direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C3×C23⋊2D4, (C2×C12)⋊24D4, C23⋊2(C3×D4), (C22×C6)⋊5D4, (C22×D4)⋊4C6, C24.5(C2×C6), C6.90C22≀C2, C22.68(C6×D4), C6.40(C4⋊1D4), (C23×C6).4C22, C6.137(C4⋊D4), C2.C42⋊12C6, C23.79(C22×C6), (C22×C6).456C23, (C22×C12).400C22, (D4×C2×C6)⋊13C2, (C2×C4)⋊2(C3×D4), (C2×C22⋊C4)⋊6C6, C2.3(C3×C4⋊1D4), C2.6(C3×C4⋊D4), (C6×C22⋊C4)⋊27C2, C2.4(C3×C22≀C2), (C2×C6).608(C2×D4), (C22×C4).9(C2×C6), C22.35(C3×C4○D4), (C2×C6).216(C4○D4), (C3×C2.C42)⋊25C2, SmallGroup(192,825)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C23⋊2D4
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, ebe-1=bcd, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef=e-1 >
Subgroups: 634 in 322 conjugacy classes, 86 normal (14 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, C2×D4, C24, C2×C12, C2×C12, C3×D4, C22×C6, C22×C6, C22×C6, C2.C42, C2×C22⋊C4, C22×D4, C3×C22⋊C4, C22×C12, C22×C12, C6×D4, C23×C6, C23⋊2D4, C3×C2.C42, C6×C22⋊C4, D4×C2×C6, C3×C23⋊2D4
Quotients: C1, C2, C3, C22, C6, D4, C23, C2×C6, C2×D4, C4○D4, C3×D4, C22×C6, C22≀C2, C4⋊D4, C4⋊1D4, C6×D4, C3×C4○D4, C23⋊2D4, C3×C22≀C2, C3×C4⋊D4, C3×C4⋊1D4, C3×C23⋊2D4
►On 96 points
Generators in S96
(1 13 9)(2 14 10)(3 15 11)(4 16 12)(5 35 31)(6 36 32)(7 33 29)(8 34 30)(17 56 21)(18 53 22)(19 54 23)(20 55 24)(25 65 69)(26 66 70)(27 67 71)(28 68 72)(37 45 41)(38 46 42)(39 47 43)(40 48 44)(49 57 61)(50 58 62)(51 59 63)(52 60 64)(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 72)(2 50)(3 70)(4 52)(5 95)(6 81)(7 93)(8 83)(9 68)(10 62)(11 66)(12 64)(13 28)(14 58)(15 26)(16 60)(17 51)(18 71)(19 49)(20 69)(21 63)(22 67)(23 61)(24 65)(25 55)(27 53)(29 85)(30 75)(31 87)(32 73)(33 89)(34 79)(35 91)(36 77)(37 76)(38 88)(39 74)(40 86)(41 80)(42 92)(43 78)(44 90)(45 84)(46 96)(47 82)(48 94)(54 57)(56 59)
(1 31)(2 32)(3 29)(4 30)(5 13)(6 14)(7 15)(8 16)(9 35)(10 36)(11 33)(12 34)(17 39)(18 40)(19 37)(20 38)(21 43)(22 44)(23 41)(24 42)(25 96)(26 93)(27 94)(28 95)(45 54)(46 55)(47 56)(48 53)(49 76)(50 73)(51 74)(52 75)(57 84)(58 81)(59 82)(60 83)(61 80)(62 77)(63 78)(64 79)(65 92)(66 89)(67 90)(68 91)(69 88)(70 85)(71 86)(72 87)
(1 37)(2 38)(3 39)(4 40)(5 54)(6 55)(7 56)(8 53)(9 41)(10 42)(11 43)(12 44)(13 45)(14 46)(15 47)(16 48)(17 29)(18 30)(19 31)(20 32)(21 33)(22 34)(23 35)(24 36)(25 81)(26 82)(27 83)(28 84)(49 87)(50 88)(51 85)(52 86)(57 95)(58 96)(59 93)(60 94)(61 91)(62 92)(63 89)(64 90)(65 77)(66 78)(67 79)(68 80)(69 73)(70 74)(71 75)(72 76)
(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 4)(6 8)(10 12)(14 16)(18 20)(22 24)(25 94)(26 93)(27 96)(28 95)(30 32)(34 36)(38 40)(42 44)(46 48)(49 76)(50 75)(51 74)(52 73)(53 55)(57 84)(58 83)(59 82)(60 81)(61 80)(62 79)(63 78)(64 77)(65 90)(66 89)(67 92)(68 91)(69 86)(70 85)(71 88)(72 87)
