direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C22×C6.D4, C25.5S3, C24⋊8Dic3, C24.90D6, (C23×C6)⋊8C4, (C24×C6).4C2, C6.52(C23×C4), C23⋊6(C2×Dic3), (C2×C6).321C24, (C2×Dic3)⋊9C23, (C22×C6).212D4, C6.175(C22×D4), (C23×Dic3)⋊10C2, C22.50(S3×C23), C2.14(C23×Dic3), C22⋊3(C22×Dic3), C23.115(C3⋊D4), C23.353(C22×S3), (C22×C6).428C23, (C23×C6).112C22, (C22×Dic3)⋊50C22, C6⋊3(C2×C22⋊C4), (C2×C6)⋊9(C22×C4), C3⋊3(C22×C22⋊C4), (C2×C6)⋊9(C22⋊C4), (C22×C6)⋊17(C2×C4), (C2×C6).592(C2×D4), C2.4(C22×C3⋊D4), C22.121(C2×C3⋊D4), SmallGroup(192,1398)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C22×C6.D4
G = < a,b,c,d,e | a2=b2=c6=d4=1, e2=c3, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=ece-1=c-1, ede-1=c3d-1 >
Subgroups: 1272 in 674 conjugacy classes, 287 normal (11 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, C6, C6, C6, C2×C4, C23, C23, Dic3, C2×C6, C2×C6, C2×C6, C22⋊C4, C22×C4, C24, C24, C24, C2×Dic3, C2×Dic3, C22×C6, C22×C6, C2×C22⋊C4, C23×C4, C25, C6.D4, C22×Dic3, C22×Dic3, C23×C6, C23×C6, C23×C6, C22×C22⋊C4, C2×C6.D4, C23×Dic3, C24×C6, C22×C6.D4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, Dic3, D6, C22⋊C4, C22×C4, C2×D4, C24, C2×Dic3, C3⋊D4, C22×S3, C2×C22⋊C4, C23×C4, C22×D4, C6.D4, C22×Dic3, C2×C3⋊D4, S3×C23, C22×C22⋊C4, C2×C6.D4, C23×Dic3, C22×C3⋊D4, C22×C6.D4
►On 96 points
Generators in S96
(1 25)(2 26)(3 27)(4 28)(5 29)(6 30)(7 82)(8 83)(9 84)(10 79)(11 80)(12 81)(13 20)(14 21)(15 22)(16 23)(17 24)(18 19)(31 53)(32 54)(33 49)(34 50)(35 51)(36 52)(37 44)(38 45)(39 46)(40 47)(41 48)(42 43)(55 77)(56 78)(57 73)(58 74)(59 75)(60 76)(61 69)(62 70)(63 71)(64 72)(65 67)(66 68)(85 93)(86 94)(87 95)(88 96)(89 91)(90 92)
(1 35)(2 36)(3 31)(4 32)(5 33)(6 34)(7 77)(8 78)(9 73)(10 74)(11 75)(12 76)(13 40)(14 41)(15 42)(16 37)(17 38)(18 39)(19 46)(20 47)(21 48)(22 43)(23 44)(24 45)(25 51)(26 52)(27 53)(28 54)(29 49)(30 50)(55 82)(56 83)(57 84)(58 79)(59 80)(60 81)(61 88)(62 89)(63 90)(64 85)(65 86)(66 87)(67 94)(68 95)(69 96)(70 91)(71 92)(72 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 74 13 70)(2 73 14 69)(3 78 15 68)(4 77 16 67)(5 76 17 72)(6 75 18 71)(7 37 94 32)(8 42 95 31)(9 41 96 36)(10 40 91 35)(11 39 92 34)(12 38 93 33)(19 63 30 59)(20 62 25 58)(21 61 26 57)(22 66 27 56)(23 65 28 55)(24 64 29 60)(43 87 53 83)(44 86 54 82)(45 85 49 81)(46 90 50 80)(47 89 51 79)(48 88 52 84)
(1 94 4 91)(2 93 5 96)(3 92 6 95)(7 16 10 13)(8 15 11 18)(9 14 12 17)(19 83 22 80)(20 82 23 79)(21 81 24 84)(25 86 28 89)(26 85 29 88)(27 90 30 87)(31 71 34 68)(32 70 35 67)(33 69 36 72)(37 74 40 77)(38 73 41 76)(39 78 42 75)(43 59 46 56)(44 58 47 55)(45 57 48 60)(49 61 52 64)(50 66 53 63)(51 65 54 62)
G:=sub<Sym(96)| (1,25)(2,26)(3,27)(4,28)(5,29)(6,30)(7,82)(8,83)(9,84)(10,79)(11,80)(12,81)(13,20)(14,21)(15,22)(16,23)(17,24)(18,19)(31,53)(32,54)(33,49)(34,50)(35,51)(36,52)(37,44)(38,45)(39,46)(40,47)(41,48)(42,43)(55,77)(56,78)(57,73)(58,74)(59,75)(60,76)(61,69)(62,70)(63,71)(64,72)(65,67)(66,68)(85,93)(86,94)(87,95)(88,96)(89,91)(90,92), (1,35)(2,36)(3,31)(4,32)(5,33)(6,34)(7,77)(8,78)(9,73)(10,74)(11,75)(12,76)(13,40)(14,41)(15,42)(16,37)(17,38)(18,39)(19,46)(20,47)(21,48)(22,43)(23,44)(24,45)(25,51)(26,52)(27,53)(28,54)(29,49)(30,50)(55,82)(56,83)(57,84)(58,79)(59,80)(60,81)(61,88)(62,89)(63,90)(64,85)(65,86)(66,87)(67,94)(68,95)(69,96)(70,91)(71,92)(72,93), (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), (1,74,13,70)(2,73,14,69)(3,78,15,68)(4,77,16,67)(5,76,17,72)(6,75,18,71)(7,37,94,32)(8,42,95,31)(9,41,96,36)(10,40,91,35)(11,39,92,34)(12,38,93,33)(19,63,30,59)(20,62,25,58)(21,61,26,57)(22,66,27,56)(23,65,28,55)(24,64,29,60)(43,87,53,83)(44,86,54,82)(45,85,49,81)(46,90,50,80)(47,89,51,79)(48,88,52,84), (1,94,4,91)(2,93,5,96)(3,92,6,95)(7,16,10,13)(8,15,11,18)(9,14,12,17)(19,83,22,80)(20,82,23,79)(21,81,24,84)(25,86,28,89)(26,85,29,88)(27,90,30,87)(31,71,34,68)(32,70,35,67)(33,69,36,72)(37,74,40,77)(38,73,41,76)(39,78,42,75)(43,59,46,56)(44,58,47,55)(45,57,48,60)(49,61,52,64)(50,66,53,63)(51,65,54,62)>;
G:=Group( (1,25)(2,26)(3,27)(4,28)(5,29)(6,30)(7,82)(8,83)(9,84)(10,79)(11,80)(12,81)(13,20)(14,21)(15,22)(16,23)(17,24)(18,19)(31,53)(32,54)(33,49)(34,50)(35,51)(36,52)(37,44)(38,45)(39,46)(40,47)(41,48)(42,43)(55,77)(56,78)(57,73)(58,74)(59,75)(60,76)(61,69)(62,70)(63,71)(64,72)(65,67)(66,68)(85,93)(86,94)(87,95)(88,96)(89,91)(90,92), (1,35)(2,36)(3,31)(4,32)(5,33)(6,34)(7,77)(8,78)(9,73)(10,74)(11,75)(12,76)(13,40)(14,41)(15,42)(16,37)(17,38)(18,39)(19,46)(20,47)(21,48)(22,43)(23,44)(24,45)(25,51)(26,52)(27,53)(28,54)(29,49)(30,50)(55,82)(56,83)(57,84)(58,79)(59,80)(60,81)(61,88)(62,89)(63,90)(64,85)(65,86)(66,87)(67,94)(68,95)(69,96)(70,91)(71,92)(72,93), (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), (1,74,13,70)(2,73,14,69)(3,78,15,68)(4,77,16,67)(5,76,17,72)(6,75,18,71)(7,37,94,32)(8,42,95,31)(9,41,96,36)(10,40,91,35)(11,39,92,34)(12,38,93,33)(19,63,30,59)(20,62,25,58)(21,61,26,57)(22,66,27,56)(23,65,28,55)(24,64,29,60)(43,87,53,83)(44,86,54,82)(45,85,49,81)(46,90,50,80)(47,89,51,79)(48,88,52,84), (1,94,4,91)(2,93,5,96)(3,92,6,95)(7,16,10,13)(8,15,11,18)(9,14,12,17)(19,83,22,80)(20,82,23,79)(21,81,24,84)(25,86,28,89)(26,85,29,88)(27,90,30,87)(31,71,34,68)(32,70,35,67)(33,69,36,72)(37,74,40,77)(38,73,41,76)(39,78,42,75)(43,59,46,56)(44,58,47,55)(45,57,48,60)(49,61,52,64)(50,66,53,63)(51,65,54,62) );
G=PermutationGroup([[(1,25),(2,26),(3,27),(4,28),(5,29),(6,30),(7,82),(8,83),(9,84),(10,79),(11,80),(12,81),(13,20),(14,21),(15,22),(16,23),(17,24),(18,19),(31,53),(32,54),(33,49),(34,50),(35,51),(36,52),(37,44),(38,45),(39,46),(40,47),(41,48),(42,43),(55,77),(56,78),(57,73),(58,74),(59,75),(60,76),(61,69),(62,70),(63,71),(64,72),(65,67),(66,68),(85,93),(86,94),(87,95),(88,96),(89,91),(90,92)], [(1,35),(2,36),(3,31),(4,32),(5,33),(6,34),(7,77),(8,78),(9,73),(10,74),(11,75),(12,76),(13,40),(14,41),(15,42),(16,37),(17,38),(18,39),(19,46),(20,47),(21,48),(22,43),(23,44),(24,45),(25,51),(26,52),(27,53),(28,54),(29,49),(30,50),(55,82),(56,83),(57,84),(58,79),(59,80),(60,81),(61,88),(62,89),(63,90),(64,85),(65,86),(66,87),(67,94),(68,95),(69,96),(70,91),(71,92),(72,93)], [(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)], [(1,74,13,70),(2,73,14,69),(3,78,15,68),(4,77,16,67),(5,76,17,72),(6,75,18,71),(7,37,94,32),(8,42,95,31),(9,41,96,36),(10,40,91,35),(11,39,92,34),(12,38,93,33),(19,63,30,59),(20,62,25,58),(21,61,26,57),(22,66,27,56),(23,65,28,55),(24,64,29,60),(43,87,53,83),(44,86,54,82),(45,85,49,81),(46,90,50,80),(47,89,51,79),(48,88,52,84)], [(1,94,4,91),(2,93,5,96),(3,92,6,95),(7,16,10,13),(8,15,11,18),(9,14,12,17),(19,83,22,80),(20,82,23,79),(21,81,24,84),(25,86,28,89),(26,85,29,88),(27,90,30,87),(31,71,34,68),(32,70,35,67),(33,69,36,72),(37,74,40,77),(38,73,41,76),(39,78,42,75),(43,59,46,56),(44,58,47,55),(45,57,48,60),(49,61,52,64),(50,66,53,63),(51,65,54,62)]])
72 conjugacy classes
| class | 1 | 2A | ··· | 2O | 2P | ··· | 2W | 3 | 4A | ··· | 4P | 6A | ··· | 6AE |
| order | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 3 | 4 | ··· | 4 | 6 | ··· | 6 |
| size | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | 6 | ··· | 6 | 2 | ··· | 2 |
72 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | - | + | ||
| image | C1 | C2 | C2 | C2 | C4 | S3 | D4 | Dic3 | D6 | C3⋊D4 |
| kernel | C22×C6.D4 | C2×C6.D4 | C23×Dic3 | C24×C6 | C23×C6 | C25 | C22×C6 | C24 | C24 | C23 |
| # reps | 1 | 12 | 2 | 1 | 16 | 1 | 8 | 8 | 7 | 16 |
Matrix representation of C22×C6.D4 ►in GL5(𝔽13)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 12 |
| 12 | 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 | 12 | 0 | 0 |
| 0 | 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 0 | 3 |
| 5 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 5 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 12 | 0 |
| 5 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 5 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(5,GF(13))| [1,0,0,0,0,0,1,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12],[12,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,1,0,0,0,0,0,1],[12,0,0,0,0,0,1,0,0,0,0,0,12,0,0,0,0,0,9,0,0,0,0,0,3],[5,0,0,0,0,0,1,0,0,0,0,0,5,0,0,0,0,0,0,12,0,0,0,1,0],[5,0,0,0,0,0,1,0,0,0,0,0,5,0,0,0,0,0,0,1,0,0,0,1,0] >;
C22×C6.D4 in GAP, Magma, Sage, TeX
C_2^2\times C_6.D_4
% in TeX
G:=Group("C2^2xC6.D4"); // GroupNames label
G:=SmallGroup(192,1398);
// by ID
G=gap.SmallGroup(192,1398);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,1123,6278]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^6=d^4=1,e^2=c^3,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,d*c*d^-1=e*c*e^-1=c^-1,e*d*e^-1=c^3*d^-1>;
// generators/relations