direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: D4×C2×C12, (C23×C4)⋊7C6, (C23×C12)⋊6C2, C12⋊7(C22×C4), C4⋊1(C22×C12), (C2×C42)⋊10C6, C23⋊5(C2×C12), C42⋊20(C2×C6), (C4×C12)⋊57C22, C2.4(C23×C12), C6.56(C23×C4), C24.36(C2×C6), C22.59(C6×D4), (C2×C6).335C24, C22⋊2(C22×C12), (C22×D4).14C6, C6.179(C22×D4), C22.8(C23×C6), (C2×C12).707C23, (C22×C12)⋊58C22, (C6×D4).330C22, (C23×C6).90C22, C23.32(C22×C6), (C22×C6).252C23, C2.3(D4×C2×C6), (C2×C4×C12)⋊20C2, (C6×C4⋊C4)⋊52C2, (C2×C4⋊C4)⋊25C6, C4⋊C4⋊19(C2×C6), (C2×C4)⋊7(C2×C12), (D4×C2×C6).26C2, C2.2(C6×C4○D4), (C2×C12)⋊32(C2×C4), (C2×C6)⋊5(C22×C4), (C6×C22⋊C4)⋊36C2, C22⋊C4⋊17(C2×C6), (C3×C4⋊C4)⋊76C22, (C2×C22⋊C4)⋊16C6, (C22×C6)⋊13(C2×C4), (C22×C4)⋊18(C2×C6), (C2×D4).76(C2×C6), C6.221(C2×C4○D4), (C2×C6).681(C2×D4), (C2×C4).54(C22×C6), C22.27(C3×C4○D4), (C2×C6).227(C4○D4), (C3×C22⋊C4)⋊71C22, SmallGroup(192,1404)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 578 in 426 conjugacy classes, 274 normal (26 characteristic)
C1, C2 [×3], C2 [×4], C2 [×8], C3, C4 [×8], C4 [×6], C22, C22 [×14], C22 [×24], C6 [×3], C6 [×4], C6 [×8], C2×C4 [×18], C2×C4 [×22], D4 [×16], C23, C23 [×12], C23 [×8], C12 [×8], C12 [×6], C2×C6, C2×C6 [×14], C2×C6 [×24], C42 [×4], C22⋊C4 [×8], C4⋊C4 [×4], C22×C4 [×3], C22×C4 [×10], C22×C4 [×8], C2×D4 [×12], C24 [×2], C2×C12 [×18], C2×C12 [×22], C3×D4 [×16], C22×C6, C22×C6 [×12], C22×C6 [×8], C2×C42, C2×C22⋊C4 [×2], C2×C4⋊C4, C4×D4 [×8], C23×C4 [×2], C22×D4, C4×C12 [×4], C3×C22⋊C4 [×8], C3×C4⋊C4 [×4], C22×C12 [×3], C22×C12 [×10], C22×C12 [×8], C6×D4 [×12], C23×C6 [×2], C2×C4×D4, C2×C4×C12, C6×C22⋊C4 [×2], C6×C4⋊C4, D4×C12 [×8], C23×C12 [×2], D4×C2×C6, D4×C2×C12
Quotients:
C1, C2 [×15], C3, C4 [×8], C22 [×35], C6 [×15], C2×C4 [×28], D4 [×4], C23 [×15], C12 [×8], C2×C6 [×35], C22×C4 [×14], C2×D4 [×6], C4○D4 [×2], C24, C2×C12 [×28], C3×D4 [×4], C22×C6 [×15], C4×D4 [×4], C23×C4, C22×D4, C2×C4○D4, C22×C12 [×14], C6×D4 [×6], C3×C4○D4 [×2], C23×C6, C2×C4×D4, D4×C12 [×4], C23×C12, D4×C2×C6, C6×C4○D4, D4×C2×C12
Generators and relations
G = < a,b,c,d | a2=b12=c4=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
►On 96 points
(1 29)(2 30)(3 31)(4 32)(5 33)(6 34)(7 35)(8 36)(9 25)(10 26)(11 27)(12 28)(13 62)(14 63)(15 64)(16 65)(17 66)(18 67)(19 68)(20 69)(21 70)(22 71)(23 72)(24 61)(37 58)(38 59)(39 60)(40 49)(41 50)(42 51)(43 52)(44 53)(45 54)(46 55)(47 56)(48 57)(73 92)(74 93)(75 94)(76 95)(77 96)(78 85)(79 86)(80 87)(81 88)(82 89)(83 90)(84 91)
(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 82 67 45)(2 83 68 46)(3 84 69 47)(4 73 70 48)(5 74 71 37)(6 75 72 38)(7 76 61 39)(8 77 62 40)(9 78 63 41)(10 79 64 42)(11 80 65 43)(12 81 66 44)(13 49 36 96)(14 50 25 85)(15 51 26 86)(16 52 27 87)(17 53 28 88)(18 54 29 89)(19 55 30 90)(20 56 31 91)(21 57 32 92)(22 58 33 93)(23 59 34 94)(24 60 35 95)