G:=sub<Sym(96)| (1,13,9)(2,14,10)(3,15,11)(4,16,12)(5,35,31)(6,36,32)(7,33,29)(8,34,30)(17,56,21)(18,53,22)(19,54,23)(20,55,24)(25,65,69)(26,66,70)(27,67,71)(28,68,72)(37,45,41)(38,46,42)(39,47,43)(40,48,44)(49,57,61)(50,58,62)(51,59,63)(52,60,64)(73,81,77)(74,82,78)(75,83,79)(76,84,80)(85,93,89)(86,94,90)(87,95,91)(88,96,92), (1,72)(2,50)(3,70)(4,52)(5,95)(6,81)(7,93)(8,83)(9,68)(10,62)(11,66)(12,64)(13,28)(14,58)(15,26)(16,60)(17,51)(18,71)(19,49)(20,69)(21,63)(22,67)(23,61)(24,65)(25,55)(27,53)(29,85)(30,75)(31,87)(32,73)(33,89)(34,79)(35,91)(36,77)(37,76)(38,88)(39,74)(40,86)(41,80)(42,92)(43,78)(44,90)(45,84)(46,96)(47,82)(48,94)(54,57)(56,59), (1,31)(2,32)(3,29)(4,30)(5,13)(6,14)(7,15)(8,16)(9,35)(10,36)(11,33)(12,34)(17,39)(18,40)(19,37)(20,38)(21,43)(22,44)(23,41)(24,42)(25,96)(26,93)(27,94)(28,95)(45,54)(46,55)(47,56)(48,53)(49,76)(50,73)(51,74)(52,75)(57,84)(58,81)(59,82)(60,83)(61,80)(62,77)(63,78)(64,79)(65,92)(66,89)(67,90)(68,91)(69,88)(70,85)(71,86)(72,87), (1,37)(2,38)(3,39)(4,40)(5,54)(6,55)(7,56)(8,53)(9,41)(10,42)(11,43)(12,44)(13,45)(14,46)(15,47)(16,48)(17,29)(18,30)(19,31)(20,32)(21,33)(22,34)(23,35)(24,36)(25,81)(26,82)(27,83)(28,84)(49,87)(50,88)(51,85)(52,86)(57,95)(58,96)(59,93)(60,94)(61,91)(62,92)(63,89)(64,90)(65,77)(66,78)(67,79)(68,80)(69,73)(70,74)(71,75)(72,76), (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), (2,4)(6,8)(10,12)(14,16)(18,20)(22,24)(25,94)(26,93)(27,96)(28,95)(30,32)(34,36)(38,40)(42,44)(46,48)(49,76)(50,75)(51,74)(52,73)(53,55)(57,84)(58,83)(59,82)(60,81)(61,80)(62,79)(63,78)(64,77)(65,90)(66,89)(67,92)(68,91)(69,86)(70,85)(71,88)(72,87)>;
G:=Group( (1,13,9)(2,14,10)(3,15,11)(4,16,12)(5,35,31)(6,36,32)(7,33,29)(8,34,30)(17,56,21)(18,53,22)(19,54,23)(20,55,24)(25,65,69)(26,66,70)(27,67,71)(28,68,72)(37,45,41)(38,46,42)(39,47,43)(40,48,44)(49,57,61)(50,58,62)(51,59,63)(52,60,64)(73,81,77)(74,82,78)(75,83,79)(76,84,80)(85,93,89)(86,94,90)(87,95,91)(88,96,92), (1,72)(2,50)(3,70)(4,52)(5,95)(6,81)(7,93)(8,83)(9,68)(10,62)(11,66)(12,64)(13,28)(14,58)(15,26)(16,60)(17,51)(18,71)(19,49)(20,69)(21,63)(22,67)(23,61)(24,65)(25,55)(27,53)(29,85)(30,75)(31,87)(32,73)(33,89)(34,79)(35,91)(36,77)(37,76)(38,88)(39,74)(40,86)(41,80)(42,92)(43,78)(44,90)(45,84)(46,96)(47,82)(48,94)(54,57)(56,59), (1,31)(2,32)(3,29)(4,30)(5,13)(6,14)(7,15)(8,16)(9,35)(10,36)(11,33)(12,34)(17,39)(18,40)(19,37)(20,38)(21,43)(22,44)(23,41)(24,42)(25,96)(26,93)(27,94)(28,95)(45,54)(46,55)(47,56)(48,53)(49,76)(50,73)(51,74)(52,75)(57,84)(58,81)(59,82)(60,83)(61,80)(62,77)(63,78)(64,79)(65,92)(66,89)(67,90)(68,91)(69,88)(70,85)(71,86)(72,87), (1,37)(2,38)(3,39)(4,40)(5,54)(6,55)(7,56)(8,53)(9,41)(10,42)(11,43)(12,44)(13,45)(14,46)(15,47)(16,48)(17,29)(18,30)(19,31)(20,32)(21,33)(22,34)(23,35)(24,36)(25,81)(26,82)(27,83)(28,84)(49,87)(50,88)(51,85)(52,86)(57,95)(58,96)(59,93)(60,94)(61,91)(62,92)(63,89)(64,90)(65,77)(66,78)(67,79)(68,80)(69,73)(70,74)(71,75)(72,76), (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), (2,4)(6,8)(10,12)(14,16)(18,20)(22,24)(25,94)(26,93)(27,96)(28,95)(30,32)(34,36)(38,40)(42,44)(46,48)(49,76)(50,75)(51,74)(52,73)(53,55)(57,84)(58,83)(59,82)(60,81)(61,80)(62,79)(63,78)(64,77)(65,90)(66,89)(67,92)(68,91)(69,86)(70,85)(71,88)(72,87) );
