direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C22×C8⋊S3, C24⋊13C23, C12.66C24, (C2×C8)⋊36D6, C3⋊C8⋊11C23, (C22×C8)⋊16S3, C8⋊10(C22×S3), C6⋊1(C2×M4(2)), (C2×C6)⋊6M4(2), (C22×C24)⋊21C2, (C2×C24)⋊50C22, (S3×C23).8C4, C23.71(C4×S3), C6.29(C23×C4), C4.65(S3×C23), C3⋊1(C22×M4(2)), (C4×S3).33C23, D6.20(C22×C4), (C22×C4).486D6, C12.145(C22×C4), (C2×C12).879C23, Dic3.21(C22×C4), (C22×Dic3).17C4, (C22×C12).567C22, (S3×C2×C4).22C4, C4.120(S3×C2×C4), (C2×C3⋊C8)⋊46C22, (C22×C3⋊C8)⋊22C2, C2.30(S3×C22×C4), C22.75(S3×C2×C4), (C4×S3).35(C2×C4), (C2×C4).187(C4×S3), (S3×C22×C4).23C2, (C2×C12).257(C2×C4), (S3×C2×C4).301C22, (C22×S3).67(C2×C4), (C2×C6).155(C22×C4), (C2×C4).823(C22×S3), (C22×C6).102(C2×C4), (C2×Dic3).104(C2×C4), SmallGroup(192,1296)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 600 in 298 conjugacy classes, 167 normal (17 characteristic)
C1, C2, C2 [×6], C2 [×4], C3, C4, C4 [×3], C4 [×4], C22 [×7], C22 [×16], S3 [×4], C6, C6 [×6], C8 [×4], C8 [×4], C2×C4 [×6], C2×C4 [×22], C23, C23 [×10], Dic3 [×4], C12, C12 [×3], D6 [×4], D6 [×12], C2×C6 [×7], C2×C8 [×6], C2×C8 [×6], M4(2) [×16], C22×C4, C22×C4 [×13], C24, C3⋊C8 [×4], C24 [×4], C4×S3 [×16], C2×Dic3 [×6], C2×C12 [×6], C22×S3 [×6], C22×S3 [×4], C22×C6, C22×C8, C22×C8, C2×M4(2) [×12], C23×C4, C8⋊S3 [×16], C2×C3⋊C8 [×6], C2×C24 [×6], S3×C2×C4 [×12], C22×Dic3, C22×C12, S3×C23, C22×M4(2), C2×C8⋊S3 [×12], C22×C3⋊C8, C22×C24, S3×C22×C4, C22×C8⋊S3
Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], S3, C2×C4 [×28], C23 [×15], D6 [×7], M4(2) [×4], C22×C4 [×14], C24, C4×S3 [×4], C22×S3 [×7], C2×M4(2) [×6], C23×C4, C8⋊S3 [×4], S3×C2×C4 [×6], S3×C23, C22×M4(2), C2×C8⋊S3 [×6], S3×C22×C4, C22×C8⋊S3
Generators and relations
G = < a,b,c,d,e | a2=b2=c8=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece=c5, ede=d-1 >
►On 96 points
Generators in S96
(1 38)(2 39)(3 40)(4 33)(5 34)(6 35)(7 36)(8 37)(9 71)(10 72)(11 65)(12 66)(13 67)(14 68)(15 69)(16 70)(17 61)(18 62)(19 63)(20 64)(21 57)(22 58)(23 59)(24 60)(25 93)(26 94)(27 95)(28 96)(29 89)(30 90)(31 91)(32 92)(41 86)(42 87)(43 88)(44 81)(45 82)(46 83)(47 84)(48 85)(49 79)(50 80)(51 73)(52 74)(53 75)(54 76)(55 77)(56 78)
(1 21)(2 22)(3 23)(4 24)(5 17)(6 18)(7 19)(8 20)(9 54)(10 55)(11 56)(12 49)(13 50)(14 51)(15 52)(16 53)(25 84)(26 85)(27 86)(28 87)(29 88)(30 81)(31 82)(32 83)(33 60)(34 61)(35 62)(36 63)(37 64)(38 57)(39 58)(40 59)(41 95)(42 96)(43 89)(44 90)(45 91)(46 92)(47 93)(48 94)(65 78)(66 79)(67 80)(68 73)(69 74)(70 75)(71 76)(72 77)
(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 86 79)(2 87 80)(3 88 73)(4 81 74)(5 82 75)(6 83 76)(7 84 77)(8 85 78)(9 62 92)(10 63 93)(11 64 94)(12 57 95)(13 58 96)(14 59 89)(15 60 90)(16 61 91)(17 31 70)(18 32 71)(19 25 72)(20 26 65)(21 27 66)(22 28 67)(23 29 68)(24 30 69)(33 44 52)(34 45 53)(35 46 54)(36 47 55)(37 48 56)(38 41 49)(39 42 50)(40 43 51)
(1 17)(2 22)(3 19)(4 24)(5 21)(6 18)(7 23)(8 20)(9 46)(10 43)(11 48)(12 45)(13 42)(14 47)(15 44)(16 41)(25 73)(26 78)(27 75)(28 80)(29 77)(30 74)(31 79)(32 76)(33 60)(34 57)(35 62)(36 59)(37 64)(38 61)(39 58)(40 63)(49 91)(50 96)(51 93)(52 90)(53 95)(54 92)(55 89)(56 94)(65 85)(66 82)(67 87)(68 84)(69 81)(70 86)(71 83)(72 88)
