direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic3×C22⋊C4, C24.54D6, (C2×C6)⋊C42, C6.40(C4×D4), C2.1(D4×Dic3), C6.D4⋊7C4, C6.19(C2×C42), C23.53(C4×S3), C22⋊2(C4×Dic3), C22.93(S3×D4), (C22×Dic3)⋊6C4, (C22×C4).322D6, C6.C42⋊37C2, (C2×Dic3).210D4, (C23×C6).24C22, (C23×Dic3).1C2, C23.17(C2×Dic3), C6.23(C42⋊C2), C2.4(Dic3⋊4D4), C23.284(C22×S3), (C22×C6).316C23, C22.40(D4⋊2S3), (C22×C12).340C22, C2.4(C23.16D6), C22.19(C22×Dic3), (C22×Dic3).182C22, C3⋊3(C4×C22⋊C4), (C2×C12)⋊20(C2×C4), C2.8(C2×C4×Dic3), (C3×C22⋊C4)⋊9C4, (C2×C4×Dic3)⋊20C2, (C2×C4)⋊6(C2×Dic3), C2.4(S3×C22⋊C4), C22.54(S3×C2×C4), (C2×C6).313(C2×D4), C6.26(C2×C22⋊C4), (C2×Dic3)⋊14(C2×C4), (C6×C22⋊C4).22C2, (C2×C22⋊C4).19S3, (C22×C6).44(C2×C4), (C2×C6).137(C4○D4), (C2×C6).102(C22×C4), (C2×C6.D4).3C2, SmallGroup(192,500)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic3×C22⋊C4
G = < a,b,c,d,e | a6=c2=d2=e4=1, b2=a3, bab-1=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, ece-1=cd=dc, de=ed >
Subgroups: 552 in 258 conjugacy classes, 111 normal (25 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, C6, C6, C6, C2×C4, C2×C4, C23, C23, C23, Dic3, Dic3, C12, C2×C6, C2×C6, C2×C6, C42, C22⋊C4, C22⋊C4, C22×C4, C22×C4, C24, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×C6, C22×C6, C22×C6, C2.C42, C2×C42, C2×C22⋊C4, C2×C22⋊C4, C23×C4, C4×Dic3, C6.D4, C3×C22⋊C4, C22×Dic3, C22×Dic3, C22×Dic3, C22×C12, C23×C6, C4×C22⋊C4, C6.C42, C2×C4×Dic3, C2×C6.D4, C6×C22⋊C4, C23×Dic3, Dic3×C22⋊C4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, Dic3, D6, C42, C22⋊C4, C22×C4, C2×D4, C4○D4, C4×S3, C2×Dic3, C22×S3, C2×C42, C2×C22⋊C4, C42⋊C2, C4×D4, C4×Dic3, S3×C2×C4, S3×D4, D4⋊2S3, C22×Dic3, C4×C22⋊C4, C23.16D6, S3×C22⋊C4, Dic3⋊4D4, C2×C4×Dic3, D4×Dic3, Dic3×C22⋊C4
►On 96 points
Generators in S96
(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 58 4 55)(2 57 5 60)(3 56 6 59)(7 64 10 61)(8 63 11 66)(9 62 12 65)(13 54 16 51)(14 53 17 50)(15 52 18 49)(19 68 22 71)(20 67 23 70)(21 72 24 69)(25 77 28 74)(26 76 29 73)(27 75 30 78)(31 83 34 80)(32 82 35 79)(33 81 36 84)(37 86 40 89)(38 85 41 88)(39 90 42 87)(43 92 46 95)(44 91 47 94)(45 96 48 93)
(1 11)(2 12)(3 7)(4 8)(5 9)(6 10)(13 79)(14 80)(15 81)(16 82)(17 83)(18 84)(19 27)(20 28)(21 29)(22 30)(23 25)(24 26)(31 53)(32 54)(33 49)(34 50)(35 51)(36 52)(37 45)(38 46)(39 47)(40 48)(41 43)(42 44)(55 63)(56 64)(57 65)(58 66)(59 61)(60 62)(67 74)(68 75)(69 76)(70 77)(71 78)(72 73)(85 95)(86 96)(87 91)(88 92)(89 93)(90 94)
(1 23)(2 24)(3 19)(4 20)(5 21)(6 22)(7 27)(8 28)(9 29)(10 30)(11 25)(12 26)(13 87)(14 88)(15 89)(16 90)(17 85)(18 86)(31 46)(32 47)(33 48)(34 43)(35 44)(36 45)(37 52)(38 53)(39 54)(40 49)(41 50)(42 51)(55 67)(56 68)(57 69)(58 70)(59 71)(60 72)(61 78)(62 73)(63 74)(64 75)(65 76)(66 77)(79 91)(80 92)(81 93)(82 94)(83 95)(84 96)
(1 42 8 32)(2 37 9 33)(3 38 10 34)(4 39 11 35)(5 40 12 36)(6 41 7 31)(13 74 94 70)(14 75 95 71)(15 76 96 72)(16 77 91 67)(17 78 92 68)(18 73 93 69)(19 53 30 43)(20 54 25 44)(21 49 26 45)(22 50 27 46)(23 51 28 47)(24 52 29 48)(55 90 66 79)(56 85 61 80)(57 86 62 81)(58 87 63 82)(59 88 64 83)(60 89 65 84)
