direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×D12.C4, M4(2)⋊25D6, C12.69C24, C24.51C23, C6⋊2(C8○D4), C4○D12.4C4, (C2×C8).279D6, C3⋊C8.35C23, (S3×C8)⋊22C22, D12.28(C2×C4), (C2×D12).16C4, C23.35(C4×S3), C4.68(S3×C23), C8.43(C22×S3), C6.32(C23×C4), C8⋊S3⋊18C22, (C6×M4(2))⋊14C2, (C2×M4(2))⋊17S3, (C4×S3).35C23, C12.92(C22×C4), Dic6.29(C2×C4), (C2×Dic6).16C4, D6.13(C22×C4), (C22×C4).390D6, (C2×C12).882C23, (C2×C24).281C22, C4○D12.49C22, (C3×M4(2))⋊30C22, Dic3.13(C22×C4), (C22×C12).264C22, C3⋊2(C2×C8○D4), (S3×C2×C8)⋊29C2, C4.123(S3×C2×C4), C22.8(S3×C2×C4), (C2×C3⋊C8)⋊33C22, (C22×C3⋊C8)⋊10C2, (C2×C4).87(C4×S3), C3⋊D4.3(C2×C4), (C2×C8⋊S3)⋊27C2, C2.33(S3×C22×C4), (C4×S3).24(C2×C4), (C2×C3⋊D4).14C4, (C2×C12).131(C2×C4), (C2×C4○D12).21C2, (S3×C2×C4).303C22, (C22×C6).78(C2×C4), (C2×C6).25(C22×C4), (C22×S3).46(C2×C4), (C2×C4).825(C22×S3), (C2×Dic3).72(C2×C4), SmallGroup(192,1303)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 504 in 266 conjugacy classes, 151 normal (27 characteristic)
C1, C2, C2 [×2], C2 [×6], C3, C4 [×2], C4 [×2], C4 [×4], C22, C22 [×2], C22 [×10], S3 [×4], C6, C6 [×2], C6 [×2], C8 [×4], C8 [×4], C2×C4 [×2], C2×C4 [×4], C2×C4 [×10], D4 [×12], Q8 [×4], C23, C23 [×2], Dic3 [×4], C12 [×2], C12 [×2], D6 [×4], D6 [×4], C2×C6, C2×C6 [×2], C2×C6 [×2], C2×C8 [×2], C2×C8 [×14], M4(2) [×4], M4(2) [×8], C22×C4, C22×C4 [×2], C2×D4 [×3], C2×Q8, C4○D4 [×8], C3⋊C8 [×4], C24 [×4], Dic6 [×4], C4×S3 [×8], D12 [×4], C2×Dic3 [×2], C3⋊D4 [×8], C2×C12 [×2], C2×C12 [×4], C22×S3 [×2], C22×C6, C22×C8 [×3], C2×M4(2), C2×M4(2) [×2], C8○D4 [×8], C2×C4○D4, S3×C8 [×8], C8⋊S3 [×8], C2×C3⋊C8 [×2], C2×C3⋊C8 [×4], C2×C24 [×2], C3×M4(2) [×4], C2×Dic6, S3×C2×C4 [×2], C2×D12, C4○D12 [×8], C2×C3⋊D4 [×2], C22×C12, C2×C8○D4, S3×C2×C8 [×2], C2×C8⋊S3 [×2], D12.C4 [×8], C22×C3⋊C8, C6×M4(2), C2×C4○D12, C2×D12.C4
Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], S3, C2×C4 [×28], C23 [×15], D6 [×7], C22×C4 [×14], C24, C4×S3 [×4], C22×S3 [×7], C8○D4 [×2], C23×C4, S3×C2×C4 [×6], S3×C23, C2×C8○D4, D12.C4 [×2], S3×C22×C4, C2×D12.C4
Generators and relations
G = < a,b,c,d | a2=b12=c2=1, d4=b6, ab=ba, ac=ca, ad=da, cbc=b-1, dbd-1=b7, dcd-1=b6c >
►On 96 points
(1 17)(2 18)(3 19)(4 20)(5 21)(6 22)(7 23)(8 24)(9 13)(10 14)(11 15)(12 16)(25 56)(26 57)(27 58)(28 59)(29 60)(30 49)(31 50)(32 51)(33 52)(34 53)(35 54)(36 55)(37 77)(38 78)(39 79)(40 80)(41 81)(42 82)(43 83)(44 84)(45 73)(46 74)(47 75)(48 76)(61 94)(62 95)(63 96)(64 85)(65 86)(66 87)(67 88)(68 89)(69 90)(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 13)(2 24)(3 23)(4 22)(5 21)(6 20)(7 19)(8 18)(9 17)(10 16)(11 15)(12 14)(25 52)(26 51)(27 50)(28 49)(29 60)(30 59)(31 58)(32 57)(33 56)(34 55)(35 54)(36 53)(37 73)(38 84)(39 83)(40 82)(41 81)(42 80)(43 79)(44 78)(45 77)(46 76)(47 75)(48 74)(61 92)(62 91)(63 90)(64 89)(65 88)(66 87)(67 86)(68 85)(69 96)(70 95)(71 94)(72 93)
(1 65 43 53 7 71 37 59)(2 72 44 60 8 66 38 54)(3 67 45 55 9 61 39 49)(4 62 46 50 10 68 40 56)(5 69 47 57 11 63 41 51)(6 64 48 52 12 70 42 58)(13 94 79 30 19 88 73 36)(14 89 80 25 20 95 74 31)(15 96 81 32 21 90 75 26)(16 91 82 27 22 85 76 33)(17 86 83 34 23 92 77 28)(18 93 84 29 24 87 78 35)
G:=sub<Sym(96)| (1,17)(2,18)(3,19)(4,20)(5,21)(6,22)(7,23)(8,24)(9,13)(10,14)(11,15)(12,16)(25,56)(26,57)(27,58)(28,59)(29,60)(30,49)(31,50)(32,51)(33,52)(34,53)(35,54)(36,55)(37,77)(38,78)(39,79)(40,80)(41,81)(42,82)(43,83)(44,84)(45,73)(46,74)(47,75)(48,76)(61,94)(62,95)(63,96)(64,85)(65,86)(66,87)(67,88)(68,89)(69,90)(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), (1,13)(2,24)(3,23)(4,22)(5,21)(6,20)(7,19)(8,18)(9,17)(10,16)(11,15)(12,14)(25,52)(26,51)(27,50)(28,49)(29,60)(30,59)(31,58)(32,57)(33,56)(34,55)(35,54)(36,53)(37,73)(38,84)(39,83)(40,82)(41,81)(42,80)(43,79)(44,78)(45,77)(46,76)(47,75)(48,74)(61,92)(62,91)(63,90)(64,89)(65,88)(66,87)(67,86)(68,85)(69,96)(70,95)(71,94)(72,93), (1,65,43,53,7,71,37,59)(2,72,44,60,8,66,38,54)(3,67,45,55,9,61,39,49)(4,62,46,50,10,68,40,56)(5,69,47,57,11,63,41,51)(6,64,48,52,12,70,42,58)(13,94,79,30,19,88,73,36)(14,89,80,25,20,95,74,31)(15,96,81,32,21,90,75,26)(16,91,82,27,22,85,76,33)(17,86,83,34,23,92,77,28)(18,93,84,29,24,87,78,35)>;
G:=Group( (1,17)(2,18)(3,19)(4,20)(5,21)(6,22)(7,23)(8,24)(9,13)(10,14)(11,15)(12,16)(25,56)(26,57)(27,58)(28,59)(29,60)(30,49)(31,50)(32,51)(33,52)(34,53)(35,54)(36,55)(37,77)(38,78)(39,79)(40,80)(41,81)(42,82)(43,83)(44,84)(45,73)(46,74)(47,75)(48,76)(61,94)(62,95)(63,96)(64,85)(65,86)(66,87)(67,88)(68,89)(69,90)(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), (1,13)(2,24)(3,23)(4,22)(5,21)(6,20)(7,19)(8,18)(9,17)(10,16)(11,15)(12,14)(25,52)(26,51)(27,50)(28,49)(29,60)(30,59)(31,58)(32,57)(33,56)(34,55)(35,54)(36,53)(37,73)(38,84)(39,83)(40,82)(41,81)(42,80)(43,79)(44,78)(45,77)(46,76)(47,75)(48,74)(61,92)(62,91)(63,90)(64,89)(65,88)(66,87)(67,86)(68,85)(69,96)(70,95)(71,94)(72,93), (1,65,43,53,7,71,37,59)(2,72,44,60,8,66,38,54)(3,67,45,55,9,61,39,49)(4,62,46,50,10,68,40,56)(5,69,47,57,11,63,41,51)(6,64,48,52,12,70,42,58)(13,94,79,30,19,88,73,36)(14,89,80,25,20,95,74,31)(15,96,81,32,21,90,75,26)(16,91,82,27,22,85,76,33)(17,86,83,34,23,92,77,28)(18,93,84,29,24,87,78,35) );
G=PermutationGroup([(1,17),(2,18),(3,19),(4,20),(5,21),(6,22),(7,23),(8,24),(9,13),(10,14),(11,15),(12,16),(25,56),(26,57),(27,58),(28,59),(29,60),(30,49),(31,50),(32,51),(33,52),(34,53),(35,54),(36,55),(37,77),(38,78),(39,79),(40,80),(41,81),(42,82),(43,83),(44,84),(45,73),(46,74),(47,75),(48,76),(61,94),(62,95),(63,96),(64,85),(65,86),(66,87),(67,88),(68,89),(69,90),(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)], [(1,13),(2,24),(3,23),(4,22),(5,21),(6,20),(7,19),(8,18),(9,17),(10,16),(11,15),(12,14),(25,52),(26,51),(27,50),(28,49),(29,60),(30,59),(31,58),(32,57),(33,56),(34,55),(35,54),(36,53),(37,73),(38,84),(39,83),(40,82),(41,81),(42,80),(43,79),(44,78),(45,77),(46,76),(47,75),(48,74),(61,92),(62,91),(63,90),(64,89),(65,88),(66,87),(67,86),(68,85),(69,96),(70,95),(71,94),(72,93)], [(1,65,43,53,7,71,37,59),(2,72,44,60,8,66,38,54),(3,67,45,55,9,61,39,49),(4,62,46,50,10,68,40,56),(5,69,47,57,11,63,41,51),(6,64,48,52,12,70,42,58),(13,94,79,30,19,88,73,36),(14,89,80,25,20,95,74,31),(15,96,81,32,21,90,75,26),(16,91,82,27,22,85,76,33),(17,86,83,34,23,92,77,28),(18,93,84,29,24,87,78,35)])
Matrix representation ►G ⊆ GL4(𝔽73) generated by
| 72 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 72 |
| 46 | 55 | 0 | 0 |
| 0 | 27 | 0 | 0 |
| 0 | 0 | 1 | 72 |
| 0 | 0 | 1 | 0 |
| 72 | 0 | 0 | 0 |
| 3 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 72 |
| 51 | 34 | 0 | 0 |
| 66 | 22 | 0 | 0 |
| 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 72 |
G:=sub<GL(4,GF(73))| [72,0,0,0,0,72,0,0,0,0,72,0,0,0,0,72],[46,0,0,0,55,27,0,0,0,0,1,1,0,0,72,0],[72,3,0,0,0,1,0,0,0,0,1,1,0,0,0,72],[51,66,0,0,34,22,0,0,0,0,72,0,0,0,0,72] >;
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 6A | 6B | 6C | 6D | 6E | 8A | ··· | 8H | 8I | ··· | 8P | 8Q | 8R | 8S | 8T | 12A | 12B | 12C | 12D | 12E | 12F | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 8 | ··· | 8 | 8 | ··· | 8 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 6 | 6 | 6 | 6 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 6 | 6 | 6 | 6 | 2 | 2 | 2 | 4 | 4 | 2 | ··· | 2 | 3 | ··· | 3 | 6 | 6 | 6 | 6 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | ··· | 4 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | ||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | S3 | D6 | D6 | D6 | C4×S3 | C4×S3 | C8○D4 | D12.C4 |
| kernel | C2×D12.C4 | S3×C2×C8 | C2×C8⋊S3 | D12.C4 | C22×C3⋊C8 | C6×M4(2) | C2×C4○D12 | C2×Dic6 | C2×D12 | C4○D12 | C2×C3⋊D4 | C2×M4(2) | C2×C8 | M4(2) | C22×C4 | C2×C4 | C23 | C6 | C2 |
| # reps | 1 | 2 | 2 | 8 | 1 | 1 | 1 | 2 | 2 | 8 | 4 | 1 | 2 | 4 | 1 | 6 | 2 | 8 | 4 |
In GAP, Magma, Sage, TeX
C_2\times D_{12}.C_4 % in TeX
G:=Group("C2xD12.C4"); // GroupNames label
G:=SmallGroup(192,1303);
// by ID
G=gap.SmallGroup(192,1303);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,758,297,80,102,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^12=c^2=1,d^4=b^6,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,d*b*d^-1=b^7,d*c*d^-1=b^6*c>;
// generators/relations