direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C3×C23.34D4, (C22×C4)⋊9C12, (C23×C4).8C6, (C22×C12)⋊13C4, (C23×C12).6C2, C24.31(C2×C6), C22.31(C6×D4), C23.39(C3×D4), C23.31(C2×C12), C2.C42⋊5C6, (C22×C6).154D4, (C23×C6).85C22, C23.58(C22×C6), C6.54(C42⋊C2), (C22×C6).445C23, C22.30(C22×C12), (C22×C12).490C22, C6.85(C22.D4), C2.6(C6×C22⋊C4), (C2×C4).56(C2×C12), (C2×C6).598(C2×D4), (C2×C22⋊C4).4C6, C6.93(C2×C22⋊C4), (C22×C4).8(C2×C6), (C2×C12).284(C2×C4), (C6×C22⋊C4).10C2, C2.6(C3×C42⋊C2), C22.16(C3×C4○D4), (C2×C6).206(C4○D4), (C2×C6).79(C22⋊C4), (C2×C6).217(C22×C4), (C3×C2.C42)⋊4C2, (C22×C6).112(C2×C4), C22.16(C3×C22⋊C4), C2.1(C3×C22.D4), SmallGroup(192,814)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C23.34D4
G = < a,b,c,d,e,f | a3=b2=c2=d2=e4=1, f2=dc=cd, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ebe-1=fbf-1=bc=cb, bd=db, ce=ec, cf=fc, de=ed, df=fd, fef-1=de-1 >
Subgroups: 354 in 218 conjugacy classes, 98 normal (14 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, C6, C6, C6, C2×C4, C2×C4, C23, C23, C23, C12, C2×C6, C2×C6, C2×C6, C22⋊C4, C22×C4, C22×C4, C24, C2×C12, C2×C12, C22×C6, C22×C6, C22×C6, C2.C42, C2×C22⋊C4, C23×C4, C3×C22⋊C4, C22×C12, C22×C12, C23×C6, C23.34D4, C3×C2.C42, C6×C22⋊C4, C23×C12, C3×C23.34D4
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D4, C23, C12, C2×C6, C22⋊C4, C22×C4, C2×D4, C4○D4, C2×C12, C3×D4, C22×C6, C2×C22⋊C4, C42⋊C2, C22.D4, C3×C22⋊C4, C22×C12, C6×D4, C3×C4○D4, C23.34D4, C6×C22⋊C4, C3×C42⋊C2, C3×C22.D4, C3×C23.34D4
►On 96 points
Generators in S96
(1 63 59)(2 64 60)(3 61 57)(4 62 58)(5 26 22)(6 27 23)(7 28 24)(8 25 21)(9 17 13)(10 18 14)(11 19 15)(12 20 16)(29 37 33)(30 38 34)(31 39 35)(32 40 36)(41 51 45)(42 52 46)(43 49 47)(44 50 48)(53 69 65)(54 70 66)(55 71 67)(56 72 68)(73 81 77)(74 82 78)(75 83 79)(76 84 80)(85 93 89)(86 94 90)(87 95 91)(88 96 92)
(1 75)(2 32)(3 73)(4 30)(5 94)(6 49)(7 96)(8 51)(9 29)(10 74)(11 31)(12 76)(13 33)(14 78)(15 35)(16 80)(17 37)(18 82)(19 39)(20 84)(21 41)(22 86)(23 43)(24 88)(25 45)(26 90)(27 47)(28 92)(34 58)(36 60)(38 62)(40 64)(42 66)(44 68)(46 70)(48 72)(50 56)(52 54)(53 93)(55 95)(57 77)(59 79)(61 81)(63 83)(65 85)(67 87)(69 89)(71 91)
(1 11)(2 12)(3 9)(4 10)(5 54)(6 55)(7 56)(8 53)(13 57)(14 58)(15 59)(16 60)(17 61)(18 62)(19 63)(20 64)(21 65)(22 66)(23 67)(24 68)(25 69)(26 70)(27 71)(28 72)(29 73)(30 74)(31 75)(32 76)(33 77)(34 78)(35 79)(36 80)(37 81)(38 82)(39 83)(40 84)(41 85)(42 86)(43 87)(44 88)(45 89)(46 90)(47 91)(48 92)(49 95)(50 96)(51 93)(52 94)