G:=sub<Sym(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), (1,58,4,55)(2,57,5,60)(3,56,6,59)(7,64,10,61)(8,63,11,66)(9,62,12,65)(13,54,16,51)(14,53,17,50)(15,52,18,49)(19,68,22,71)(20,67,23,70)(21,72,24,69)(25,77,28,74)(26,76,29,73)(27,75,30,78)(31,83,34,80)(32,82,35,79)(33,81,36,84)(37,86,40,89)(38,85,41,88)(39,90,42,87)(43,92,46,95)(44,91,47,94)(45,96,48,93), (1,11)(2,12)(3,7)(4,8)(5,9)(6,10)(13,79)(14,80)(15,81)(16,82)(17,83)(18,84)(19,27)(20,28)(21,29)(22,30)(23,25)(24,26)(31,53)(32,54)(33,49)(34,50)(35,51)(36,52)(37,45)(38,46)(39,47)(40,48)(41,43)(42,44)(55,63)(56,64)(57,65)(58,66)(59,61)(60,62)(67,74)(68,75)(69,76)(70,77)(71,78)(72,73)(85,95)(86,96)(87,91)(88,92)(89,93)(90,94), (1,23)(2,24)(3,19)(4,20)(5,21)(6,22)(7,27)(8,28)(9,29)(10,30)(11,25)(12,26)(13,87)(14,88)(15,89)(16,90)(17,85)(18,86)(31,46)(32,47)(33,48)(34,43)(35,44)(36,45)(37,52)(38,53)(39,54)(40,49)(41,50)(42,51)(55,67)(56,68)(57,69)(58,70)(59,71)(60,72)(61,78)(62,73)(63,74)(64,75)(65,76)(66,77)(79,91)(80,92)(81,93)(82,94)(83,95)(84,96), (1,42,8,32)(2,37,9,33)(3,38,10,34)(4,39,11,35)(5,40,12,36)(6,41,7,31)(13,74,94,70)(14,75,95,71)(15,76,96,72)(16,77,91,67)(17,78,92,68)(18,73,93,69)(19,53,30,43)(20,54,25,44)(21,49,26,45)(22,50,27,46)(23,51,28,47)(24,52,29,48)(55,90,66,79)(56,85,61,80)(57,86,62,81)(58,87,63,82)(59,88,64,83)(60,89,65,84)>;
G:=Group( (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), (1,58,4,55)(2,57,5,60)(3,56,6,59)(7,64,10,61)(8,63,11,66)(9,62,12,65)(13,54,16,51)(14,53,17,50)(15,52,18,49)(19,68,22,71)(20,67,23,70)(21,72,24,69)(25,77,28,74)(26,76,29,73)(27,75,30,78)(31,83,34,80)(32,82,35,79)(33,81,36,84)(37,86,40,89)(38,85,41,88)(39,90,42,87)(43,92,46,95)(44,91,47,94)(45,96,48,93), (1,11)(2,12)(3,7)(4,8)(5,9)(6,10)(13,79)(14,80)(15,81)(16,82)(17,83)(18,84)(19,27)(20,28)(21,29)(22,30)(23,25)(24,26)(31,53)(32,54)(33,49)(34,50)(35,51)(36,52)(37,45)(38,46)(39,47)(40,48)(41,43)(42,44)(55,63)(56,64)(57,65)(58,66)(59,61)(60,62)(67,74)(68,75)(69,76)(70,77)(71,78)(72,73)(85,95)(86,96)(87,91)(88,92)(89,93)(90,94), (1,23)(2,24)(3,19)(4,20)(5,21)(6,22)(7,27)(8,28)(9,29)(10,30)(11,25)(12,26)(13,87)(14,88)(15,89)(16,90)(17,85)(18,86)(31,46)(32,47)(33,48)(34,43)(35,44)(36,45)(37,52)(38,53)(39,54)(40,49)(41,50)(42,51)(55,67)(56,68)(57,69)(58,70)(59,71)(60,72)(61,78)(62,73)(63,74)(64,75)(65,76)(66,77)(79,91)(80,92)(81,93)(82,94)(83,95)(84,96), (1,42,8,32)(2,37,9,33)(3,38,10,34)(4,39,11,35)(5,40,12,36)(6,41,7,31)(13,74,94,70)(14,75,95,71)(15,76,96,72)(16,77,91,67)(17,78,92,68)(18,73,93,69)(19,53,30,43)(20,54,25,44)(21,49,26,45)(22,50,27,46)(23,51,28,47)(24,52,29,48)(55,90,66,79)(56,85,61,80)(57,86,62,81)(58,87,63,82)(59,88,64,83)(60,89,65,84) );
G=PermutationGroup([[(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)], [(1,58,4,55),(2,57,5,60),(3,56,6,59),(7,64,10,61),(8,63,11,66),(9,62,12,65),(13,54,16,51),(14,53,17,50),(15,52,18,49),(19,68,22,71),(20,67,23,70),(21,72,24,69),(25,77,28,74),(26,76,29,73),(27,75,30,78),(31,83,34,80),(32,82,35,79),(33,81,36,84),(37,86,40,89),(38,85,41,88),(39,90,42,87),(43,92,46,95),(44,91,47,94),(45,96,48,93)], [(1,11),(2,12),(3,7),(4,8),(5,9),(6,10),(13,79),(14,80),(15,81),(16,82),(17,83),(18,84),(19,27),(20,28),(21,29),(22,30),(23,25),(24,26),(31,53),(32,54),(33,49),(34,50),(35,51),(36,52),(37,45),(38,46),(39,47),(40,48),(41,43),(42,44),(55,63),(56,64),(57,65),(58,66),(59,61),(60,62),(67,74),(68,75),(69,76),(70,77),(71,78),(72,73),(85,95),(86,96),(87,91),(88,92),(89,93),(90,94)], [(1,23),(2,24),(3,19),(4,20),(5,21),(6,22),(7,27),(8,28),(9,29),(10,30),(11,25),(12,26),(13,87),(14,88),(15,89),(16,90),(17,85),(18,86),(31,46),(32,47),(33,48),(34,43),(35,44),(36,45),(37,52),(38,53),(39,54),(40,49),(41,50),(42,51),(55,67),(56,68),(57,69),(58,70),(59,71),(60,72),(61,78),(62,73),(63,74),(64,75),(65,76),(66,77),(79,91),(80,92),(81,93),(82,94),(83,95),(84,96)], [(1,42,8,32),(2,37,9,33),(3,38,10,34),(4,39,11,35),(5,40,12,36),(6,41,7,31),(13,74,94,70),(14,75,95,71),(15,76,96,72),(16,77,91,67),(17,78,92,68),(18,73,93,69),(19,53,30,43),(20,54,25,44),(21,49,26,45),(22,50,27,46),(23,51,28,47),(24,52,29,48),(55,90,66,79),(56,85,61,80),(57,86,62,81),(58,87,63,82),(59,88,64,83),(60,89,65,84)]])
60 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 3 | 4A | ··· | 4H | 4I | ··· | 4P | 4Q | ··· | 4AB | 6A | ··· | 6G | 6H | 6I | 6J | 6K | 12A | ··· | 12H |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 3 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 12 | ··· | 12 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 3 | ··· | 3 | 6 | ··· | 6 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | + | + | + | - | |||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | S3 | D4 | Dic3 | D6 | D6 | C4○D4 | C4×S3 | S3×D4 | D4⋊2S3 |
| kernel | Dic3×C22⋊C4 | C6.C42 | C2×C4×Dic3 | C2×C6.D4 | C6×C22⋊C4 | C23×Dic3 | C6.D4 | C3×C22⋊C4 | C22×Dic3 | C2×C22⋊C4 | C2×Dic3 | C22⋊C4 | C22×C4 | C24 | C2×C6 | C23 | C22 | C22 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 8 | 8 | 8 | 1 | 4 | 4 | 2 | 1 | 4 | 8 | 2 | 2 |
Matrix representation of Dic3×C22⋊C4 ►in GL6(𝔽13)
| 1 | 12 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 10 | 0 | 0 | 0 |
| 0 | 0 | 6 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 8 | 0 | 0 | 0 | 0 | 0 |
| 8 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 6 | 0 | 0 |
| 0 | 0 | 9 | 6 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 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 |
| 5 | 0 | 0 | 0 | 0 | 0 |
| 0 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 12 | 0 |
G:=sub<GL(6,GF(13))| [1,1,0,0,0,0,12,0,0,0,0,0,0,0,10,6,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[8,8,0,0,0,0,0,5,0,0,0,0,0,0,7,9,0,0,0,0,6,6,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1],[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],[5,0,0,0,0,0,0,5,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,1,0] >;
Dic3×C22⋊C4 in GAP, Magma, Sage, TeX
{\rm Dic}_3\times C_2^2\rtimes C_4 % in TeX
G:=Group("Dic3xC2^2:C4"); // GroupNames label
G:=SmallGroup(192,500);
// by ID
G=gap.SmallGroup(192,500);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,56,387,100,6278]);
// Polycyclic
G:=Group<a,b,c,d,e|a^6=c^2=d^2=e^4=1,b^2=a^3,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,e*c*e^-1=c*d=d*c,d*e=e*d>;
// generators/relations