direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C2×C6×C22⋊C4, C24⋊8C12, C25.6C6, (C23×C4)⋊6C6, (C23×C6)⋊6C4, (C23×C12)⋊5C2, C23⋊6(C2×C12), (C24×C6).2C2, (C2×C12)⋊13C23, C6.53(C23×C4), C24.34(C2×C6), C2.1(C23×C12), C22.57(C6×D4), C23.63(C3×D4), (C2×C6).332C24, C22⋊3(C22×C12), (C22×C6).219D4, C6.177(C22×D4), C22.5(C23×C6), (C22×C12)⋊57C22, C23.69(C22×C6), (C23×C6).88C22, (C22×C6).251C23, C2.1(D4×C2×C6), (C2×C6)⋊8(C22×C4), (C2×C4)⋊3(C22×C6), (C22×C6)⋊16(C2×C4), (C22×C4)⋊17(C2×C6), (C2×C6).679(C2×D4), SmallGroup(192,1401)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1010 in 674 conjugacy classes, 338 normal (12 characteristic)
C1, C2, C2 [×14], C2 [×8], C3, C4 [×8], C22, C22 [×42], C22 [×56], C6, C6 [×14], C6 [×8], C2×C4 [×8], C2×C4 [×24], C23 [×43], C23 [×56], C12 [×8], C2×C6, C2×C6 [×42], C2×C6 [×56], C22⋊C4 [×16], C22×C4 [×12], C22×C4 [×8], C24, C24 [×14], C24 [×8], C2×C12 [×8], C2×C12 [×24], C22×C6 [×43], C22×C6 [×56], C2×C22⋊C4 [×12], C23×C4 [×2], C25, C3×C22⋊C4 [×16], C22×C12 [×12], C22×C12 [×8], C23×C6, C23×C6 [×14], C23×C6 [×8], C22×C22⋊C4, C6×C22⋊C4 [×12], C23×C12 [×2], C24×C6, C2×C6×C22⋊C4
Quotients:
C1, C2 [×15], C3, C4 [×8], C22 [×35], C6 [×15], C2×C4 [×28], D4 [×8], C23 [×15], C12 [×8], C2×C6 [×35], C22⋊C4 [×16], C22×C4 [×14], C2×D4 [×12], C24, C2×C12 [×28], C3×D4 [×8], C22×C6 [×15], C2×C22⋊C4 [×12], C23×C4, C22×D4 [×2], C3×C22⋊C4 [×16], C22×C12 [×14], C6×D4 [×12], C23×C6, C22×C22⋊C4, C6×C22⋊C4 [×12], C23×C12, D4×C2×C6 [×2], C2×C6×C22⋊C4
Generators and relations
G = < a,b,c,d,e | a2=b6=c2=d2=e4=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, ece-1=cd=dc, de=ed >
►On 96 points
Generators in S96
(1 50)(2 51)(3 52)(4 53)(5 54)(6 49)(7 75)(8 76)(9 77)(10 78)(11 73)(12 74)(13 89)(14 90)(15 85)(16 86)(17 87)(18 88)(19 70)(20 71)(21 72)(22 67)(23 68)(24 69)(25 66)(26 61)(27 62)(28 63)(29 64)(30 65)(31 47)(32 48)(33 43)(34 44)(35 45)(36 46)(37 56)(38 57)(39 58)(40 59)(41 60)(42 55)(79 95)(80 96)(81 91)(82 92)(83 93)(84 94)
(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 34)(2 35)(3 36)(4 31)(5 32)(6 33)(7 16)(8 17)(9 18)(10 13)(11 14)(12 15)(19 94)(20 95)(21 96)(22 91)(23 92)(24 93)(25 38)(26 39)(27 40)(28 41)(29 42)(30 37)(43 49)(44 50)(45 51)(46 52)(47 53)(48 54)(55 64)(56 65)(57 66)(58 61)(59 62)(60 63)(67 81)(68 82)(69 83)(70 84)(71 79)(72 80)(73 90)(74 85)(75 86)(76 87)(77 88)(78 89)
(1 39)(2 40)(3 41)(4 42)(5 37)(6 38)(7 93)(8 94)(9 95)(10 96)(11 91)(12 92)(13 21)(14 22)(15 23)(16 24)(17 19)(18 20)(25 33)(26 34)(27 35)(28 36)(29 31)(30 32)(43 66)(44 61)(45 62)(46 63)(47 64)(48 65)(49 57)(50 58)(51 59)(52 60)(53 55)(54 56)(67 90)(68 85)(69 86)(70 87)(71 88)(72 89)(73 81)(74 82)(75 83)(76 84)(77 79)(78 80)
(1 20 31 12)(2 21 32 7)(3 22 33 8)(4 23 34 9)(5 24 35 10)(6 19 36 11)(13 30 93 40)(14 25 94 41)(15 26 95 42)(16 27 96 37)(17 28 91 38)(18 29 92 39)(43 76 52 67)(44 77 53 68)(45 78 54 69)(46 73 49 70)(47 74 50 71)(48 75 51 72)(55 85 61 79)(56 86 62 80)(57 87 63 81)(58 88 64 82)(59 89 65 83)(60 90 66 84)
G:=sub<Sym(96)| (1,50)(2,51)(3,52)(4,53)(5,54)(6,49)(7,75)(8,76)(9,77)(10,78)(11,73)(12,74)(13,89)(14,90)(15,85)(16,86)(17,87)(18,88)(19,70)(20,71)(21,72)(22,67)(23,68)(24,69)(25,66)(26,61)(27,62)(28,63)(29,64)(30,65)(31,47)(32,48)(33,43)(34,44)(35,45)(36,46)(37,56)(38,57)(39,58)(40,59)(41,60)(42,55)(79,95)(80,96)(81,91)(82,92)(83,93)(84,94), (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), (1,34)(2,35)(3,36)(4,31)(5,32)(6,33)(7,16)(8,17)(9,18)(10,13)(11,14)(12,15)(19,94)(20,95)(21,96)(22,91)(23,92)(24,93)(25,38)(26,39)(27,40)(28,41)(29,42)(30,37)(43,49)(44,50)(45,51)(46,52)(47,53)(48,54)(55,64)(56,65)(57,66)(58,61)(59,62)(60,63)(67,81)(68,82)(69,83)(70,84)(71,79)(72,80)(73,90)(74,85)(75,86)(76,87)(77,88)(78,89), (1,39)(2,40)(3,41)(4,42)(5,37)(6,38)(7,93)(8,94)(9,95)(10,96)(11,91)(12,92)(13,21)(14,22)(15,23)(16,24)(17,19)(18,20)(25,33)(26,34)(27,35)(28,36)(29,31)(30,32)(43,66)(44,61)(45,62)(46,63)(47,64)(48,65)(49,57)(50,58)(51,59)(52,60)(53,55)(54,56)(67,90)(68,85)(69,86)(70,87)(71,88)(72,89)(73,81)(74,82)(75,83)(76,84)(77,79)(78,80), (1,20,31,12)(2,21,32,7)(3,22,33,8)(4,23,34,9)(5,24,35,10)(6,19,36,11)(13,30,93,40)(14,25,94,41)(15,26,95,42)(16,27,96,37)(17,28,91,38)(18,29,92,39)(43,76,52,67)(44,77,53,68)(45,78,54,69)(46,73,49,70)(47,74,50,71)(48,75,51,72)(55,85,61,79)(56,86,62,80)(57,87,63,81)(58,88,64,82)(59,89,65,83)(60,90,66,84)>;
G:=Group( (1,50)(2,51)(3,52)(4,53)(5,54)(6,49)(7,75)(8,76)(9,77)(10,78)(11,73)(12,74)(13,89)(14,90)(15,85)(16,86)(17,87)(18,88)(19,70)(20,71)(21,72)(22,67)(23,68)(24,69)(25,66)(26,61)(27,62)(28,63)(29,64)(30,65)(31,47)(32,48)(33,43)(34,44)(35,45)(36,46)(37,56)(38,57)(39,58)(40,59)(41,60)(42,55)(79,95)(80,96)(81,91)(82,92)(83,93)(84,94), (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), (1,34)(2,35)(3,36)(4,31)(5,32)(6,33)(7,16)(8,17)(9,18)(10,13)(11,14)(12,15)(19,94)(20,95)(21,96)(22,91)(23,92)(24,93)(25,38)(26,39)(27,40)(28,41)(29,42)(30,37)(43,49)(44,50)(45,51)(46,52)(47,53)(48,54)(55,64)(56,65)(57,66)(58,61)(59,62)(60,63)(67,81)(68,82)(69,83)(70,84)(71,79)(72,80)(73,90)(74,85)(75,86)(76,87)(77,88)(78,89), (1,39)(2,40)(3,41)(4,42)(5,37)(6,38)(7,93)(8,94)(9,95)(10,96)(11,91)(12,92)(13,21)(14,22)(15,23)(16,24)(17,19)(18,20)(25,33)(26,34)(27,35)(28,36)(29,31)(30,32)(43,66)(44,61)(45,62)(46,63)(47,64)(48,65)(49,57)(50,58)(51,59)(52,60)(53,55)(54,56)(67,90)(68,85)(69,86)(70,87)(71,88)(72,89)(73,81)(74,82)(75,83)(76,84)(77,79)(78,80), (1,20,31,12)(2,21,32,7)(3,22,33,8)(4,23,34,9)(5,24,35,10)(6,19,36,11)(13,30,93,40)(14,25,94,41)(15,26,95,42)(16,27,96,37)(17,28,91,38)(18,29,92,39)(43,76,52,67)(44,77,53,68)(45,78,54,69)(46,73,49,70)(47,74,50,71)(48,75,51,72)(55,85,61,79)(56,86,62,80)(57,87,63,81)(58,88,64,82)(59,89,65,83)(60,90,66,84) );