(1 73)(2 74)(3 75)(4 76)(5 50)(6 51)(7 52)(8 49)(9 31)(10 32)(11 29)(12 30)(13 35)(14 36)(15 33)(16 34)(17 39)(18 40)(19 37)(20 38)(21 43)(22 44)(23 41)(24 42)(25 47)(26 48)(27 45)(28 46)(53 95)(54 96)(55 93)(56 94)(57 79)(58 80)(59 77)(60 78)(61 83)(62 84)(63 81)(64 82)(65 87)(66 88)(67 85)(68 86)(69 91)(70 92)(71 89)(72 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 44 29 66)(2 21 30 87)(3 42 31 68)(4 23 32 85)(5 19 96 81)(6 40 93 62)(7 17 94 83)(8 38 95 64)(9 86 75 24)(10 67 76 41)(11 88 73 22)(12 65 74 43)(13 90 79 28)(14 71 80 45)(15 92 77 26)(16 69 78 47)(18 55 84 51)(20 53 82 49)(25 34 91 60)(27 36 89 58)(33 70 59 48)(35 72 57 46)(37 54 63 50)(39 56 61 52)
G:=sub<Sym(96)| (1,63,59)(2,64,60)(3,61,57)(4,62,58)(5,26,22)(6,27,23)(7,28,24)(8,25,21)(9,17,13)(10,18,14)(11,19,15)(12,20,16)(29,37,33)(30,38,34)(31,39,35)(32,40,36)(41,51,45)(42,52,46)(43,49,47)(44,50,48)(53,69,65)(54,70,66)(55,71,67)(56,72,68)(73,81,77)(74,82,78)(75,83,79)(76,84,80)(85,93,89)(86,94,90)(87,95,91)(88,96,92), (1,75)(2,32)(3,73)(4,30)(5,94)(6,49)(7,96)(8,51)(9,29)(10,74)(11,31)(12,76)(13,33)(14,78)(15,35)(16,80)(17,37)(18,82)(19,39)(20,84)(21,41)(22,86)(23,43)(24,88)(25,45)(26,90)(27,47)(28,92)(34,58)(36,60)(38,62)(40,64)(42,66)(44,68)(46,70)(48,72)(50,56)(52,54)(53,93)(55,95)(57,77)(59,79)(61,81)(63,83)(65,85)(67,87)(69,89)(71,91), (1,11)(2,12)(3,9)(4,10)(5,54)(6,55)(7,56)(8,53)(13,57)(14,58)(15,59)(16,60)(17,61)(18,62)(19,63)(20,64)(21,65)(22,66)(23,67)(24,68)(25,69)(26,70)(27,71)(28,72)(29,73)(30,74)(31,75)(32,76)(33,77)(34,78)(35,79)(36,80)(37,81)(38,82)(39,83)(40,84)(41,85)(42,86)(43,87)(44,88)(45,89)(46,90)(47,91)(48,92)(49,95)(50,96)(51,93)(52,94), (1,73)(2,74)(3,75)(4,76)(5,50)(6,51)(7,52)(8,49)(9,31)(10,32)(11,29)(12,30)(13,35)(14,36)(15,33)(16,34)(17,39)(18,40)(19,37)(20,38)(21,43)(22,44)(23,41)(24,42)(25,47)(26,48)(27,45)(28,46)(53,95)(54,96)(55,93)(56,94)(57,79)(58,80)(59,77)(60,78)(61,83)(62,84)(63,81)(64,82)(65,87)(66,88)(67,85)(68,86)(69,91)(70,92)(71,89)(72,90), (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), (1,44,29,66)(2,21,30,87)(3,42,31,68)(4,23,32,85)(5,19,96,81)(6,40,93,62)(7,17,94,83)(8,38,95,64)(9,86,75,24)(10,67,76,41)(11,88,73,22)(12,65,74,43)(13,90,79,28)(14,71,80,45)(15,92,77,26)(16,69,78,47)(18,55,84,51)(20,53,82,49)(25,34,91,60)(27,36,89,58)(33,70,59,48)(35,72,57,46)(37,54,63,50)(39,56,61,52)>;
G:=Group( (1,63,59)(2,64,60)(3,61,57)(4,62,58)(5,26,22)(6,27,23)(7,28,24)(8,25,21)(9,17,13)(10,18,14)(11,19,15)(12,20,16)(29,37,33)(30,38,34)(31,39,35)(32,40,36)(41,51,45)(42,52,46)(43,49,47)(44,50,48)(53,69,65)(54,70,66)(55,71,67)(56,72,68)(73,81,77)(74,82,78)(75,83,79)(76,84,80)(85,93,89)(86,94,90)(87,95,91)(88,96,92), (1,75)(2,32)(3,73)(4,30)(5,94)(6,49)(7,96)(8,51)(9,29)(10,74)(11,31)(12,76)(13,33)(14,78)(15,35)(16,80)(17,37)(18,82)(19,39)(20,84)(21,41)(