [×16], C23, Dic3 [×2], Dic3 [×4], C12 [×2], C12 [×4], D6 [×6], C2×C6, C2×C8, C2×C8 [×5], Q16 [×4], Q16 [×12], C22×C4 [×3], C2×Q8 [×2], C2×Q8 [×16], C3⋊C8 [×2], C24 [×2], Dic6 [×4], Dic6 [×10], C4×S3 [×4], C4×S3 [×8], C2×Dic3, C2×Dic3 [×2], C2×C12, C2×C12 [×2], C3×Q8 [×4], C3×Q8 [×2], C22×S3, C22×C8, C2×Q16, C2×Q16 [×11], C22×Q8 [×2], S3×C8 [×4], Dic12 [×4], C2×C3⋊C8, C3⋊Q16 [×8], C2×C24, C3×Q16 [×4], C2×Dic6 [×2], C2×Dic6 [×2], S3×C2×C4, S3×C2×C4 [×2], S3×Q8 [×8], S3×Q8 [×4], C6×Q8 [×2], C22×Q16, S3×C2×C8, C2×Dic12, S3×Q16 [×8], C2×C3⋊Q16 [×2], C6×Q16, C2×S3×Q8 [×2], C2×S3×Q16
Quotients:
C1, C2 [×15], C22 [×35], S3, D4 [×4], C23 [×15], D6 [×7], Q16 [×4], C2×D4 [×6], C24, C22×S3 [×7], C2×Q16 [×6], C22×D4, S3×D4 [×2], S3×C23, C22×Q16, S3×Q16 [×2], C2×S3×D4, C2×S3×Q16
Generators and relations
G = < a,b,c,d,e | a2=b3=c2=d8=1, e2=d4, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >
►On 96 points
(1 58)(2 59)(3 60)(4 61)(5 62)(6 63)(7 64)(8 57)(9 70)(10 71)(11 72)(12 65)(13 66)(14 67)(15 68)(16 69)(17 46)(18 47)(19 48)(20 41)(21 42)(22 43)(23 44)(24 45)(25 89)(26 90)(27 91)(28 92)(29 93)(30 94)(31 95)(32 96)(33 85)(34 86)(35 87)(36 88)(37 81)(38 82)(39 83)(40 84)(49 73)(50 74)(51 75)(52 76)(53 77)(54 78)(55 79)(56 80)
(1 13 82)(2 14 83)(3 15 84)(4 16 85)(5 9 86)(6 10 87)(7 11 88)(8 12 81)(17 26 80)(18 27 73)(19 28 74)(20 29 75)(21 30 76)(22 31 77)(23 32 78)(24 25 79)(33 61 69)(34 62 70)(35 63 71)(36 64 72)(37 57 65)(38 58 66)(39 59 67)(40 60 68)(41 93 51)(42 94 52)(43 95 53)(44 96 54)(45 89 55)(46 90 56)(47 91 49)(48 92 50)
(9 86)(10 87)(11 88)(12 81)(13 82)(14 83)(15 84)(16 85)(25 79)(26 80)(27 73)(28 74)(29 75)(30 76)(31 77)(32 78)(33 69)(34 70)(35 71)(36 72)(37 65)(38 66)(39 67)(40 68)(49 91)(50 92)(51 93)(52 94)(53 95)(54 96)(55 89)(56 90)
(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 20 5 24)(2 19 6 23)(3 18 7 22)(4 17 8 21)(9 25 13 29)(10 32 14 28)(11 31 15 27)(12 30 16 26)(33 56 37 52)(34 55 38 51)(35 54 39 50)(36 53 40 49)(41 62 45 58)(42 61 46 57)(43 60 47 64)(44 59 48 63)(65 94 69 90)(66 93 70 89)(67 92 71 96)(68 91 72 95)(73 88 77 84)(74 87 78 83)(75 86 79 82)(76 85 80 81)
G:=sub<Sym(96)| (1,58)(2,59)(3,60)(4,61)(5,62)(6,63)(7,64)(8,57)(9,70)(10,71)(11,72)(12,65)(13,66)(14,67)(15,68)(16,69)(17,46)(18,47)(19,48)(20,41)(21,42)(22,43)(23,44)(24,45)(25,89)(26,90)(27,91)(28,92)(29,93)(30,94)(31,95)(32,96)(33,85)(34,86)(35,87)(36,88)(37,81)(38,82)(39,83)(40,84)(49,73)(50,74)(51,75)(52,76)(53,77)(54,78)(55,79)(56,80), (1,13,82)(2,14,83)(3,15,84)(4,16,85)(5,9,86)(6,10,87)(7,11,88)(8,12,81)(17,26,80)(18,27,73)(19,28,74)(20,29,75)(21,30,76)(22,31,77)(23,32,78)(24,25,79)(33,61,69)(34,62,70)(35,63,71)(36,64,72)(