direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: S3×C42.C2, C42.235D6, (C4×S3).7Q8, C4.37(S3×Q8), C4⋊C4.203D6, D6.15(C2×Q8), C12.48(C2×Q8), (S3×C42).7C2, Dic3.5(C2×Q8), D6.41(C4○D4), C6.40(C22×Q8), Dic3.Q8⋊31C2, (C2×C6).231C24, (C2×C12).85C23, C12.3Q8⋊32C2, C12.6Q8⋊21C2, (C4×C12).191C22, Dic3⋊C4.71C22, C4⋊Dic3.238C22, C22.252(S3×C23), (C22×S3).257C23, (C4×Dic3).297C22, (C2×Dic3).257C23, C2.23(C2×S3×Q8), (S3×C4⋊C4).10C2, C3⋊4(C2×C42.C2), C2.83(S3×C4○D4), C6.194(C2×C4○D4), (C3×C42.C2)⋊4C2, (S3×C2×C4).248C22, (C2×C4).76(C22×S3), (C3×C4⋊C4).186C22, SmallGroup(192,1246)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 480 in 226 conjugacy classes, 111 normal (19 characteristic)
C1, C2, C2 [×2], C2 [×4], C3, C4 [×2], C4 [×14], C22, C22 [×6], S3 [×4], C6, C6 [×2], C2×C4, C2×C4 [×6], C2×C4 [×23], C23, Dic3 [×2], Dic3 [×6], C12 [×2], C12 [×6], D6 [×6], C2×C6, C42, C42 [×3], C4⋊C4 [×6], C4⋊C4 [×18], C22×C4 [×7], C4×S3 [×4], C4×S3 [×12], C2×Dic3, C2×Dic3 [×6], C2×C12, C2×C12 [×6], C22×S3, C2×C42, C2×C4⋊C4 [×6], C42.C2, C42.C2 [×7], C4×Dic3, C4×Dic3 [×2], Dic3⋊C4 [×12], C4⋊Dic3 [×6], C4×C12, C3×C4⋊C4 [×6], S3×C2×C4, S3×C2×C4 [×6], C2×C42.C2, C12.6Q8, S3×C42, Dic3.Q8 [×4], C12.3Q8 [×2], S3×C4⋊C4 [×6], C3×C42.C2, S3×C42.C2
Quotients:
C1, C2 [×15], C22 [×35], S3, Q8 [×4], C23 [×15], D6 [×7], C2×Q8 [×6], C4○D4 [×4], C24, C22×S3 [×7], C42.C2 [×4], C22×Q8, C2×C4○D4 [×2], S3×Q8 [×2], S3×C23, C2×C42.C2, C2×S3×Q8, S3×C4○D4 [×2], S3×C42.C2
Generators and relations
G = < a,b,c,d,e | a3=b2=c4=d4=1, e2=d2, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece-1=cd2, ede-1=c2d >
►On 96 points
Generators in S96
(1 63 5)(2 64 6)(3 61 7)(4 62 8)(9 72 46)(10 69 47)(11 70 48)(12 71 45)(13 81 39)(14 82 40)(15 83 37)(16 84 38)(17 87 75)(18 88 76)(19 85 73)(20 86 74)(21 59 33)(22 60 34)(23 57 35)(24 58 36)(25 93 51)(26 94 52)(27 95 49)(28 96 50)(29 79 53)(30 80 54)(31 77 55)(32 78 56)(41 91 65)(42 92 66)(43 89 67)(44 90 68)
(1 3)(2 4)(5 61)(6 62)(7 63)(8 64)(9 70)(10 71)(11 72)(12 69)(13 15)(14 16)(17 73)(18 74)(19 75)(20 76)(21 57)(22 58)(23 59)(24 60)(25 27)(26 28)(29 77)(30 78)(31 79)(32 80)(33 35)(34 36)(37 81)(38 82)(39 83)(40 84)(41 89)(42 90)(43 91)(44 92)(45 47)(46 48)(49 93)(50 94)(51 95)(52 96)(53 55)(54 56)(65 67)(66 68)(85 87)(86 88)
(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 65 53 87)(2 66 54 88)(3 67 55 85)(4 68 56 86)(5 91 79 17)(6 92 80 18)(7 89 77 19)(8 90 78 20)(9 22 96 84)(10 23 93 81)(11 24 94 82)(12 21 95 83)(13 47 35 25)(14 48 36 26)(15 45 33 27)(16 46 34 28)(29 75 63 41)(30 76 64 42)(31 73 61 43)(32 74 62 44)(37 71 59 49)(38 72 60 50)(39 69 57 51)(40 70 58 52)
(1 14 53 36)(2 33 54 15)(3 16 55 34)(4 35 56 13)(5 40 79 58)(6 59 80 37)(7 38 77 60)(8 57 78 39)(9 75 96 41)(10 42 93 76)(11 73 94 43)(12 44 95 74)(17 50 91 72)(18 69 92 51)(19 52 89 70)(20 71 90 49)(21 30 83 64)(22 61 84 31)(23 32 81 62)(24 63 82 29)(25 88 47 66)(26 67 48 85)(27 86 45 68)(28 65 46 87)
