direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C3×C23.23D4, (C2×D4)⋊3C12, (C6×D4)⋊15C4, (C2×C12)⋊36D4, (C23×C4)⋊5C6, C2.5(D4×C12), C23⋊2(C2×C12), (C23×C12)⋊2C2, C6.106(C4×D4), C6.89C22≀C2, C24.33(C2×C6), (C22×D4).2C6, C23.27(C3×D4), C22.35(C6×D4), (C22×C6).156D4, C6.133(C4⋊D4), C2.C42⋊11C6, (C23×C6).87C22, C23.63(C22×C6), (C22×C6).450C23, C22.35(C22×C12), (C22×C12).494C22, C6.87(C22.D4), (C2×C4)⋊9(C3×D4), (C2×C4)⋊3(C2×C12), (D4×C2×C6).13C2, (C2×C12)⋊24(C2×C4), (C6×C22⋊C4)⋊6C2, (C2×C22⋊C4)⋊2C6, (C22×C6)⋊4(C2×C4), C2.2(C3×C4⋊D4), C2.7(C6×C22⋊C4), (C2×C6)⋊4(C22⋊C4), C2.3(C3×C22≀C2), (C2×C6).602(C2×D4), C6.94(C2×C22⋊C4), C22⋊2(C3×C22⋊C4), (C22×C4).94(C2×C6), C22.20(C3×C4○D4), (C2×C6).210(C4○D4), (C2×C6).222(C22×C4), C2.3(C3×C22.D4), (C3×C2.C42)⋊24C2, SmallGroup(192,819)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C23.23D4
G = < a,b,c,d,e,f | a3=b2=c2=d2=e4=f2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, fbf=bc=cb, bd=db, be=eb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef=de-1 >
Subgroups: 498 in 286 conjugacy classes, 106 normal (30 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, C6, C6, C6, C2×C4, C2×C4, D4, C23, C23, C23, C12, C2×C6, C2×C6, C2×C6, C22⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C24, C2×C12, C2×C12, C3×D4, C22×C6, C22×C6, C22×C6, C2.C42, C2×C22⋊C4, C2×C22⋊C4, C23×C4, C22×D4, C3×C22⋊C4, C22×C12, C22×C12, C22×C12, C6×D4, C6×D4, C23×C6, C23.23D4, C3×C2.C42, C6×C22⋊C4, C6×C22⋊C4, C23×C12, D4×C2×C6, C3×C23.23D4
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, C4×D4, C22≀C2, C4⋊D4, C22.D4, C3×C22⋊C4, C22×C12, C6×D4, C3×C4○D4, C23.23D4, C6×C22⋊C4, D4×C12, C3×C22≀C2, C3×C4⋊D4, C3×C22.D4, C3×C23.23D4
►On 96 points
Generators in S96
(1 71 66)(2 72 67)(3 69 68)(4 70 65)(5 95 90)(6 96 91)(7 93 92)(8 94 89)(9 17 14)(10 18 15)(11 19 16)(12 20 13)(21 29 28)(22 30 25)(23 31 26)(24 32 27)(33 41 38)(34 42 39)(35 43 40)(36 44 37)(45 55 50)(46 56 51)(47 53 52)(48 54 49)(57 63 74)(58 64 75)(59 61 76)(60 62 73)(77 85 84)(78 86 81)(79 87 82)(80 88 83)
(1 74)(2 75)(3 76)(4 73)(5 87)(6 88)(7 85)(8 86)(9 22)(10 23)(11 24)(12 21)(13 28)(14 25)(15 26)(16 27)(17 30)(18 31)(19 32)(20 29)(33 46)(34 47)(35 48)(36 45)(37 50)(38 51)(39 52)(40 49)(41 56)(42 53)(43 54)(44 55)(57 71)(58 72)(59 69)(60 70)(61 68)(62 65)(63 66)(64 67)(77 92)(78 89)(79 90)(80 91)(81 94)(82 95)(83 96)(84 93)
(1 34)(2 35)(3 36)(4 33)(5 31)(6 32)(7 29)(8 30)(9 78)(10 79)(11 80)(12 77)(13 84)(14 81)(15 82)(16 83)(17 86)(18 87)(19 88)(20 85)(21 92)(22 89)(23 90)(24 91)(25 94)(26 95)(27 96)(28 93)(37 68)(38 65)(39 66)(40 67)(41 70)(42 71)(43 72)(44 69)(45 76)(46 73)(47 74)(48 75)(49 64)(50 61)(51 62)(52 63)(53 57)(54 58)(55 59)(56 60)