G:=sub<Sym(96)| (1,38)(2,39)(3,40)(4,33)(5,34)(6,35)(7,36)(8,37)(9,71)(10,72)(11,65)(12,66)(13,67)(14,68)(15,69)(16,70)(17,61)(18,62)(19,63)(20,64)(21,57)(22,58)(23,59)(24,60)(25,93)(26,94)(27,95)(28,96)(29,89)(30,90)(31,91)(32,92)(41,86)(42,87)(43,88)(44,81)(45,82)(46,83)(47,84)(48,85)(49,79)(50,80)(51,73)(52,74)(53,75)(54,76)(55,77)(56,78), (1,21)(2,22)(3,23)(4,24)(5,17)(6,18)(7,19)(8,20)(9,54)(10,55)(11,56)(12,49)(13,50)(14,51)(15,52)(16,53)(25,84)(26,85)(27,86)(28,87)(29,88)(30,81)(31,82)(32,83)(33,60)(34,61)(35,62)(36,63)(37,64)(38,57)(39,58)(40,59)(41,95)(42,96)(43,89)(44,90)(45,91)(46,92)(47,93)(48,94)(65,78)(66,79)(67,80)(68,73)(69,74)(70,75)(71,76)(72,77), (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), (1,86,79)(2,87,80)(3,88,73)(4,81,74)(5,82,75)(6,83,76)(7,84,77)(8,85,78)(9,62,92)(10,63,93)(11,64,94)(12,57,95)(13,58,96)(14,59,89)(15,60,90)(16,61,91)(17,31,70)(18,32,71)(19,25,72)(20,26,65)(21,27,66)(22,28,67)(23,29,68)(24,30,69)(33,44,52)(34,45,53)(35,46,54)(36,47,55)(37,48,56)(38,41,49)(39,42,50)(40,43,51), (1,17)(2,22)(3,19)(4,24)(5,21)(6,18)(7,23)(8,20)(9,46)(10,43)(11,48)(12,45)(13,42)(14,47)(15,44)(16,41)(25,73)(26,78)(27,75)(28,80)(29,77)(30,74)(31,79)(32,76)(33,60)(34,57)(35,62)(36,59)(37,64)(38,61)(39,58)(40,63)(49,91)(50,96)(51,93)(52,90)(53,95)(54,92)(55,89)(56,94)(65,85)(66,82)(67,87)(68,84)(69,81)(70,86)(71,83)(72,88)>;
G:=Group( (1,38)(2,39)(3,40)(4,33)(5,34)(6,35)(7,36)(8,37)(9,71)(10,72)(11,65)(12,66)(13,67)(14,68)(15,69)(16,70)(17,61)(18,62)(19,63)(20,64)(21,57)(22,58)(23,59)(24,60)(25,93)(26,94)(27,95)(28,96)(29,89)(30,90)(31,91)(32,92)(41,86)(42,87)(43,88)(44,81)(45,82)(46,83)(47,84)(48,85)(49,79)(50,80)(51,73)(52,74)(53,75)(54,76)(55,77)(56,78), (1,21)(2,22)(3,23)(4,24)(5,17)(6,18)(7,19)(8,20)(9,54)(10,55)(11,56)(12,49)(13,50)(14,51)(15,52)(16,53)(25,84)(26,85)(27,86)(28,87)(29,88)(30,81)(31,82)(32,83)(33,60)(34,61)(35,62)(36,63)(37,64)(38,57)(39,58)(40,59)(41,95)(42,96)(43,89)(44,90)(45,91)(46,92)(47,93)(48,94)(65,78)(66,79)(67,80)(68,73)(69,74)(70,75)(71,76)(72,77), (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), (1,86,79)(2,87,80)(3,88,73)(4,81,74)(5,82,75)(6,83,76)(7,84,77)(8,85,78)(9,62,92)(10,63,93)(11,64,94)(12,57,95)(13,58,96)(14,59,89)(15,60,90)(16,61,91)(17,31,70)(18,32,71)(19,25,72)(20,26,65)(21,27,66)(22,28,67)(23,29,68)(24,30,69)(33,44,52)(34,45,53)(35,46,54)(36,47,55)(37,48,56)(38,41,49)(39,42,50)(40,43,51), (1,17)(2,22)(3,19)(4,24)(5,21)(6,18)(7,23)(8,20)(9,46)(10,43)(11,48)(12,45)(13,42)(14,47)(15,44)(16,41)(25,73)(26,78)(27,75)(28,80)(29,77)(30,74)(31,79)(32,76)(33,60)(34,57)(35,62)(36,59)(37,64)(38,61)(39,58)(40,63)(49,91)(50,96)(51,93)(52,90)(53,95)(54,92)(55,89)(56,94)(65,85)(66,82)(67,87)(68,84)(69,81)(70,86)(71,83)(72,88) );
G=PermutationGroup([(1,38),(2,39),(3,40),(4,33),(5,34),(6,35),(7,36),(8,37),(9,71),(10,72),(11,65),(12,66),(13,67),(14,68),(15,69),(16,70),(17,61),(18,62),(19,63),(20,64),(21,57),(22,58),(23,59),(24,60),(25,93),(26,94),(27,95),(28,96),(29,89),(30,90),(31,91),(32,92),(41,86),(42,87),(43,88),(44,81),(45,82),(46,83),(47,84),(48,85),(49,79),(50,80),(51,73),(52,74),(53,75),(54,76),(55,77),(56,78)], [(1,21),(2,22),(3,23),(4,24),(5,17),(6,18),(7,19),(8,20),(9,54),(10,55),(11,56),(12,49),(13,50),(14,51),(15,52),(16,53),(25,84),(26,85),(27,86),(28,87),(29,88),(30,81),(31,82),(32,83),(33,60),(34,61),(35,62),(36,63),(37,64),(38,57),(39,58),(40,59),(41,95),(42,96),(43,89),(44,90),(45,91),(46,92),(47,93),(48,94),(65,78),(66,79),(67,80),(68,73),(69,74),(70,75),(71,76),(72,77)], [(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)], [(1,86,79),(2,87,80),(3,88,73),(4,81,74),(5,82,75),(6,83,76),(7,84,77),(8,85,78),(9,62,92),(10,63,93),(11,64,94),(12,57,95),(13,58,96),(14,59,89),(15,60,90),(16,61,91),(17,31,70),(18,32,71),(19,25,72),(20,26,65),(21,27,66),(22,28,67),(23,29,68),(24,30,69),(33,44,52),(34,45,53),(35,46,54),(36,47,55),(37,48,56),(38,41,49),(39,42,50),(40,43,51)], [(1,17),(2,22),(3,19),(4,24),(5,21),(6,18),(7,23),(8,20),(9,46),(10,43),(11,48),(12,45),(13,42),(14,47),(15,44),(16,41),(25,73),(26,78),(27,75),(28,80),(29,77),(30,74),(31,79),(32,76),(33,60),(34,57),(35,62),(36,59),(37,64),(38,61),(39,58),(40,63),(49,91),(50,96),(51,93),(52,90),(53,95),(54,92),(55,89),(56,94),(65,85),(66,82),(67,87),(68,84),(69,81),(70,86),(71,83),(72,88)])
Matrix representation ►G ⊆ GL5(𝔽73)
| 72 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 72 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 72 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 45 | 36 |
| 0 | 0 | 0 | 54 | 28 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 72 | 72 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 72 | 72 | 0 | 0 |
| 0 | 0 | 0 | 72 | 26 |
| 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(5,GF(73))| [72,0,0,0,0,0,72,0,0,0,0,0,72,0,0,0,0,0,1,0,0,0,0,0,1],[72,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[72,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,45,54,0,0,0,36,28],[1,0,0,0,0,0,0,72,0,0,0,1,72,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,72,0,0,0,0,72,0,0,0,0,0,72,0,0,0,0,26,1] >;
72 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 3 | 4A | ··· | 4H | 4I | 4J | 4K | 4L | 6A | ··· | 6G | 8A | ··· | 8H | 8I | ··· | 8P | 12A | ··· | 12H | 24A | ··· | 24P |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 3 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 8 | ··· | 8 | 8 | ··· | 8 | 12 | ··· | 12 | 24 | ··· | 24 |
| size | 1 | 1 | ··· | 1 | 6 | 6 | 6 | 6 | 2 | 1 | ··· | 1 | 6 | 6 | 6 | 6 | 2 | ··· | 2 | 2 | ··· | 2 | 6 | ··· | 6 | 2 | ··· | 2 | 2 | ··· | 2 |
72 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | S3 | D6 | D6 | M4(2) | C4×S3 | C4×S3 | C8⋊S3 |
| kernel | C22×C8⋊S3 | C2×C8⋊S3 | C22×C3⋊C8 | C22×C24 | S3×C22×C4 | S3×C2×C4 | C22×Dic3 | S3×C23 | C22×C8 | C2×C8 | C22×C4 | C2×C6 | C2×C4 | C23 | C22 |
| # reps | 1 | 12 | 1 | 1 | 1 | 12 | 2 | 2 | 1 | 6 | 1 | 8 | 6 | 2 | 16 |
In GAP, Magma, Sage, TeX
C_2^2\times C_8\rtimes S_3
% in TeX
G:=Group("C2^2xC8:S3"); // GroupNames label
G:=SmallGroup(192,1296);
// by ID
G=gap.SmallGroup(192,1296);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,1123,80,102,6278]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^8=d^3=e^2=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,c*d=d*c,e*c*e=c^5,e*d*e=d^-1>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