22,86)(23,43)(24,88)(25,45)(26,90)(27,47)(28,92)(34,58)(36,60)(38,62)(40,64)(42,66)(44,68)(46,70)(48,72)(50,56)(52,54)(53,93)(55,95)(57,77)(59,79)(61,81)(63,83)(65,85)(67,87)(69,89)(71,91), (1,11)(2,12)(3,9)(4,10)(5,54)(6,55)(7,56)(8,53)(13,57)(14,58)(15,59)(16,60)(17,61)(18,62)(19,63)(20,64)(21,65)(22,66)(23,67)(24,68)(25,69)(26,70)(27,71)(28,72)(29,73)(30,74)(31,75)(32,76)(33,77)(34,78)(35,79)(36,80)(37,81)(38,82)(39,83)(40,84)(41,85)(42,86)(43,87)(44,88)(45,89)(46,90)(47,91)(48,92)(49,95)(50,96)(51,93)(52,94), (1,73)(2,74)(3,75)(4,76)(5,50)(6,51)(7,52)(8,49)(9,31)(10,32)(11,29)(12,30)(13,35)(14,36)(15,33)(16,34)(17,39)(18,40)(19,37)(20,38)(21,43)(22,44)(23,41)(24,42)(25,47)(26,48)(27,45)(28,46)(53,95)(54,96)(55,93)(56,94)(57,79)(58,80)(59,77)(60,78)(61,83)(62,84)(63,81)(64,82)(65,87)(66,88)(67,85)(68,86)(69,91)(70,92)(71,89)(72,90), (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), (1,44,29,66)(2,21,30,87)(3,42,31,68)(4,23,32,85)(5,19,96,81)(6,40,93,62)(7,17,94,83)(8,38,95,64)(9,86,75,24)(10,67,76,41)(11,88,73,22)(12,65,74,43)(13,90,79,28)(14,71,80,45)(15,92,77,26)(16,69,78,47)(18,55,84,51)(20,53,82,49)(25,34,91,60)(27,36,89,58)(33,70,59,48)(35,72,57,46)(37,54,63,50)(39,56,61,52) );
G=PermutationGroup([[(1,63,59),(2,64,60),(3,61,57),(4,62,58),(5,26,22),(6,27,23),(7,28,24),(8,25,21),(9,17,13),(10,18,14),(11,19,15),(12,20,16),(29,37,33),(30,38,34),(31,39,35),(32,40,36),(41,51,45),(42,52,46),(43,49,47),(44,50,48),(53,69,65),(54,70,66),(55,71,67),(56,72,68),(73,81,77),(74,82,78),(75,83,79),(76,84,80),(85,93,89),(86,94,90),(87,95,91),(88,96,92)], [(1,75),(2,32),(3,73),(4,30),(5,94),(6,49),(7,96),(8,51),(9,29),(10,74),(11,31),(12,76),(13,33),(14,78),(15,35),(16,80),(17,37),(18,82),(19,39),(20,84),(21,41),(22,86),(23,43),(24,88),(25,45),(26,90),(27,47),(28,92),(34,58),(36,60),(38,62),(40,64),(42,66),(44,68),(46,70),(48,72),(50,56),(52,54),(53,93),(55,95),(57,77),(59,79),(61,81),(63,83),(65,85),(67,87),(69,89),(71,91)], [(1,11),(2,12),(3,9),(4,10),(5,54),(6,55),(7,56),(8,53),(13,57),(14,58),(15,59),(16,60),(17,61),(18,62),(19,63),(20,64),(21,65),(22,66),(23,67),(24,68),(25,69),(26,70),(27,71),(28,72),(29,73),(30,74),(31,75),(32,76),(33,77),(34,78),(35,79),(36,80),(37,81),(38,82),(39,83),(40,84),(41,85),(42,86),(43,87),(44,88),(45,89),(46,90),(47,91),(48,92),(49,95),(50,96),(51,93),(52,94)], [(1,73),(2,74),(3,75),(4,76),(5,50),(6,51),(7,52),(8,49),(9,31),(10,32),(11,29),(12,30),(13,35),(14,36),(15,33),(16,34),(17,39),(18,40),(19,37),(20,38),(21,43),(22,44),(23,41),(24,42),(25,47),(26,48),(27,45),(28,46),(53,95),(54,96),(55,93),(56,94),(57,79),(58,80),(59,77),(60,78),(61,83),(62,84),(63,81),(64,82),(65,87),(66,88),(67