direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C3×C23.20D4, C2.D8⋊9C6, C4.Q8⋊12C6, Q8⋊C4⋊8C6, C22⋊C8.5C6, C22⋊Q8.7C6, (C2×C12).462D4, C23.19(C3×D4), (C22×C6).37D4, C6.129(C4○D8), C42⋊C2.9C6, C22.106(C6×D4), C12.322(C4○D4), (C2×C24).192C22, (C2×C12).941C23, (C6×Q8).171C22, C6.144(C8.C22), (C22×C12).433C22, C6.100(C22.D4), C4⋊C4.62(C2×C6), (C2×C8).11(C2×C6), (C3×C2.D8)⋊24C2, (C3×C4.Q8)⋊27C2, C4.34(C3×C4○D4), C2.16(C3×C4○D8), (C2×C4).108(C3×D4), (C2×C6).662(C2×D4), (C2×Q8).16(C2×C6), (C3×Q8⋊C4)⋊36C2, (C3×C22⋊C8).11C2, (C22×C4).56(C2×C6), C2.19(C3×C8.C22), (C3×C22⋊Q8).17C2, (C3×C4⋊C4).385C22, (C2×C4).116(C22×C6), (C3×C42⋊C2).23C2, C2.16(C3×C22.D4), SmallGroup(192,918)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C23.20D4
G = < a,b,c,d,e,f | a3=b2=c2=d2=1, e4=f2=d, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ebe-1=bc=cb, bd=db, fbf-1=bcd, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=ce3 >
Subgroups: 162 in 96 conjugacy classes, 50 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C6, C8, C2×C4, C2×C4, Q8, C23, C12, C12, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C2×Q8, C24, C2×C12, C2×C12, C3×Q8, C22×C6, C22⋊C8, Q8⋊C4, C4.Q8, C2.D8, C42⋊C2, C22⋊Q8, C4×C12, C3×C22⋊C4, C3×C4⋊C4, C3×C4⋊C4, C2×C24, C22×C12, C6×Q8, C23.20D4, C3×C22⋊C8, C3×Q8⋊C4, C3×C4.Q8, C3×C2.D8, C3×C42⋊C2, C3×C22⋊Q8, C3×C23.20D4
Quotients: C1, C2, C3, C22, C6, D4, C23, C2×C6, C2×D4, C4○D4, C3×D4, C22×C6, C22.D4, C4○D8, C8.C22, C6×D4, C3×C4○D4, C23.20D4, C3×C22.D4, C3×C4○D8, C3×C8.C22, C3×C23.20D4
►On 96 points
Generators in S96
(1 79 95)(2 80 96)(3 73 89)(4 74 90)(5 75 91)(6 76 92)(7 77 93)(8 78 94)(9 65 81)(10 66 82)(11 67 83)(12 68 84)(13 69 85)(14 70 86)(15 71 87)(16 72 88)(17 49 25)(18 50 26)(19 51 27)(20 52 28)(21 53 29)(22 54 30)(23 55 31)(24 56 32)(33 41 61)(34 42 62)(35 43 63)(36 44 64)(37 45 57)(38 46 58)(39 47 59)(40 48 60)
(2 56)(4 50)(6 52)(8 54)(9 13)(10 61)(11 15)(12 63)(14 57)(16 59)(18 90)(20 92)(22 94)(24 96)(26 74)(28 76)(30 78)(32 80)(33 66)(34 38)(35 68)(36 40)(37 70)(39 72)(41 82)(42 46)(43 84)(44 48)(45 86)(47 88)(58 62)(60 64)(65 69)(67 71)(81 85)(83 87)
(1 55)(2 56)(3 49)(4 50)(5 51)(6 52)(7 53)(8 54)(9 64)(10 57)(11 58)(12 59)(13 60)(14 61)(15 62)(16 63)(17 89)(18 90)(19 91)(20 92)(21 93)(22 94)(23 95)(24 96)(25 73)(26 74)(27 75)(28 76)(29 77)(30 78)(31 79)(32 80)(33 70)(34 71)(35 72)(36 65)(37 66)(38 67)(39 68)(40 69)(41 86)(42 87)(43 88)(44 81)(45 82)(46 83)(47 84)(48 85)
(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)(17 21)(18 22)(19 23)(20 24)(25 29)(26 30)(27 31)(28 32)(33 37)(34 38)(35 39)(36 40)(41 45)(42 46)(43 47)(44 48)(49 53)(50 54)(51 55)(52 56)(57 61)(58 62)(59 63)(60 64)(65 69)(66 70)(67 71)(68 72)(73 77)(74 78)(75 79)(76 80)(81 85)(82 86)(83 87)(84 88)(89 93)(90 94)(91 95)(92 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 70 5 66)(2 36 6 40)(3 68 7 72)(4 34 8 38)(9 20 13 24)(10 95 14 91)(11 18 15 22)(12 93 16 89)(17 59 21 63)(19 57 23 61)(25 47 29 43)(26 87 30 83)(27 45 31 41)(28 85 32 81)(33 51 37 55)(35 49 39 53)(42 78 46 74)(44 76 48 80)(50 71 54 67)(52 69 56 65)(58 90 62 94)(60 96 64 92)(73 84 77 88)(75 82 79 86)
G:=sub<Sym(96)| (1,79,95)(2,80,96)(3,73,89)(4,74,90)(5,75,91)(6,76,92)(7,77,93)(8,78,94)(9,65,81)(10,66,82)(11,67,83)(12,68,84)(13,69,85)(14,70,86)(15,71,87)(16,72,88)(17,49,25)(18,50,26)(19,51,27)(20,52,28)(21,53,29)(22,54,30)(23,55,31)(24,56,32)(33,41,61)(34,42,62)(35,43,63)(36,44,64)(37,45,57)(38,46,58)(39,47,59)(40,48,60), (2,56)(4,50)(6,52)(8,54)(9,13)(10,61)(11,15)(12,63)(14,57)(16,59)(18,90)(20,92)(22,94)(24,96)(26,74)(28,76)(30,78)(32,80)(33,66)(34,38)(35,68)(36,40)(37,70)(39,72)(41,82)(42,46)(43,84)(44,48)(45,86)(47,88)(58,62)(60,64)(65,69)(67,71)(81,85)(83,87), (1,55)(2,56)(3,49)(4,50)(5,51)(6,52)(7,53)(8,54)(9,64)(10,57)(11,58)(12,59)(13,60)(14,61)(15,62)(16,63)(17,89)(18,90)(19,91)(20,92)(21,93)(22,94)(23,95)(24,96)(25,73)(26,74)(27,75)(28,76)(29,77)(30,78)(31,79)(32,80)(33,70)(34,71)(35,72)(36,65)(37,66)(38,67)(39,68)(40,69)(41,86)(42,87)(43,88)(44,81)(45,82)(46,83)(47,84)(48,85), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32)(33,37)(34,38)(35,39)(36,40)(41,45)(42,46)(43,47)(44,48)(49,53)(50,54)(51,55)(52,56)(57,61)(58,62)(59,63)(60,64)(65,69)(66,70)(67,71)(68,72)(73,77)(74,78)(75,79)(76,80)(81,85)(82,86)(83,87)(84,88)(89,93)(90,94)(91,95)(92,96), (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), (1,70,5,66)(2,36,6,40)(3,68,7,72)(4,34,8,38)(9,20,13,24)(10,95,14,91)(11,18,15,22)(12,93,16,89)(17,59,21,63)(19,57,23,61)(25,47,29,43)(26,87,30,83)(27,45,31,41)(28,85,32,81)(33,51,37,55)(35,49,39,53)(42,78,46,74)(44,76,48,80)(50,71,54,67)(52,69,56,65)(58,90,62,94)(60,96,64,92)(73,84,77,88)(75,82,79,86)>;
G:=Group( (1,79,95)(2,80,96)(3,73,89)(4,74,90)(5,75,91)(6,76,92)(7,77,93)(8,78,94)(9,65,81)(10,66,82)(11,67,83)(12,68,84)(13,69,85)(14,70,86)(15,71,87)(16,72,88)(17,49,25)(18,50,26)(19,51,27)(20,52,28)(21,53,29)(22,54,30)(23,55,31)(24,56,32)(33,41,61)(34,42,62)(35,43,63)(36,44,64)(37,45,57)(38,46,58)(39,47,59)(40,48,60), (2,56)(4,50)(6,52)(8,54)(9,13)(10,61)(11,15)(12,63)(14,57)(16,59)(18,90)(20,92)(22,94)(24,96)(26,74)(28,76)(30,78)(32,80)(33,66)(34,38)(35,68)(36,40)(37,70)(39,72)(41,82)(42,46)(43,84)(44,48)(45,86)(47,88)(58,62)(60,64)(65,69)(67,71)(81,85)(83,87), (1,55)(2,56)(3,49)(4,50)(5,51)(6,52)(7,53)(8,54)(9,64)(10,57)(11,58)(12,59)(13,60)(14,61)(15,62)(16,63)(17,89)(18,90)(19,91)(20,92)(21,93)(22,94)(23,95)(24,96)(25,73)(26,74)(27,75)(28,76)(29,77)(30,78)(31,79)(32,80)(33,70)(34,71)(35,72)(36,65)(37,66)(38,67)(39,68)(40,69)(41,86)(42,87)(43,88)(44,81)(45,82)(46,83)(47,84)(48,85), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32)(33,37)(34,38)(35,39)(36,40)(41,45)(42,46)(43,47)(44,48)(49,53)(50,54)(51,55)(52,56)(57,61)(58,62)(59,63)(60,64)(65,69)(66,70)(67,71)(68,72)(73,77)(74,78)(75,79)(76,80)(81,85)(82,86)(83,87)(84,88)(89,93)(90,94)(91,95)(92,96), (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), (1,70,5,66)(2,36,6,40)(3,68,7,72)(4,34,8,38)(9,20,13,24)(10,95,14,91)(11,18,15,22)(12,93,16,89)(17,59,21,63)(19,57,23,61)(25,47,29,43)(26,87,30,83)(27,45,31,41)(28,85,32,81)(33,51,37,55)(35,49,39,53)(42,78,46,74)(44,76,48,80)(50,71,54,67)(52,69,56,65)(58,90,62,94)(60,96,64,92)(73,84,77,88)(75,82,79,86) );
G=PermutationGroup([[(1,79,95),(2,80,96),(3,73,89),(4,74,90),(5,75,91),(6,76,92),(7,77,93),(8,78,94),(9,65,81),(10,66,82),(11,67,83),(12,68,84),(13,69,85),(14,70,86),(15,71,87),(16,72,88),(17,49,25),(18,50,26),(19,51,27),(20,52,28),(21,53,29),(22,54,30),(23,55,31),(24,56,32),(33,41,61),(34,42,62),(35,43,63),(36,44,64),(37,45,57),(38,46,58),(39,47,59),(40,48,60)], [(2,56),(4,50),(6,52),(8,54),(9,13),(10,61),(11,15),(12,63),(14,57),(16,59),(18,90),(20,92),(22,94),(24,96),(26,74),(28,76),(30,78),(32,80),(33,66),(34,38),(35,68),(36,40),(37,70),(39,72),(41,82),(42,46),(43,84),(44,48),(45,86),(47,88),(58,62),(60,64),(65,69),(67,71),(81,85),(83,87)], [(1,55),(2,56),(3,49),(4,50),(5,51),(6,52),(7,53),(8,54),(9,64),(10,57),(11,58),(12,59),(13,60),(14,61),(15,62),(16,63),(17,89),(18,90),(19,91),(20,92),(21,93),(22,94),(23,95),(24,96),(25,73),(26,74),(27,75),(28,76),(29,77),(30,78),(31,79),(32,80),(33,70),(34,71),(35,72),(36,65),(37,66),(38,67),(39,68),(40,69),(41,86),(42,87),(43,88),(44,81),(45,82),(46,83),(47,84),(48,85)], [(1,5),(2,6),(3,7),(4,8),(9,13),(10,14),(11,15),(12,16),(17,21),(18,22),(19,23),(20,24),(25,29),(26,30),(27,31),(28,32),(33,37),(34,38),(35,39),(36,40),(41,45),(42,46),(43,47),(44,48),(49,53),(50,54),(51,55),(52,56),(57,61),(58,62),(59,63),(60,64),(65,69),(66,70),(67,71),(68,72),(73,77),(74,78),(75,79),(76,80),(81,85),(82,86),(83,87),(84,88),(89,93),(90,94),(91,95),(92,96)], [(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)], [(1,70,5,66),(2,36,6,40),(3,68,7,72),(4,34,8,38),(9,20,13,24),(10,95,14,91),(11,18,15,22),(12,93,16,89),(17,59,21,63),(19,57,23,61),(25,47,29,43),(26,87,30,83),(27,45,31,41),(28,85,32,81),(33,51,37,55),(35,49,39,53),(42,78,46,74),(44,76,48,80),(50,71,54,67),(52,69,56,65),(58,90,62,94),(60,96,64,92),(73,84,77,88),(75,82,79,86)]])
57 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 6A | ··· | 6F | 6G | 6H | 8A | 8B | 8C | 8D | 12A | ··· | 12H | 12I | ··· | 12P | 12Q | 12R | 12S | 12T | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | ··· | 12 | 12 | ··· | 12 | 12 | 12 | 12 | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 1 | ··· | 1 | 4 | 4 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 | 4 | ··· | 4 |
57 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | ||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C6 | C6 | C6 | D4 | D4 | C4○D4 | C3×D4 | C3×D4 | C4○D8 | C3×C4○D4 | C3×C4○D8 | C8.C22 | C3×C8.C22 |
| kernel | C3×C23.20D4 | C3×C22⋊C8 | C3×Q8⋊C4 | C3×C4.Q8 | C3×C2.D8 | C3×C42⋊C2 | C3×C22⋊Q8 | C23.20D4 | C22⋊C8 | Q8⋊C4 | C4.Q8 | C2.D8 | C42⋊C2 | C22⋊Q8 | C2×C12 | C22×C6 | C12 | C2×C4 | C23 | C6 | C4 | C2 | C6 | C2 |
| # reps | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 2 | 2 | 2 | 2 | 1 | 1 | 4 | 2 | 2 | 4 | 8 | 8 | 1 | 2 |
Matrix representation of C3×C23.20D4 ►in GL4(𝔽73) generated by
| 64 | 0 | 0 | 0 |
| 0 | 64 | 0 | 0 |
| 0 | 0 | 64 | 0 |
| 0 | 0 | 0 | 64 |
| 1 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 72 | 72 |
| 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 |
| 51 | 0 | 0 | 0 |
| 0 | 10 | 0 | 0 |
| 0 | 0 | 46 | 19 |
| 0 | 0 | 27 | 27 |
| 0 | 10 | 0 | 0 |
| 51 | 0 | 0 | 0 |
| 0 | 0 | 72 | 71 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(73))| [64,0,0,0,0,64,0,0,0,0,64,0,0,0,0,64],[1,0,0,0,0,72,0,0,0,0,1,72,0,0,0,72],[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],[51,0,0,0,0,10,0,0,0,0,46,27,0,0,19,27],[0,51,0,0,10,0,0,0,0,0,72,0,0,0,71,1] >;
C3×C23.20D4 in GAP, Magma, Sage, TeX
C_3\times C_2^3._{20}D_4 % in TeX
G:=Group("C3xC2^3.20D4"); // GroupNames label
G:=SmallGroup(192,918);
// by ID
G=gap.SmallGroup(192,918);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,672,365,680,1094,142,6053,1531,124]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^3=b^2=c^2=d^2=1,e^4=f^2=d,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,e*b*e^-1=b*c=c*b,b*d=d*b,f*b*f^-1=b*c*d,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=c*e^3>;
// generators/relations