(1 12)(2 9)(3 10)(4 11)(5 55)(6 56)(7 53)(8 54)(13 66)(14 67)(15 68)(16 65)(17 72)(18 69)(19 70)(20 71)(21 74)(22 75)(23 76)(24 73)(25 64)(26 61)(27 62)(28 63)(29 57)(30 58)(31 59)(32 60)(33 80)(34 77)(35 78)(36 79)(37 82)(38 83)(39 84)(40 81)(41 88)(42 85)(43 86)(44 87)(45 90)(46 91)(47 92)(48 89)(49 94)(50 95)(51 96)(52 93)
(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 10)(3 12)(5 57)(6 32)(7 59)(8 30)(13 68)(15 66)(18 71)(20 69)(21 45)(22 89)(23 47)(24 91)(25 94)(26 52)(27 96)(28 50)(29 55)(31 53)(34 79)(36 77)(37 84)(39 82)(42 87)(44 85)(46 73)(48 75)(49 64)(51 62)(54 58)(56 60)(61 93)(63 95)(74 90)(76 92)
G:=sub<Sym(96)| (1,71,66)(2,72,67)(3,69,68)(4,70,65)(5,95,90)(6,96,91)(7,93,92)(8,94,89)(9,17,14)(10,18,15)(11,19,16)(12,20,13)(21,29,28)(22,30,25)(23,31,26)(24,32,27)(33,41,38)(34,42,39)(35,43,40)(36,44,37)(45,55,50)(46,56,51)(47,53,52)(48,54,49)(57,63,74)(58,64,75)(59,61,76)(60,62,73)(77,85,84)(78,86,81)(79,87,82)(80,88,83), (1,74)(2,75)(3,76)(4,73)(5,87)(6,88)(7,85)(8,86)(9,22)(10,23)(11,24)(12,21)(13,28)(14,25)(15,26)(16,27)(17,30)(18,31)(19,32)(20,29)(33,46)(34,47)(35,48)(36,45)(37,50)(38,51)(39,52)(40,49)(41,56)(42,53)(43,54)(44,55)(57,71)(58,72)(59,69)(60,70)(61,68)(62,65)(63,66)(64,67)(77,92)(78,89)(79,90)(80,91)(81,94)(82,95)(83,96)(84,93), (1,34)(2,35)(3,36)(4,33)(5,31)(6,32)(7,29)(8,30)(9,78)(10,79)(11,80)(12,77)(13,84)(14,81)(15,82)(16,83)(17,86)(18,87)(19,88)(20,85)(21,92)(22,89)(23,90)(24,91)(25,94)(26,95)(27,96)(28,93)(37,68)(38,65)(39,66)(40,67)(41,70)(42,71)(43,72)(44,69)(45,76)(46,73)(47,74)(48,75)(49,64)(50,61)(51,62)(52,63)(53,57)(54,58)(55,59)(56,60), (1,12)(2,9)(3,10)(4,11)(5,55)(6,56)(7,53)(8,54)(13,66)(14,67)(15,68)(16,65)(17,72)(18,69)(19,70)(20,71)(21,74)(22,75)(23,76)(24,73)(25,64)(26,61)(27,62)(28,63)(29,57)(30,58)(31,59)(32,60)(33,80)(34,77)(35,78)(36,79)(37,82)(38,83)(39,84)(40,81)(41,88)(42,85)(43,86)(44,87)(45,90)(46,91)(47,92)(48,89)(49,94)(50,95)(51,96)(52,93), (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), (1,10)(3,12)(5,57)(6,32)(7,59)(8,30)(13,68)(15,66)(18,71)(20,69)(21,45)(22,89)(23,47)(24,91)(25,94)(26,52)(27,96)(28,50)(29,55)(31,53)(34,79)(36,77)(37,84)(39,82)(42,87)(44,85)(46,73)(48,75)(49,64)(51,62)(54,58)(56,60)(61,93)(63,95)(74,90)(76,92)>;
G:=Group( (1,71,66)(2,72,67)(3,69,68)(4,70,65)(5,95,90)(6,96,91)(7,93,92)(8,94,89)(9,17,14)(10,18,15)(11,19,16)(12,20,13)(21,29,28)(22,30,25)(23,31,26)(24,32,27)(33,41,38)(34,42,39)(35,43,40)(36,44,37)(45,55,50)(46,56,51)(47,53,52)(48,54,49)(57,63,74)(58,64,75)(59,61,76)(60,62,73)(77,85,84)(78,86,81)(79,87,82)(80,88,83), (1,74)(2,75)(3,76)(4,73)(5,87)(6,88)(7,85)(8,86)(9,22)(10,23)(11,24)(12,21)(13,28)(14,25)(15,26)(16,27)(17,30)(18,31)(19,32)(20,29)(33,46)(34,47)(35,48)(36,45)(37,50)(38,51)(39,52)(40,49)(41,56)(42,53)(43,54)(44,55)(57,71)(58,72)(59,69)(60,70)(61,68)(62,65)(63,66)(64,67)(77,92)(78,89)(79,90)(80,91)(81,94)(82,95)(83,96)(84,93), (1,34)(2,35)(3,36)(4,33)(5,31)(6,32)(7,29)(8,30)(9,78)(10,79)(11,80)(12,77)(13,84)(14,81)(15,82)(16,83)(17,86)(18,87)(19,88)(20,85)(21,92)(22,89)(23,90)(24,91)(25,94)(26,95)(27,96)(28,93)(37,68)(38,65)(39,66)(40,67)(41,70)(42,71)(43,72)(44,69)(45,76)(46,73)(47,74)(48,75)(49,64)(50,61)(51,62)(52,63)(53,57)(54,58)(55,59)(56,60), (1,12)(2,9)(3,10)(4,11)(5,55)(6,56)(7,53)(8,54)(13,66)(14,67)(15,68)(16,65)(17,72)(18,69)(19,70)(20,71)(21,74)(22,75)(23,76)(24,73)(25,64)(26,61)(27,62)(28,63)(29,57)(30,58)(31,59)(32,60)(33,80)(34,77)(35,78)(36,79)(37,82)(38,83)(39,84)(40,81)(41,88)(42,85)(43,86)(44,87)(45,90)(46,91)(47,92)(48,89)(49,94)(50,95)(51,96)(52,93), (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), (1,10)(3,12)(5,57)(6,32)(7,59)(8,30)(13,68)(15,66)(18,71)(20,69)(21,45)(22,89)(23,47)(24,91)(25,94)(26,52)(27,96)(28,50)(29,55)(31,53)(34,79)(36,77)(37,84)(39,82)(42,87)(44,85)(46,73)(48,75)(49,64)(51,62)(54,58)(56,60)(61,93)(63,95)(74,90)(76,92) );
G=PermutationGroup([[(1,71,66),(2,72,67),(3,69,68),(4,70,65),(5,95,90),(6,96,91),(7,93,92),(8,94,89),(9,17,14),(10,18,15),(11,19,16),(12,20,13),(21,29,28),(22,30,25),(23,31,26),(24,32,27),(33,41,38),(34,42,39),(35,43,40),(36,44,37),(45,55,50),(46,56,51),(47,53,52),(48,54,49),(57,63,74),(58,64,75),(59,61,76),(60,62,73),(77,85,84),(78,86,81),(79,87,82),(80,88,83)], [(1,74),(2,75),(3,76),(4,73),(5,87),(6,88),(7,85),(8,86),(9,22),(10,23),(11,24),(12,21),(13,28),(14,25),(15,26),(16,27),(17,30),(18,31),(19,32),(20,29),(33,46),(34,47),(35,48),(36,45),(37,50),(38,51),(39,52),(40,49),(41,56),(42,53),(43,54),(44,55),(57,71),(58,72),(59,69),(60,70),(61,68),(62,65),(63,66),(64,67),(77,92),(78,89),(79,90),(80,91),(81,94),(82,95),(83,96),(84,93)], [(1,34),(2,35),(3,36),(4,33),(5,31),(6,32),(7,29),(8,30),(9,78),(10,79),(11,80),(12,77),(13,84),(14,81),(15,82),(16,83),(17,86),(18,87),(19,88),(20,85),(21,92),(22,89),(23,90),(24,91),(25,94),(26,95),(27,96),(28,93),(37,68),(38,65),(39,66),(40,67),(41,70),(42,71),(43,72),(44,69),(45,76),(46,73),(47,74),(48,75),(49,64),(50,61),(51,62),(52,63),(53,57),(54,58),(55,59),(56,60)], [(1,12),(2,9),(3,10),(4,11),(5,55),(6,56),(7,53),(8,54),(13,66),(14,67),(15,68),(16,65),(17,72),(18,69),(19,70),(20,71),(21,74),(22,75),(23,76),(24,73),(25,64),(26,61),(27,62),(28,63),(29,57),(30,58),(31,59),(32,60),(33,80),(34,77),(35,78),(36,79),(37,82),(38,83),(39,84),(40,81),(41,88),(42,85),(43,86),(44,87),(45,90),(46,91),(47,92),(48,89),(49,94),(50,95),(51,96),(52,93)], [(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)], [(1,10),(3,12),(5,57),(6,32),(7,59),(8,30),(13,68),(15,66),(18,71),(20,69),(21,45),(22,89),(23,47),(24,91),(25,94),(26,52),(27,96),(28,50),(29,55),(31,53),(34,79),(36,77),(37,84),(39,82),(42,87),(44,85),(46,73),(48,75),(49,64),(51,62),(54,58),(56,60),(61,93),(63,95),(74,90),(76,92)]])
84 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 2L | 2M | 3A | 3B | 4A | ··· | 4H | 4I | ··· | 4N | 6A | ··· | 6N | 6O | ··· | 6V | 6W | 6X | 6Y | 6Z | 12A | ··· | 12P | 12Q | ··· | 12AB |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | ··· | 4 | 4 | ··· | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 12 | ··· | 12 | 12 | ··· | 12 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 1 | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
84 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | |||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C3 | C4 | C6 | C6 | C6 | C6 | C12 | D4 | D4 | C4○D4 | C3×D4 | C3×D4 | C3×C4○D4 |
| kernel | C3×C23.23D4 | C3×C2.C42 | C6×C22⋊C4 | C23×C12 | D4×C2×C6 | C23.23D4 | C6×D4 | C2.C42 | C2×C22⋊C4 | C23×C4 | C22×D4 | C2×D4 | C2×C12 | C22×C6 | C2×C6 | C2×C4 | C23 | C22 |
| # reps | 1 | 2 | 3 | 1 | 1 | 2 | 8 | 4 | 6 | 2 | 2 | 16 | 4 | 4 | 4 | 8 | 8 | 8 |
Matrix representation of C3×C23.23D4 ►in GL5(𝔽13)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 | 0 |
| 0 | 0 | 3 | 0 | 0 |
| 0 | 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 0 | 3 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 2 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 12 |
| 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 |
| 8 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 8 | 10 |
| 0 | 0 | 0 | 0 | 5 |
| 12 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 | 1 |
G:=sub<GL(5,GF(13))| [1,0,0,0,0,0,3,0,0,0,0,0,3,0,0,0,0,0,3,0,0,0,0,0,3],[1,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,2,1],[1,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12],[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],[8,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,8,0,0,0,0,10,5],[12,0,0,0,0,0,1,0,0,0,0,0,12,0,0,0,0,0,12,12,0,0,0,0,1] >;
C3×C23.23D4 in GAP, Magma, Sage, TeX
C_3\times C_2^3._{23}D_4 % in TeX
G:=Group("C3xC2^3.23D4"); // GroupNames label
G:=SmallGroup(192,819);
// by ID
G=gap.SmallGroup(192,819);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,672,365,848,1094]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^3=b^2=c^2=d^2=e^4=f^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,f*b*f=b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f=d*e^-1>;
// generators/relations