G:=sub<Sym(96)| (1,63,5)(2,64,6)(3,61,7)(4,62,8)(9,72,46)(10,69,47)(11,70,48)(12,71,45)(13,81,39)(14,82,40)(15,83,37)(16,84,38)(17,87,75)(18,88,76)(19,85,73)(20,86,74)(21,59,33)(22,60,34)(23,57,35)(24,58,36)(25,93,51)(26,94,52)(27,95,49)(28,96,50)(29,79,53)(30,80,54)(31,77,55)(32,78,56)(41,91,65)(42,92,66)(43,89,67)(44,90,68), (1,3)(2,4)(5,61)(6,62)(7,63)(8,64)(9,70)(10,71)(11,72)(12,69)(13,15)(14,16)(17,73)(18,74)(19,75)(20,76)(21,57)(22,58)(23,59)(24,60)(25,27)(26,28)(29,77)(30,78)(31,79)(32,80)(33,35)(34,36)(37,81)(38,82)(39,83)(40,84)(41,89)(42,90)(43,91)(44,92)(45,47)(46,48)(49,93)(50,94)(51,95)(52,96)(53,55)(54,56)(65,67)(66,68)(85,87)(86,88), (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), (1,65,53,87)(2,66,54,88)(3,67,55,85)(4,68,56,86)(5,91,79,17)(6,92,80,18)(7,89,77,19)(8,90,78,20)(9,22,96,84)(10,23,93,81)(11,24,94,82)(12,21,95,83)(13,47,35,25)(14,48,36,26)(15,45,33,27)(16,46,34,28)(29,75,63,41)(30,76,64,42)(31,73,61,43)(32,74,62,44)(37,71,59,49)(38,72,60,50)(39,69,57,51)(40,70,58,52), (1,14,53,36)(2,33,54,15)(3,16,55,34)(4,35,56,13)(5,40,79,58)(6,59,80,37)(7,38,77,60)(8,57,78,39)(9,75,96,41)(10,42,93,76)(11,73,94,43)(12,44,95,74)(17,50,91,72)(18,69,92,51)(19,52,89,70)(20,71,90,49)(21,30,83,64)(22,61,84,31)(23,32,81,62)(24,63,82,29)(25,88,47,66)(26,67,48,85)(27,86,45,68)(28,65,46,87)>;
G:=Group( (1,63,5)(2,64,6)(3,61,7)(4,62,8)(9,72,46)(10,69,47)(11,70,48)(12,71,45)(13,81,39)(14,82,40)(15,83,37)(16,84,38)(17,87,75)(18,88,76)(19,85,73)(20,86,74)(21,59,33)(22,60,34)(23,57,35)(24,58,36)(25,93,51)(26,94,52)(27,95,49)(28,96,50)(29,79,53)(30,80,54)(31,77,55)(32,78,56)(41,91,65)(42,92,66)(43,89,67)(44,90,68), (1,3)(2,4)(5,61)(6,62)(7,63)(8,64)(9,70)(10,71)(11,72)(12,69)(13,15)(14,16)(17,73)(18,74)(19,75)(20,76)(21,57)(22,58)(23,59)(24,60)(25,27)(26,28)(29,77)(30,78)(31,79)(32,80)(33,35)(34,36)(37,81)(38,82)(39,83)(40,84)(41,89)(42,90)(43,91)(44,92)(45,47)(46,48)(49,93)(50,94)(51,95)(52,96)(53,55)(54,56)(65,67)(66,68)(85,87)(86,88), (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), (1,65,53,87)(2,66,54,88)(3,67,55,85)(4,68,56,86)(5,91,79,17)(6,92,80,18)(7,89,77,19)(8,90,78,20)(9,22,96,84)(10,23,93,81)(11,24,94,82)(12,21,95,83)(13,47,35,25)(14,48,36,26)(15,45,33,27)(16,46,34,28)(29,75,63,41)(30,76,64,42)(31,73,61,43)(32,74,62,44)(37,71,59,49)(38,72,60,50)(39,69,57,51)(40,70,58,52), (1,14,53,36)(2,33,54,15)(3,16,55,34)(4,35,56,13)(5,40,79,58)(6,59,80,37)(7,38,77,60)(8,57,78,39)(9,75,96,41)(10,42,93,76)(11,73,94,43)(12,44,95,74)(17,50,91,72)(18,69,92,51)(19,52,89,70)(20,71,90,49)(21,30,83,64)(22,61,84,31)(23,32,81,62)(24,63,82,29)(25,88,47,66)(26,67,48,85)(27,86,45,68)(28,65,46,87) );
G=PermutationGroup([(1,63,5),(2,64,6),(3,61,7),(4,62,8),(9,72,46),(10,69,47),(11,70,48),(12,71,45),(13,81,39),(14,82,40),(15,83,37),(16,84,38),(17,87,75),(18,88,76),(19,85,73),(20,86,74),(21,59,33),(22,60,34),(23,57,35),(24,58,36),(25,93,51),(26,94,52),(27,95,49),(28,96,50),(29,79,53),(30,80,54),(31,77,55),(32,78,56),(41,91,65),(42,92,66),(43,89,67),(44,90,68)], [(1,3),(2,4),(5,61),(6,62),(7,63),(8,64),(9,70),(10,71),(11,72),(12,69),(13,15),(14,16),(17,73),(18,74),(19,75),(20,76),(21,57),(22,58),(23,59),(24,60),(25,27),(26,28),(29,77),(30,78),(31,79),(32,80),(33,35),(34,36),(37,81),(38,82),(39,83),(40,84),(41,89),(42,90),(43,91),(44,92),(45,47),(46,48),(49,93),(50,94),(51,95),(52,96),(53,55),(54,56),(65,67),(66,68),(85,87),(86,88)], [(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)], [(1,65,53,87),(2,66,54,88),(3,67,55,85),(4,68,56,86),(5,91,79,17),(6,92,80,18),(7,89,77,19),(8,90,78,20),(9,22,96,84),(10,23,93,81),(11,24,94,82),(12,21,95,83),(13,47,35,25),(14,48,36,26),(15,45,33,27),(16,46,34,28),(29,75,63,41),(30,76,64,42),(31,73,61,43),(32,74,62,44),(37,71,59,49),(38,72,60,50),(39,69,57,51),(40,70,58,52)], [(1,14,53,36),(2,33,54,15),(3,16,55,34),(4,35,56,13),(5,40,79,58),(6,59,80,37),(7,38,77,60),(8,57,78,39),(9,75,96,41),(10,42,93,76),(11,73,94,43),(12,44,95,74),(17,50,91,72),(18,69,92,51),(19,52,89,70),(20,71,90,49),(21,30,83,64),(22,61,84,31),(23,32,81,62),(24,63,82,29),(25,88,47,66),(26,67,48,85),(27,86,45,68),(28,65,46,87)])
Matrix representation ►G ⊆ GL6(𝔽13)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 12 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 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 | 12 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 11 | 4 | 0 | 0 | 0 | 0 |
| 9 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 10 |
| 0 | 0 | 0 | 0 | 10 | 4 |
| 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 | 4 | 3 |
| 0 | 0 | 0 | 0 | 3 | 9 |
| 9 | 2 | 0 | 0 | 0 | 0 |
| 11 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 9 |
| 0 | 0 | 0 | 0 | 9 | 10 |
G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,1,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,12,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[11,9,0,0,0,0,4,2,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,9,10,0,0,0,0,10,4],[5,0,0,0,0,0,0,5,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,4,3,0,0,0,0,3,9],[9,11,0,0,0,0,2,4,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,3,9,0,0,0,0,9,10] >;
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 4K | ··· | 4P | 4Q | 4R | 4S | 4T | 6A | 6B | 6C | 12A | ··· | 12F | 12G | 12H | 12I | 12J |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 12 | ··· | 12 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | Q8 | D6 | D6 | C4○D4 | S3×Q8 | S3×C4○D4 |
| kernel | S3×C42.C2 | C12.6Q8 | S3×C42 | Dic3.Q8 | C12.3Q8 | S3×C4⋊C4 | C3×C42.C2 | C42.C2 | C4×S3 | C42 | C4⋊C4 | D6 | C4 | C2 |
| # reps | 1 | 1 | 1 | 4 | 2 | 6 | 1 | 1 | 4 | 1 | 6 | 8 | 2 | 4 |
In GAP, Magma, Sage, TeX
S_3\times C_4^2.C_2
% in TeX
G:=Group("S3xC4^2.C2"); // GroupNames label
G:=SmallGroup(192,1246);
// by ID
G=gap.SmallGroup(192,1246);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,100,346,297,136,6278]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^2=c^4=d^4=1,e^2=d^2,b*a*b=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,c*d=d*c,e*c*e^-1=c*d^2,e*d*e^-1=c^2*d>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