,85),(68,86),(69,91),(70,92),(71,89),(72,90)], [(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)], [(1,44,29,66),(2,21,30,87),(3,42,31,68),(4,23,32,85),(5,19,96,81),(6,40,93,62),(7,17,94,83),(8,38,95,64),(9,86,75,24),(10,67,76,41),(11,88,73,22),(12,65,74,43),(13,90,79,28),(14,71,80,45),(15,92,77,26),(16,69,78,47),(18,55,84,51),(20,53,82,49),(25,34,91,60),(27,36,89,58),(33,70,59,48),(35,72,57,46),(37,54,63,50),(39,56,61,52)]])
84 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 3A | 3B | 4A | ··· | 4H | 4I | ··· | 4P | 6A | ··· | 6N | 6O | ··· | 6V | 12A | ··· | 12P | 12Q | ··· | 12AF |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | ··· | 4 | 4 | ··· | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 12 | ··· | 12 | 12 | ··· | 12 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
84 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | |||||||||
| image | C1 | C2 | C2 | C2 | C3 | C4 | C6 | C6 | C6 | C12 | D4 | C4○D4 | C3×D4 | C3×C4○D4 |
| kernel | C3×C23.34D4 | C3×C2.C42 | C6×C22⋊C4 | C23×C12 | C23.34D4 | C22×C12 | C2.C42 | C2×C22⋊C4 | C23×C4 | C22×C4 | C22×C6 | C2×C6 | C23 | C22 |
| # reps | 1 | 4 | 2 | 1 | 2 | 8 | 8 | 4 | 2 | 16 | 4 | 8 | 8 | 16 |
Matrix representation of C3×C23.34D4 ►in GL5(𝔽13)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 0 | 3 |
| 12 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 6 | 12 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 12 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 12 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 5 | 0 | 0 | 0 | 0 |
| 0 | 7 | 2 | 0 | 0 |
| 0 | 2 | 6 | 0 | 0 |
| 0 | 0 | 0 | 3 | 6 |
| 0 | 0 | 0 | 7 | 10 |
| 8 | 0 | 0 | 0 | 0 |
| 0 | 9 | 10 | 0 | 0 |
| 0 | 10 | 4 | 0 | 0 |
| 0 | 0 | 0 | 7 | 10 |
| 0 | 0 | 0 | 3 | 6 |
G:=sub<GL(5,GF(13))| [1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,3,0,0,0,0,0,3],[12,0,0,0,0,0,1,6,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12],[1,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,1,0,0,0,0,0,1],[12,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[5,0,0,0,0,0,7,2,0,0,0,2,6,0,0,0,0,0,3,7,0,0,0,6,10],[8,0,0,0,0,0,9,10,0,0,0,10,4,0,0,0,0,0,7,3,0,0,0,10,6] >;
C3×C23.34D4 in GAP, Magma, Sage, TeX
C_3\times C_2^3._{34}D_4 % in TeX
G:=Group("C3xC2^3.34D4"); // GroupNames label
G:=SmallGroup(192,814);
// by ID
G=gap.SmallGroup(192,814);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,672,365,1094,142]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^3=b^2=c^2=d^2=e^4=1,f^2=d*c=c*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=f*b*f^-1=b*c=c*b,b*d=d*b,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=d*e^-1>;
// generators/relations