G=PermutationGroup([(1,50),(2,51),(3,52),(4,53),(5,54),(6,49),(7,75),(8,76),(9,77),(10,78),(11,73),(12,74),(13,89),(14,90),(15,85),(16,86),(17,87),(18,88),(19,70),(20,71),(21,72),(22,67),(23,68),(24,69),(25,66),(26,61),(27,62),(28,63),(29,64),(30,65),(31,47),(32,48),(33,43),(34,44),(35,45),(36,46),(37,56),(38,57),(39,58),(40,59),(41,60),(42,55),(79,95),(80,96),(81,91),(82,92),(83,93),(84,94)], [(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)], [(1,34),(2,35),(3,36),(4,31),(5,32),(6,33),(7,16),(8,17),(9,18),(10,13),(11,14),(12,15),(19,94),(20,95),(21,96),(22,91),(23,92),(24,93),(25,38),(26,39),(27,40),(28,41),(29,42),(30,37),(43,49),(44,50),(45,51),(46,52),(47,53),(48,54),(55,64),(56,65),(57,66),(58,61),(59,62),(60,63),(67,81),(68,82),(69,83),(70,84),(71,79),(72,80),(73,90),(74,85),(75,86),(76,87),(77,88),(78,89)], [(1,39),(2,40),(3,41),(4,42),(5,37),(6,38),(7,93),(8,94),(9,95),(10,96),(11,91),(12,92),(13,21),(14,22),(15,23),(16,24),(17,19),(18,20),(25,33),(26,34),(27,35),(28,36),(29,31),(30,32),(43,66),(44,61),(45,62),(46,63),(47,64),(48,65),(49,57),(50,58),(51,59),(52,60),(53,55),(54,56),(67,90),(68,85),(69,86),(70,87),(71,88),(72,89),(73,81),(74,82),(75,83),(76,84),(77,79),(78,80)], [(1,20,31,12),(2,21,32,7),(3,22,33,8),(4,23,34,9),(5,24,35,10),(6,19,36,11),(13,30,93,40),(14,25,94,41),(15,26,95,42),(16,27,96,37),(17,28,91,38),(18,29,92,39),(43,76,52,67),(44,77,53,68),(45,78,54,69),(46,73,49,70),(47,74,50,71),(48,75,51,72),(55,85,61,79),(56,86,62,80),(57,87,63,81),(58,88,64,82),(59,89,65,83),(60,90,66,84)])
Matrix representation ►G ⊆ GL5(𝔽13)
| 12 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 0 | 3 |
| 12 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 12 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 12 |
| 12 | 0 | 0 | 0 | 0 |
| 0 | 5 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 |
| 0 | 0 | 0 | 12 | 0 |
G:=sub<GL(5,GF(13))| [12,0,0,0,0,0,1,0,0,0,0,0,12,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,3,0,0,0,0,0,3],[12,0,0,0,0,0,12,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,12],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,12,0,0,0,0,0,12],[12,0,0,0,0,0,5,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,12,0] >;
120 conjugacy classes
| class | 1 | 2A | ··· | 2O | 2P | ··· | 2W | 3A | 3B | 4A | ··· | 4P | 6A | ··· | 6AD | 6AE | ··· | 6AT | 12A | ··· | 12AF |
| order | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 3 | 3 | 4 | ··· | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
120 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C3 | C4 | C6 | C6 | C6 | C12 | D4 | C3×D4 |
| kernel | C2×C6×C22⋊C4 | C6×C22⋊C4 | C23×C12 | C24×C6 | C22×C22⋊C4 | C23×C6 | C2×C22⋊C4 | C23×C4 | C25 | C24 | C22×C6 | C23 |
| # reps | 1 | 12 | 2 | 1 | 2 | 16 | 24 | 4 | 2 | 32 | 8 | 16 |
In GAP, Magma, Sage, TeX
C_2\times C_6\times C_2^2\rtimes C_4
% in TeX
G:=Group("C2xC6xC2^2:C4"); // GroupNames label
G:=SmallGroup(192,1401);
// by ID
G=gap.SmallGroup(192,1401);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-2,672,701]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^6=c^2=d^2=e^4=1,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,e*c*e^-1=c*d=d*c,d*e=e*d>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