37,57,65)(38,58,66)(39,59,67)(40,60,68)(41,93,51)(42,94,52)(43,95,53)(44,96,54)(45,89,55)(46,90,56)(47,91,49)(48,92,50), (9,86)(10,87)(11,88)(12,81)(13,82)(14,83)(15,84)(16,85)(25,79)(26,80)(27,73)(28,74)(29,75)(30,76)(31,77)(32,78)(33,69)(34,70)(35,71)(36,72)(37,65)(38,66)(39,67)(40,68)(49,91)(50,92)(51,93)(52,94)(53,95)(54,96)(55,89)(56,90), (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), (1,20,5,24)(2,19,6,23)(3,18,7,22)(4,17,8,21)(9,25,13,29)(10,32,14,28)(11,31,15,27)(12,30,16,26)(33,56,37,52)(34,55,38,51)(35,54,39,50)(36,53,40,49)(41,62,45,58)(42,61,46,57)(43,60,47,64)(44,59,48,63)(65,94,69,90)(66,93,70,89)(67,92,71,96)(68,91,72,95)(73,88,77,84)(74,87,78,83)(75,86,79,82)(76,85,80,81)>;
G:=Group( (1,58)(2,59)(3,60)(4,61)(5,62)(6,63)(7,64)(8,57)(9,70)(10,71)(11,72)(12,65)(13,66)(14,67)(15,68)(16,69)(17,46)(18,47)(19,48)(20,41)(21,42)(22,43)(23,44)(24,45)(25,89)(26,90)(27,91)(28,92)(29,93)(30,94)(31,95)(32,96)(33,85)(34,86)(35,87)(36,88)(37,81)(38,82)(39,83)(40,84)(49,73)(50,74)(51,75)(52,76)(53,77)(54,78)(55,79)(56,80), (1,13,82)(2,14,83)(3,15,84)(4,16,85)(5,9,86)(6,10,87)(7,11,88)(8,12,81)(17,26,80)(18,27,73)(19,28,74)(20,29,75)(21,30,76)(22,31,77)(23,32,78)(24,25,79)(33,61,69)(34,62,70)(35,63,71)(36,64,72)(37,57,65)(38,58,66)(39,59,67)(40,60,68)(41,93,51)(42,94,52)(43,95,53)(44,96,54)(45,89,55)(46,90,56)(47,91,49)(48,92,50), (9,86)(10,87)(11,88)(12,81)(13,82)(14,83)(15,84)(16,85)(25,79)(26,80)(27,73)(28,74)(29,75)(30,76)(31,77)(32,78)(33,69)(34,70)(35,71)(36,72)(37,65)(38,66)(39,67)(40,68)(49,91)(50,92)(51,93)(52,94)(53,95)(54,96)(55,89)(56,90), (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), (1,20,5,24)(2,19,6,23)(3,18,7,22)(4,17,8,21)(9,25,13,29)(10,32,14,28)(11,31,15,27)(12,30,16,26)(33,56,37,52)(34,55,38,51)(35,54,39,50)(36,53,40,49)(41,62,45,58)(42,61,46,57)(43,60,47,64)(44,59,48,63)(65,94,69,90)(66,93,70,89)(67,92,71,96)(68,91,72,95)(73,88,77,84)(74,87,78,83)(75,86,79,82)(76,85,80,81) );
G=PermutationGroup([(1,58),(2,59),(3,60),(4,61),(5,62),(6,63),(7,64),(8,57),(9,70),(10,71),(11,72),(12,65),(13,66),(14,67),(15,68),(16,69),(17,46),(18,47),(19,48),(20,41),(21,42),(22,43),(23,44),(24,45),(25,89),(26,90),(27,91),(28,92),(29,93),(30,94),(31,95),(32,96),(33,85),(34,86),(35,87),(36,88),(37,81),(38,82),(39,83),(40,84),(49,73),(50,74),(51,75),(52,76),(53,77),(54,78),(55,79),(56,80)], [(1,13,82),(2,14,83),(3,15,84),(4,16,85),(5,9,86),(6,10,87),(7,11,88),(8,12,81),(17,26,80),(18,27,73),(19,28,74),(20,29,75),(21,30,76),(22,31,77),(23,32,78),(24,25,79),(33,61,69),(34,62,70),(35,63,71),(36,64,72),(37,57,65),(38,58,66),(39,59,67),(40,60,68),(41,93,51),(42,94,52),(43,95,53),(44,96,54),(45,89,55),(46,90,56),(47,91,49),(48,92,50)], [(9,86),(10,87),(11,88),(12,81),(13,82),(14,83),(15,84),(16,85),(25,79),(26,80),(27,73),(28,74),(29,75),(30,76),(31,77),(32,78),(33,69),(34,70),(35,71),(36,72),(37,65),(38,66),(39,67),(40,68),(49,91),(50,92),(51,93),(52,94),(53,95),(54,96),(55,89),(56,90)], [(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)], [(1,20,5,24),(2,19,6,23),(3,18,7,22),(4,17,8,21),(9,25,13,29),(10,32,14,28),(11,31,15,27),(12,30,16,26),(33,56,37,52),(34,55,38,51),(35,54,39,50),(36,53,40,49),(41,62,45,58),(42,61,46,57),(43,60,47,64),(44,59,48,63),(65,94,69,90),(66,93,70,89),(67,92,71,96),(68,91,72,95),(73,88,77,84),(74,87,78,83),(75,86,79,82),(76,85,80,81)])
Matrix representation ►G ⊆ GL5(𝔽73)
| 72 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 72 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 72 | 72 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 72 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 72 | 72 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 72 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 41 | 25 |
| 0 | 0 | 0 | 35 | 0 |
| 72 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 49 | 23 |
| 0 | 0 | 0 | 13 | 24 |
G:=sub<GL(5,GF(73))| [72,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,72,0,0,0,0,0,72],[1,0,0,0,0,0,72,1,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,1],[72,0,0,0,0,0,1,72,0,0,0,0,72,0,0,0,0,0,1,0,0,0,0,0,1],[72,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,41,35,0,0,0,25,0],[72,0,0,0,0,0,72,0,0,0,0,0,72,0,0,0,0,0,49,13,0,0,0,23,24] >;
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 12A | 12B | 12C | 12D | 12E | 12F | 24A | 24B | 24C | 24D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 24 | 24 | 24 | 24 |
| size | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 6 | 6 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | 4 | 4 | 4 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | + | - |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D4 | D6 | D6 | D6 | Q16 | S3×D4 | S3×D4 | S3×Q16 |
| kernel | C2×S3×Q16 | S3×C2×C8 | C2×Dic12 | S3×Q16 | C2×C3⋊Q16 | C6×Q16 | C2×S3×Q8 | C2×Q16 | C4×S3 | C2×Dic3 | C22×S3 | C2×C8 | Q16 | C2×Q8 | D6 | C4 | C22 | C2 |
| # reps | 1 | 1 | 1 | 8 | 2 | 1 | 2 | 1 | 2 | 1 | 1 | 1 | 4 | 2 | 8 | 1 | 1 | 4 |
In GAP, Magma, Sage, TeX
C_2\times S_3\times Q_{16} % in TeX
G:=Group("C2xS3xQ16"); // GroupNames label
G:=SmallGroup(192,1322);
// by ID
G=gap.SmallGroup(192,1322);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,184,185,136,438,235,102,6278]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^3=c^2=d^8=1,e^2=d^4,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations