direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C3×C23.19D4, C2.D8⋊7C6, C22⋊C8⋊7C6, C4.Q8⋊10C6, C4⋊D4.7C6, D4⋊C4⋊13C6, C42⋊C2⋊5C6, (C2×C12).461D4, C23.18(C3×D4), (C22×C6).36D4, C6.128(C4○D8), C22.103(C6×D4), C12.319(C4○D4), C6.143(C8⋊C22), (C2×C24).306C22, (C2×C12).938C23, (C6×D4).197C22, (C22×C12).430C22, C6.97(C22.D4), C4⋊C4.59(C2×C6), (C2×C8).43(C2×C6), (C3×C4.Q8)⋊25C2, (C3×C2.D8)⋊22C2, C4.31(C3×C4○D4), C2.15(C3×C4○D8), (C3×C22⋊C8)⋊17C2, (C2×D4).20(C2×C6), (C2×C4).107(C3×D4), (C2×C6).659(C2×D4), C2.18(C3×C8⋊C22), (C3×D4⋊C4)⋊19C2, (C3×C4⋊D4).17C2, (C22×C4).53(C2×C6), (C3×C42⋊C2)⋊26C2, (C3×C4⋊C4).382C22, (C2×C4).113(C22×C6), C2.13(C3×C22.D4), SmallGroup(192,915)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C23.19D4
G = < a,b,c,d,e,f | a3=b2=c2=d2=f2=1, e4=d, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ebe-1=bc=cb, bd=db, fbf=bcd, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef=cde3 >
Subgroups: 210 in 106 conjugacy classes, 50 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C6, C8, C2×C4, C2×C4, D4, C23, C23, C12, C12, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C2×C8, C22×C4, C2×D4, C2×D4, C24, C2×C12, C2×C12, C3×D4, C22×C6, C22×C6, C22⋊C8, D4⋊C4, C4.Q8, C2.D8, C42⋊C2, C4⋊D4, C4×C12, C3×C22⋊C4, C3×C4⋊C4, C2×C24, C22×C12, C6×D4, C6×D4, C23.19D4, C3×C22⋊C8, C3×D4⋊C4, C3×C4.Q8, C3×C2.D8, C3×C42⋊C2, C3×C4⋊D4, C3×C23.19D4
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, C23.19D4, C3×C22.D4, C3×C4○D8, C3×C8⋊C22, C3×C23.19D4
►On 96 points
Generators in S96
(1 34 16)(2 35 9)(3 36 10)(4 37 11)(5 38 12)(6 39 13)(7 40 14)(8 33 15)(17 53 43)(18 54 44)(19 55 45)(20 56 46)(21 49 47)(22 50 48)(23 51 41)(24 52 42)(25 92 70)(26 93 71)(27 94 72)(28 95 65)(29 96 66)(30 89 67)(31 90 68)(32 91 69)(57 76 87)(58 77 88)(59 78 81)(60 79 82)(61 80 83)(62 73 84)(63 74 85)(64 75 86)
(1 61)(2 66)(3 63)(4 68)(5 57)(6 70)(7 59)(8 72)(9 96)(10 85)(11 90)(12 87)(13 92)(14 81)(15 94)(16 83)(17 71)(18 60)(19 65)(20 62)(21 67)(22 64)(23 69)(24 58)(25 39)(26 53)(27 33)(28 55)(29 35)(30 49)(31 37)(32 51)(34 80)(36 74)(38 76)(40 78)(41 91)(42 88)(43 93)(44 82)(45 95)(46 84)(47 89)(48 86)(50 75)(52 77)(54 79)(56 73)
(1 19)(2 20)(3 21)(4 22)(5 23)(6 24)(7 17)(8 18)(9 46)(10 47)(11 48)(12 41)(13 42)(14 43)(15 44)(16 45)(25 77)(26 78)(27 79)(28 80)(29 73)(30 74)(31 75)(32 76)(33 54)(34 55)(35 56)(36 49)(37 50)(38 51)(39 52)(40 53)(57 69)(58 70)(59 71)(60 72)(61 65)(62 66)(63 67)(64 68)(81 93)(82 94)(83 95)(84 96)(85 89)(86 90)(87 91)(88 92)
(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)
(2 18)(3 7)(4 24)(6 22)(8 20)(9 44)(10 14)(11 42)(13 48)(15 46)(17 21)(25 27)(26 78)(28 76)(29 31)(30 74)(32 80)(33 56)(35 54)(36 40)(37 52)(39 50)(43 47)(49 53)(57 65)(58 60)(59 71)(61 69)(62 64)(63 67)(66 68)(70 72)(73 75)(77 79)(81 93)(82 88)(83 91)(84 86)(85 89)(87 95)(90 96)(92 94)
G:=sub<Sym(96)| (1,34,16)(2,35,9)(3,36,10)(4,37,11)(5,38,12)(6,39,13)(7,40,14)(8,33,15)(17,53,43)(18,54,44)(19,55,45)(20,56,46)(21,49,47)(22,50,48)(23,51,41)(24,52,42)(25,92,70)(26,93,71)(27,94,72)(28,95,65)(29,96,66)(30,89,67)(31,90,68)(32,91,69)(57,76,87)(58,77,88)(59,78,81)(60,79,82)(61,80,83)(62,73,84)(63,74,85)(64,75,86), (1,61)(2,66)(3,63)(4,68)(5,57)(6,70)(7,59)(8,72)(9,96)(10,85)(11,90)(12,87)(13,92)(14,81)(15,94)(16,83)(17,71)(18,60)(19,65)(20,62)(21,67)(22,64)(23,69)(24,58)(25,39)(26,53)(27,33)(28,55)(29,35)(30,49)(31,37)(32,51)(34,80)(36,74)(38,76)(40,78)(41,91)(42,88)(43,93)(44,82)(45,95)(46,84)(47,89)(48,86)(50,75)(52,77)(54,79)(56,73), (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,17)(8,18)(9,46)(10,47)(11,48)(12,41)(13,42)(14,43)(15,44)(16,45)(25,77)(26,78)(27,79)(28,80)(29,73)(30,74)(31,75)(32,76)(33,54)(34,55)(35,56)(36,49)(37,50)(38,51)(39,52)(40,53)(57,69)(58,70)(59,71)(60,72)(61,65)(62,66)(63,67)(64,68)(81,93)(82,94)(83,95)(84,96)(85,89)(86,90)(87,91)(88,92), (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), (2,18)(3,7)(4,24)(6,22)(8,20)(9,44)(10,14)(11,42)(13,48)(15,46)(17,21)(25,27)(26,78)(28,76)(29,31)(30,74)(32,80)(33,56)(35,54)(36,40)(37,52)(39,50)(43,47)(49,53)(57,65)(58,60)(59,71)(61,69)(62,64)(63,67)(66,68)(70,72)(73,75)(77,79)(81,93)(82,88)(83,91)(84,86)(85,89)(87,95)(90,96)(92,94)>;
G:=Group( (1,34,16)(2,35,9)(3,36,10)(4,37,11)(5,38,12)(6,39,13)(7,40,14)(8,33,15)(17,53,43)(18,54,44)(19,55,45)(20,56,46)(21,49,47)(22,50,48)(23,51,41)(24,52,42)(25,92,70)(26,93,71)(27,94,72)(28,95,65)(29,96,66)(30,89,67)(31,90,68)(32,91,69)(57,76,87)(58,77,88)(59,78,81)(60,79,82)(61,80,83)(62,73,84)(63,74,85)(64,75,86), (1,61)(2,66)(3,63)(4,68)(5,57)(6,70)(7,59)(8,72)(9,96)(10,85)(11,90)(12,87)(13,92)(14,81)(15,94)(16,83)(17,71)(18,60)(19,65)(20,62)(21,67)(22,64)(23,69)(24,58)(25,39)(26,53)(27,33)(28,55)(29,35)(30,49)(31,37)(32,51)(34,80)(36,74)(38,76)(40,78)(41,91)(42,88)(43,93)(44,82)(45,95)(46,84)(47,89)(48,86)(50,75)(52,77)(54,79)(56,73), (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,17)(8,18)(9,46)(10,47)(11,48)(12,41)(13,42)(14,43)(15,44)(16,45)(25,77)(26,78)(27,79)(28,80)(29,73)(30,74)(31,75)(32,76)(33,54)(34,55)(35,56)(36,49)(37,50)(38,51)(39,52)(40,53)(57,69)(58,70)(59,71)(60,72)(61,65)(62,66)(63,67)(64,68)(81,93)(82,94)(83,95)(84,96)(85,89)(86,90)(87,91)(88,92), (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), (2,18)(3,7)(4,24)(6,22)(8,20)(9,44)(10,14)(11,42)(13,48)(15,46)(17,21)(25,27)(26,78)(28,76)(29,31)(30,74)(32,80)(33,56)(35,54)(36,40)(37,52)(39,50)(43,47)(49,53)(57,65)(58,60)(59,71)(61,69)(62,64)(63,67)(66,68)(70,72)(73,75)(77,79)(81,93)(82,88)(83,91)(84,86)(85,89)(87,95)(90,96)(92,94) );
G=PermutationGroup([[(1,34,16),(2,35,9),(3,36,10),(4,37,11),(5,38,12),(6,39,13),(7,40,14),(8,33,15),(17,53,43),(18,54,44),(19,55,45),(20,56,46),(21,49,47),(22,50,48),(23,51,41),(24,52,42),(25,92,70),(26,93,71),(27,94,72),(28,95,65),(29,96,66),(30,89,67),(31,90,68),(32,91,69),(57,76,87),(58,77,88),(59,78,81),(60,79,82),(61,80,83),(62,73,84),(63,74,85),(64,75,86)], [(1,61),(2,66),(3,63),(4,68),(5,57),(6,70),(7,59),(8,72),(9,96),(10,85),(11,90),(12,87),(13,92),(14,81),(15,94),(16,83),(17,71),(18,60),(19,65),(20,62),(21,67),(22,64),(23,69),(24,58),(25,39),(26,53),(27,33),(28,55),(29,35),(30,49),(31,37),(32,51),(34,80),(36,74),(38,76),(40,78),(41,91),(42,88),(43,93),(44,82),(45,95),(46,84),(47,89),(48,86),(50,75),(52,77),(54,79),(56,73)], [(1,19),(2,20),(3,21),(4,22),(5,23),(6,24),(7,17),(8,18),(9,46),(10,47),(11,48),(12,41),(13,42),(14,43),(15,44),(16,45),(25,77),(26,78),(27,79),(28,80),(29,73),(30,74),(31,75),(32,76),(33,54),(34,55),(35,56),(36,49),(37,50),(38,51),(39,52),(40,53),(57,69),(58,70),(59,71),(60,72),(61,65),(62,66),(63,67),(64,68),(81,93),(82,94),(83,95),(84,96),(85,89),(86,90),(87,91),(88,92)], [(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)], [(2,18),(3,7),(4,24),(6,22),(8,20),(9,44),(10,14),(11,42),(13,48),(15,46),(17,21),(25,27),(26,78),(28,76),(29,31),(30,74),(32,80),(33,56),(35,54),(36,40),(37,52),(39,50),(43,47),(49,53),(57,65),(58,60),(59,71),(61,69),(62,64),(63,67),(66,68),(70,72),(73,75),(77,79),(81,93),(82,88),(83,91),(84,86),(85,89),(87,95),(90,96),(92,94)]])
57 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 6A | ··· | 6F | 6G | 6H | 6I | 6J | 8A | 8B | 8C | 8D | 12A | ··· | 12H | 12I | ··· | 12P | 12Q | 12R | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | ··· | 12 | 12 | ··· | 12 | 12 | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 4 | 8 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 1 | ··· | 1 | 4 | 4 | 8 | 8 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 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×C23.19D4 | C3×C22⋊C8 | C3×D4⋊C4 | C3×C4.Q8 | C3×C2.D8 | C3×C42⋊C2 | C3×C4⋊D4 | C23.19D4 | C22⋊C8 | D4⋊C4 | C4.Q8 | C2.D8 | C42⋊C2 | C4⋊D4 | 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×C23.19D4 ►in GL4(𝔽73) generated by
| 8 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 |
| 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 8 |
| 1 | 2 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 27 | 19 |
| 0 | 0 | 27 | 46 |
| 72 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 72 |
| 27 | 0 | 0 | 0 |
| 46 | 46 | 0 | 0 |
| 0 | 0 | 0 | 41 |
| 0 | 0 | 16 | 41 |
| 1 | 0 | 0 | 0 |
| 72 | 72 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 72 |
G:=sub<GL(4,GF(73))| [8,0,0,0,0,8,0,0,0,0,8,0,0,0,0,8],[1,0,0,0,2,72,0,0,0,0,27,27,0,0,19,46],[72,0,0,0,0,72,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,72,0,0,0,0,72],[27,46,0,0,0,46,0,0,0,0,0,16,0,0,41,41],[1,72,0,0,0,72,0,0,0,0,1,1,0,0,0,72] >;
C3×C23.19D4 in GAP, Magma, Sage, TeX
C_3\times C_2^3._{19}D_4 % in TeX
G:=Group("C3xC2^3.19D4"); // GroupNames label
G:=SmallGroup(192,915);
// by ID
G=gap.SmallGroup(192,915);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,365,680,1094,142,6053,1531,124]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^3=b^2=c^2=d^2=f^2=1,e^4=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=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=c*d*e^3>;
// generators/relations