G=PermutationGroup([[(1,13,9),(2,14,10),(3,15,11),(4,16,12),(5,35,31),(6,36,32),(7,33,29),(8,34,30),(17,56,21),(18,53,22),(19,54,23),(20,55,24),(25,65,69),(26,66,70),(27,67,71),(28,68,72),(37,45,41),(38,46,42),(39,47,43),(40,48,44),(49,57,61),(50,58,62),(51,59,63),(52,60,64),(73,81,77),(74,82,78),(75,83,79),(76,84,80),(85,93,89),(86,94,90),(87,95,91),(88,96,92)], [(1,72),(2,50),(3,70),(4,52),(5,95),(6,81),(7,93),(8,83),(9,68),(10,62),(11,66),(12,64),(13,28),(14,58),(15,26),(16,60),(17,51),(18,71),(19,49),(20,69),(21,63),(22,67),(23,61),(24,65),(25,55),(27,53),(29,85),(30,75),(31,87),(32,73),(33,89),(34,79),(35,91),(36,77),(37,76),(38,88),(39,74),(40,86),(41,80),(42,92),(43,78),(44,90),(45,84),(46,96),(47,82),(48,94),(54,57),(56,59)], [(1,31),(2,32),(3,29),(4,30),(5,13),(6,14),(7,15),(8,16),(9,35),(10,36),(11,33),(12,34),(17,39),(18,40),(19,37),(20,38),(21,43),(22,44),(23,41),(24,42),(25,96),(26,93),(27,94),(28,95),(45,54),(46,55),(47,56),(48,53),(49,76),(50,73),(51,74),(52,75),(57,84),(58,81),(59,82),(60,83),(61,80),(62,77),(63,78),(64,79),(65,92),(66,89),(67,90),(68,91),(69,88),(70,85),(71,86),(72,87)], [(1,37),(2,38),(3,39),(4,40),(5,54),(6,55),(7,56),(8,53),(9,41),(10,42),(11,43),(12,44),(13,45),(14,46),(15,47),(16,48),(17,29),(18,30),(19,31),(20,32),(21,33),(22,34),(23,35),(24,36),(25,81),(26,82),(27,83),(28,84),(49,87),(50,88),(51,85),(52,86),(57,95),(58,96),(59,93),(60,94),(61,91),(62,92),(63,89),(64,90),(65,77),(66,78),(67,79),(68,80),(69,73),(70,74),(71,75),(72,76)], [(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)], [(2,4),(6,8),(10,12),(14,16),(18,20),(22,24),(25,94),(26,93),(27,96),(28,95),(30,32),(34,36),(38,40),(42,44),(46,48),(49,76),(50,75),(51,74),(52,73),(53,55),(57,84),(58,83),(59,82),(60,81),(61,80),(62,79),(63,78),(64,77),(65,90),(66,89),(67,92),(68,91),(69,86),(70,85),(71,88),(72,87)]])
66 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | ··· | 2M | 3A | 3B | 4A | ··· | 4H | 6A | ··· | 6N | 6O | ··· | 6Z | 12A | ··· | 12P |
| order | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 3 | 3 | 4 | ··· | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | ··· | 1 | 4 | ··· | 4 | 1 | 1 | 4 | ··· | 4 | 1 | ··· | 1 | 4 | ··· | 4 | 4 | ··· | 4 |
66 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | ||||||||
| image | C1 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | D4 | D4 | C4○D4 | C3×D4 | C3×D4 | C3×C4○D4 |
| kernel | C3×C23⋊2D4 | C3×C2.C42 | C6×C22⋊C4 | D4×C2×C6 | C23⋊2D4 | C2.C42 | C2×C22⋊C4 | C22×D4 | C2×C12 | C22×C6 | C2×C6 | C2×C4 | C23 | C22 |
| # reps | 1 | 1 | 3 | 3 | 2 | 2 | 6 | 6 | 6 | 6 | 2 | 12 | 12 | 4 |
Matrix representation of C3×C23⋊2D4 ►in GL6(𝔽13)
| 3 | 0 | 0 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 0 | 0 | 0 |
| 0 | 0 | 0 | 9 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 12 | 12 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 11 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
G:=sub<GL(6,GF(13))| [3,0,0,0,0,0,0,3,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,0,0,0,0,0,12,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,12,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,11,0,0,0,0,1,12,0,0,0,0,0,0,0,1,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,12,0],[1,0,0,0,0,0,1,12,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,12] >;
C3×C23⋊2D4 in GAP, Magma, Sage, TeX
C_3\times C_2^3\rtimes_2D_4
% in TeX
G:=Group("C3xC2^3:2D4"); // GroupNames label
G:=SmallGroup(192,825);
// by ID
G=gap.SmallGroup(192,825);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,365,1094,1059]);
// 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,e*b*e^-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=e^-1>;
// generators/relations