(1 18)(2 19)(3 20)(4 21)(5 22)(6 23)(7 24)(8 13)(9 14)(10 15)(11 16)(12 17)(25 63)(26 64)(27 65)(28 66)(29 67)(30 68)(31 69)(32 70)(33 71)(34 72)(35 61)(36 62)(37 58)(38 59)(39 60)(40 49)(41 50)(42 51)(43 52)(44 53)(45 54)(46 55)(47 56)(48 57)(73 92)(74 93)(75 94)(76 95)(77 96)(78 85)(79 86)(80 87)(81 88)(82 89)(83 90)(84 91)
G:=sub<Sym(96)| (1,29)(2,30)(3,31)(4,32)(5,33)(6,34)(7,35)(8,36)(9,25)(10,26)(11,27)(12,28)(13,62)(14,63)(15,64)(16,65)(17,66)(18,67)(19,68)(20,69)(21,70)(22,71)(23,72)(24,61)(37,58)(38,59)(39,60)(40,49)(41,50)(42,51)(43,52)(44,53)(45,54)(46,55)(47,56)(48,57)(73,92)(74,93)(75,94)(76,95)(77,96)(78,85)(79,86)(80,87)(81,88)(82,89)(83,90)(84,91), (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), (1,82,67,45)(2,83,68,46)(3,84,69,47)(4,73,70,48)(5,74,71,37)(6,75,72,38)(7,76,61,39)(8,77,62,40)(9,78,63,41)(10,79,64,42)(11,80,65,43)(12,81,66,44)(13,49,36,96)(14,50,25,85)(15,51,26,86)(16,52,27,87)(17,53,28,88)(18,54,29,89)(19,55,30,90)(20,56,31,91)(21,57,32,92)(22,58,33,93)(23,59,34,94)(24,60,35,95), (1,18)(2,19)(3,20)(4,21)(5,22)(6,23)(7,24)(8,13)(9,14)(10,15)(11,16)(12,17)(25,63)(26,64)(27,65)(28,66)(29,67)(30,68)(31,69)(32,70)(33,71)(34,72)(35,61)(36,62)(37,58)(38,59)(39,60)(40,49)(41,50)(42,51)(43,52)(44,53)(45,54)(46,55)(47,56)(48,57)(73,92)(74,93)(75,94)(76,95)(77,96)(78,85)(79,86)(80,87)(81,88)(82,89)(83,90)(84,91)>;
G:=Group( (1,29)(2,30)(3,31)(4,32)(5,33)(6,34)(7,35)(8,36)(9,25)(10,26)(11,27)(12,28)(13,62)(14,63)(15,64)(16,65)(17,66)(18,67)(19,68)(20,69)(21,70)(22,71)(23,72)(24,61)(37,58)(38,59)(39,60)(40,49)(41,50)(42,51)(43,52)(44,53)(45,54)(46,55)(47,56)(48,57)(73,92)(74,93)(75,94)(76,95)(77,96)(78,85)(79,86)(80,87)(81,88)(82,89)(83,90)(84,91), (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), (1,82,67,45)(2,83,68,46)(3,84,69,47)(4,73,70,48)(5,74,71,37)(6,75,72,38)(7,76,61,39)(8,77,62,40)(9,78,63,41)(10,79,64,42)(11,80,65,43)(12,81,66,44)(13,49,36,96)(14,50,25,85)(15,51,26,86)(16,52,27,87)(17,53,28,88)(18,54,29,89)(19,55,30,90)(20,56,31,91)(21,57,32,92)(22,58,33,93)(23,59,34,94)(24,60,35,95), (1,18)(2,19)(3,20)(4,21)(5,22)(6,23)(7,24)(8,13)(9,14)(10,15)(11,16)(12,17)(25,63)(26,64)(27,65)(28,66)(29,67)(30,68)(31,69)(32,70)(33,71)(34,72)(35,61)(36,62)(37,58)(38,59)(39,60)(40,49)(41,50)(42,51)(43,52)(44,53)(45,54)(46,55)(47,56)(48,57)(73,92)(74,93)(75,94)(76,95)(77,96)(78,85)(79,86)(80,87)(81,88)(82,89)(83,90)(84,91) );
G=PermutationGroup([(1,29),(2,30),(3,31),(4,32),(5,33),(6,34),(7,35),(8,36),(9,25),(10,26),(11,27),(12,28),(13,62),(14,63),(15,64),(16,65),(17,66),(18,67),(19,68),(20,69),(21,70),(22,71),(23,72),(24,61),(37,58),(38,59),(39,60),(40,49),(41,50),(42,51),(43,52),(44,53),(45,54),(46,55),(47,56),(48,57),(73,92),(74,93),(75,94),(76,95),(77,96),(78,85),(79,86),(80,87),(81,88),(82,89),(83,90),(84,91)], [(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)], [(1,82,67,45),(2,83,68,46),(3,84,69,47),(4,73,70,48),(5,74,71,37),(6,75,72,38),(7,76,61,39),(8,77,62,40),(9,78,63,41),(10,79,64,42),(11,80,65,43),(12,81,66,44),(13,49,36,96),(14,50,25,85),(15,51,26,86),(16,52,27,87),(17,53,28,88),(18,54,29,89),(19,55,30,90),(20,56,31,91),(21,57,32,92),(22,58,33,93),(23,59,34,94),(24,60,35,95)], [(1,18),(2,19),(3,20),(4,21),(5,22),(6,23),(7,24),(8,13),(9,14),(10,15),(11,16),(12,17),(25,63),(26,64),(27,65),(28,66),(29,67),(30,68),(31,69),(32,70),(33,71),(34,72),(35,61),(36,62),(37,58),(38,59),(39,60),(40,49),(41,50),(42,51),(43,52),(44,53),(45,54),(46,55),(47,56),(48,57),(73,92),(74,93),(75,94),(76,95),(77,96),(78,85),(79,86),(80,87),(81,88),(82,89),(83,90),(84,91)])
Matrix representation ►G ⊆ GL4(𝔽13) generated by
| 1 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 |
| 2 | 0 | 0 | 0 |
| 0 | 5 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 9 |
| 12 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 12 |
| 0 | 0 | 1 | 0 |
| 12 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 12 |
G:=sub<GL(4,GF(13))| [1,0,0,0,0,12,0,0,0,0,12,0,0,0,0,12],[2,0,0,0,0,5,0,0,0,0,9,0,0,0,0,9],[12,0,0,0,0,1,0,0,0,0,0,1,0,0,12,0],[12,0,0,0,0,1,0,0,0,0,1,0,0,0,0,12] >;
120 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | ··· | 2O | 3A | 3B | 4A | ··· | 4H | 4I | ··· | 4X | 6A | ··· | 6N | 6O | ··· | 6AD | 12A | ··· | 12P | 12Q | ··· | 12AV |
| order | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 3 | 3 | 4 | ··· | 4 | 4 | ··· | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 12 | ··· | 12 | 12 | ··· | 12 |
| size | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 |
120 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | ||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C3 | C4 | C6 | C6 | C6 | C6 | C6 | C6 | C12 | D4 | C4○D4 | C3×D4 | C3×C4○D4 |
| kernel | D4×C2×C12 | C2×C4×C12 | C6×C22⋊C4 | C6×C4⋊C4 | D4×C12 | C23×C12 | D4×C2×C6 | C2×C4×D4 | C6×D4 | C2×C42 | C2×C22⋊C4 | C2×C4⋊C4 | C4×D4 | C23×C4 | C22×D4 | C2×D4 | C2×C12 | C2×C6 | C2×C4 | C22 |
| # reps | 1 | 1 | 2 | 1 | 8 | 2 | 1 | 2 | 16 | 2 | 4 | 2 | 16 | 4 | 2 | 32 | 4 | 4 | 8 | 8 |
In GAP, Magma, Sage, TeX
D_4\times C_2\times C_{12} % in TeX
G:=Group("D4xC2xC12"); // GroupNames label
G:=SmallGroup(192,1404);
// by ID
G=gap.SmallGroup(192,1404);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-2,672,701,520]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^12=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations