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), (C23×C6).112C22, (C22×C6).428C23, (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
Subgroups: 1272 in 674 conjugacy classes, 287 normal (11 characteristic)
C1, C2, C2 [×14], C2 [×8], C3, C4 [×8], C22, C22 [×42], C22 [×56], C6, C6 [×14], C6 [×8], C2×C4 [×32], C23 [×43], C23 [×56], Dic3 [×8], C2×C6, C2×C6 [×42], C2×C6 [×56], C22⋊C4 [×16], C22×C4 [×20], C24, C24 [×14], C24 [×8], C2×Dic3 [×8], C2×Dic3 [×24], C22×C6 [×43], C22×C6 [×56], C2×C22⋊C4 [×12], C23×C4 [×2], C25, C6.D4 [×16], C22×Dic3 [×12], C22×Dic3 [×8], C23×C6, C23×C6 [×14], C23×C6 [×8], C22×C22⋊C4, C2×C6.D4 [×12], C23×Dic3 [×2], C24×C6, C22×C6.D4
Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], S3, C2×C4 [×28], D4 [×8], C23 [×15], Dic3 [×8], D6 [×7], C22⋊C4 [×16], C22×C4 [×14], C2×D4 [×12], C24, C2×Dic3 [×28], C3⋊D4 [×8], C22×S3 [×7], C2×C22⋊C4 [×12], C23×C4, C22×D4 [×2], C6.D4 [×16], C22×Dic3 [×14], C2×C3⋊D4 [×12], S3×C23, C22×C22⋊C4, C2×C6.D4 [×12], C23×Dic3, C22×C3⋊D4 [×2], C22×C6.D4
Generators and relations
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 >
►On 96 points
Generators in S96
(1 25)(2 26)(3 27)(4 28)(5 29)(6 30)(7 84)(8 79)(9 80)(10 81)(11 82)(12 83)(13 24)(14 19)(15 20)(16 21)(17 22)(18 23)(31 49)(32 50)(33 51)(34 52)(35 53)(36 54)(37 48)(38 43)(39 44)(40 45)(41 46)(42 47)(55 75)(56 76)(57 77)(58 78)(59 73)(60 74)(61 71)(62 72)(63 67)(64 68)(65 69)(66 70)(85 95)(86 96)(87 91)(88 92)(89 93)(90 94)
(1 33)(2 34)(3 35)(4 36)(5 31)(6 32)(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 72)(2 73 14 71)(3 78 15 70)(4 77 16 69)(5 76 17 68)(6 75 18 67)(7 37 96 36)(8 42 91 35)(9 41 92 34)(10 40 93 33)(11 39 94 32)(12 38 95 31)(19 61 26 59)(20 66 27 58)(21 65 28 57)(22 64 29 56)(23 63 30 55)(24 62 25 60)(43 85 49 83)(44 90 50 82)(45 89 51 81)(46 88 52 80)(47 87 53 79)(48 86 54 84)
(1 96 4 93)(2 95 5 92)(3 94 6 91)(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)(7,84)(8,79)(9,80)(10,81)(11,82)(12,83)(13,24)(14,19)(15,20)(16,21)(17,22)(18,23)(31,49)(32,50)(33,51)(34,52)(35,53)(36,54)(37,48)(38,43)(39,44)(40,45)(41,46)(42,47)(55,75)(56,76)(57,77)(58,78)(59,73)(60,74)(61,71)(62,72)(63,67)(64,68)(65,69)(66,70)(85,95)(86,96)(87,91)(88,92)(89,93)(90,94), (1,33)(2,34)(3,35)(4,36)(5,31)(6,32)(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), (1,74,13,72)(2,73,14,71)(3,78,15,70)(4,77,16,69)(5,76,17,68)(6,75,18,67)(7,37,96,36)(8,42,91,35)(9,41,92,34)(10,40,93,33)(11,39,94,32)(12,38,95,31)(19,61,26,59)(20,66,27,58)(21,65,28,57)(22,64,29,56)(23,63,30,55)(24,62,25,60)(43,85,49,83)(44,90,50,82)(45,89,51,81)(46,88,52,80)(47,87,53,79)(48,86,54,84), (1,96,4,93)(2,95,5,92)(3,94,6,91)(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)(7,84)(8,79)(9,80)(10,81)(11,82)(12,83)(13,24)(14,19)(15,20)(16,21)(17,22)(18,23)(31,49)(32,50)(33,51)(34,52)(35,53)(36,54)(37,48)(38,43)(39,44)(40,45)(41,46)(42,47)(55,75)(56,76)(57,77)(58,78)(59,73)(60,74)(61,71)(62,72)(63,67)(64,68)(65,69)(66,70)(85,95)(86,96)(87,91)(88,92)(89,93)(90,94), (1,33)(2,34)(3,35)(4,36)(5,31)(6,32)(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), (1,74,13,72)(2,73,14,71)(3,78,15,70)(4,77,16,69)(5,76,17,68)(6,75,18,67)(7,37,96,36)(8,42,91,35)(9,41,92,34)(10,40,93,33)(11,39,94,32)(12,38,95,31)(19,61,26,59)(20,66,27,58)(21,65,28,57)(22,64,29,56)(23,63,30,55)(24,62,25,60)(43,85,49,83)(44,90,50,82)(45,89,51,81)(46,88,52,80)(47,87,53,79)(48,86,54,84), (1,96,4,93)(2,95,5,92)(3,94,6,91)(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),(7,84),(8,79),(9,80),(10,81),(11,82),(12,83),(13,24),(14,19),(15,20),(16,21),(17,22),(18,23),(31,49),(32,50),(33,51),(34,52),(35,53),(36,54),(37,48),(38,43),(39,44),(40,45),(41,46),(42,47),(55,75),(56,76),(57,77),(58,78),(59,73),(60,74),(61,71),(62,72),(63,67),(64,68),(65,69),(66,70),(85,95),(86,96),(87,91),(88,92),(89,93),(90,94)], [(1,33),(2,34),(3,35),(4,36),(5,31),(6,32),(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)], [(1,74,13,72),(2,73,14,71),(3,78,15,70),(4,77,16,69),(5,76,17,68),(6,75,18,67),(7,37,96,36),(8,42,91,35),(9,41,92,34),(10,40,93,33),(11,39,94,32),(12,38,95,31),(19,61,26,59),(20,66,27,58),(21,65,28,57),(22,64,29,56),(23,63,30,55),(24,62,25,60),(43,85,49,83),(44,90,50,82),(45,89,51,81),(46,88,52,80),(47,87,53,79),(48,86,54,84)], [(1,96,4,93),(2,95,5,92),(3,94,6,91),(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)])
Matrix representation ►G ⊆ 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] >;
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 |
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
×
⋊
ℤ
𝔽
○
≀