direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C2×C6×SD16, C24⋊14C23, C12.79C24, C8⋊3(C22×C6), C4.19(C6×D4), (C22×C8)⋊14C6, C4.2(C23×C6), Q8⋊2(C22×C6), (C22×C24)⋊24C2, (C2×C24)⋊52C22, (C2×C12).433D4, C12.326(C2×D4), (C22×Q8)⋊16C6, (C3×Q8)⋊11C23, (C6×Q8)⋊53C22, D4.1(C22×C6), C22.66(C6×D4), C23.66(C3×D4), (C22×D4).13C6, (C3×D4).34C23, (C22×C6).222D4, C6.200(C22×D4), (C2×C12).972C23, (C6×D4).327C22, (C22×C12).602C22, (Q8×C2×C6)⋊20C2, C2.24(D4×C2×C6), (C2×C8)⋊14(C2×C6), (D4×C2×C6).25C2, (C2×Q8)⋊15(C2×C6), (C2×C4).89(C3×D4), (C2×D4).73(C2×C6), (C2×C6).687(C2×D4), (C2×C4).142(C22×C6), (C22×C4).136(C2×C6), SmallGroup(192,1459)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 498 in 298 conjugacy classes, 178 normal (18 characteristic)
C1, C2, C2 [×6], C2 [×4], C3, C4, C4 [×3], C4 [×4], C22 [×7], C22 [×16], C6, C6 [×6], C6 [×4], C8 [×4], C2×C4 [×6], C2×C4 [×6], D4 [×4], D4 [×6], Q8 [×4], Q8 [×6], C23, C23 [×10], C12, C12 [×3], C12 [×4], C2×C6 [×7], C2×C6 [×16], C2×C8 [×6], SD16 [×16], C22×C4, C22×C4, C2×D4 [×6], C2×D4 [×3], C2×Q8 [×6], C2×Q8 [×3], C24, C24 [×4], C2×C12 [×6], C2×C12 [×6], C3×D4 [×4], C3×D4 [×6], C3×Q8 [×4], C3×Q8 [×6], C22×C6, C22×C6 [×10], C22×C8, C2×SD16 [×12], C22×D4, C22×Q8, C2×C24 [×6], C3×SD16 [×16], C22×C12, C22×C12, C6×D4 [×6], C6×D4 [×3], C6×Q8 [×6], C6×Q8 [×3], C23×C6, C22×SD16, C22×C24, C6×SD16 [×12], D4×C2×C6, Q8×C2×C6, C2×C6×SD16
Quotients:
C1, C2 [×15], C3, C22 [×35], C6 [×15], D4 [×4], C23 [×15], C2×C6 [×35], SD16 [×4], C2×D4 [×6], C24, C3×D4 [×4], C22×C6 [×15], C2×SD16 [×6], C22×D4, C3×SD16 [×4], C6×D4 [×6], C23×C6, C22×SD16, C6×SD16 [×6], D4×C2×C6, C2×C6×SD16
Generators and relations
G = < a,b,c,d | a2=b6=c8=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c3 >
►On 96 points
(1 58)(2 59)(3 60)(4 61)(5 62)(6 63)(7 64)(8 57)(9 32)(10 25)(11 26)(12 27)(13 28)(14 29)(15 30)(16 31)(17 48)(18 41)(19 42)(20 43)(21 44)(22 45)(23 46)(24 47)(33 86)(34 87)(35 88)(36 81)(37 82)(38 83)(39 84)(40 85)(49 74)(50 75)(51 76)(52 77)(53 78)(54 79)(55 80)(56 73)(65 89)(66 90)(67 91)(68 92)(69 93)(70 94)(71 95)(72 96)
(1 14 82 41 95 54)(2 15 83 42 96 55)(3 16 84 43 89 56)(4 9 85 44 90 49)(5 10 86 45 91 50)(6 11 87 46 92 51)(7 12 88 47 93 52)(8 13 81 48 94 53)(17 70 78 57 28 36)(18 71 79 58 29 37)(19 72 80 59 30 38)(20 65 73 60 31 39)(21 66 74 61 32 40)(22 67 75 62 25 33)(23 68 76 63 26 34)(24 69 77 64 27 35)
(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 41)(2 44)(3 47)(4 42)(5 45)(6 48)(7 43)(8 46)(9 96)(10 91)(11 94)(12 89)(13 92)(14 95)(15 90)(16 93)(17 63)(18 58)(19 61)(20 64)(21 59)(22 62)(23 57)(24 60)(25 67)(26 70)(27 65)(28 68)(29 71)(30 66)(31 69)(32 72)(33 75)(34 78)(35 73)(36 76)(37 79)(38 74)(39 77)(40 80)(49 83)(50 86)(51 81)(52 84)(53 87)(54 82)(55 85)(56 88)
G:=sub<Sym(96)| (1,58)(2,59)(3,60)(4,61)(5,62)(6,63)(7,64)(8,57)(9,32)(10,25)(11,26)(12,27)(13,28)(14,29)(15,30)(16,31)(17,48)(18,41)(19,42)(20,43)(21,44)(22,45)(23,46)(24,47)(33,86)(34,87)(35,88)(36,81)(37,82)(38,83)(39,84)(40,85)(49,74)(50,75)(51,76)(52,77)(53,78)(54,79)(55,80)(56,73)(65,89)(66,90)(67,91)(68,92)(69,93)(70,94)(71,95)(72,96), (1,14,82,41,95,54)(2,15,83,42,96,55)(3,16,84,43,89,56)(4,9,85,44,90,49)(5,10,86,45,91,50)(6,11,87,46,92,51)(7,12,88,47,93,52)(8,13,81,48,94,53)(17,70,78,57,28,36)(18,71,79,58,29,37)(19,72,80,59,30,38)(20,65,73,60,31,39)(21,66,74,61,32,40)(22,67,75,62,25,33)(23,68,76,63,26,34)(24,69,77,64,27,35), (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), (1,41)(2,44)(3,47)(4,42)(5,45)(6,48)(7,43)(8,46)(9,96)(10,91)(11,94)(12,89)(13,92)(14,95)(15,90)(16,93)(17,63)(18,58)(19,61)(20,64)(21,59)(22,62)(23,57)(24,60)(25,67)(26,70)(27,65)(28,68)(29,71)(30,66)(31,69)(32,72)(33,75)(34,78)(35,73)(36,76)(37,79)(38,74)(39,77)(40,80)(49,83)(50,86)(51,81)(52,84)(53,87)(54,82)(55,85)(56,88)>;
G:=Group( (1,58)(2,59)(3,60)(4,61)(5,62)(6,63)(7,64)(8,57)(9,32)(10,25)(11,26)(12,27)(13,28)(14,29)(15,30)(16,31)(17,48)(18,41)(19,42)(20,43)(21,44)(22,45)(23,46)(24,47)(33,86)(34,87)(35,88)(36,81)(37,82)(38,83)(39,84)(40,85)(49,74)(50,75)(51,76)(52,77)(53,78)(54,79)(55,80)(56,73)(65,89)(66,90)(67,91)(68,92)(69,93)(70,94)(71,95)(72,96), (1,14,82,41,95,54)(2,15,83,42,96,55)(3,16,84,43,89,56)(4,9,85,44,90,49)(5,10,86,45,91,50)(6,11,87,46,92,51)(7,12,88,47,93,52)(8,13,81,48,94,53)(17,70,78,57,28,36)(18,71,79,58,29,37)(19,72,80,59,30,38)(20,65,73,60,31,39)(21,66,74,61,32,40)(22,67,75,62,25,33)(23,68,76,63,26,34)(24,69,77,64,27,35), (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), (1,41)(2,44)(3,47)(4,42)(5,45)(6,48)(7,43)(8,46)(9,96)(10,91)(11,94)(12,89)(13,92)(14,95)(15,90)(16,93)(17,63)(18,58)(19,61)(20,64)(21,59)(22,62)(23,57)(24,60)(25,67)(26,70)(27,65)(28,68)(29,71)(30,66)(31,69)(32,72)(33,75)(34,78)(35,73)(36,76)(37,79)(38,74)(39,77)(40,80)(49,83)(50,86)(51,81)(52,84)(53,87)(54,82)(55,85)(56,88) );
G=PermutationGroup([(1,58),(2,59),(3,60),(4,61),(5,62),(6,63),(7,64),(8,57),(9,32),(10,25),(11,26),(12,27),(13,28),(14,29),(15,30),(16,31),(17,48),(18,41),(19,42),(20,43),(21,44),(22,45),(23,46),(24,47),(33,86),(34,87),(35,88),(36,81),(37,82),(38,83),(39,84),(40,85),(49,74),(50,75),(51,76),(52,77),(53,78),(54,79),(55,80),(56,73),(65,89),(66,90),(67,91),(68,92),(69,93),(70,94),(71,95),(72,96)], [(1,14,82,41,95,54),(2,15,83,42,96,55),(3,16,84,43,89,56),(4,9,85,44,90,49),(5,10,86,45,91,50),(6,11,87,46,92,51),(7,12,88,47,93,52),(8,13,81,48,94,53),(17,70,78,57,28,36),(18,71,79,58,29,37),(19,72,80,59,30,38),(20,65,73,60,31,39),(21,66,74,61,32,40),(22,67,75,62,25,33),(23,68,76,63,26,34),(24,69,77,64,27,35)], [(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)], [(1,41),(2,44),(3,47),(4,42),(5,45),(6,48),(7,43),(8,46),(9,96),(10,91),(11,94),(12,89),(13,92),(14,95),(15,90),(16,93),(17,63),(18,58),(19,61),(20,64),(21,59),(22,62),(23,57),(24,60),(25,67),(26,70),(27,65),(28,68),(29,71),(30,66),(31,69),(32,72),(33,75),(34,78),(35,73),(36,76),(37,79),(38,74),(39,77),(40,80),(49,83),(50,86),(51,81),(52,84),(53,87),(54,82),(55,85),(56,88)])
Matrix representation ►G ⊆ GL4(𝔽73) generated by
| 72 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 72 |
| 72 | 0 | 0 | 0 |
| 0 | 64 | 0 | 0 |
| 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 72 |
| 72 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 61 |
| 0 | 0 | 6 | 61 |
| 1 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 72 | 0 |
| 0 | 0 | 72 | 1 |
G:=sub<GL(4,GF(73))| [72,0,0,0,0,72,0,0,0,0,72,0,0,0,0,72],[72,0,0,0,0,64,0,0,0,0,72,0,0,0,0,72],[72,0,0,0,0,72,0,0,0,0,0,6,0,0,61,61],[1,0,0,0,0,72,0,0,0,0,72,72,0,0,0,1] >;
84 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | ··· | 6N | 6O | ··· | 6V | 8A | ··· | 8H | 12A | ··· | 12H | 12I | ··· | 12P | 24A | ··· | 24P |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 8 | ··· | 8 | 12 | ··· | 12 | 12 | ··· | 12 | 24 | ··· | 24 |
| size | 1 | 1 | ··· | 1 | 4 | 4 | 4 | 4 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 4 | ··· | 4 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 |
84 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | |||||||||
| image | C1 | C2 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C6 | D4 | D4 | SD16 | C3×D4 | C3×D4 | C3×SD16 |
| kernel | C2×C6×SD16 | C22×C24 | C6×SD16 | D4×C2×C6 | Q8×C2×C6 | C22×SD16 | C22×C8 | C2×SD16 | C22×D4 | C22×Q8 | C2×C12 | C22×C6 | C2×C6 | C2×C4 | C23 | C22 |
| # reps | 1 | 1 | 12 | 1 | 1 | 2 | 2 | 24 | 2 | 2 | 3 | 1 | 8 | 6 | 2 | 16 |
In GAP, Magma, Sage, TeX
C_2\times C_6\times SD_{16} % in TeX
G:=Group("C2xC6xSD16"); // GroupNames label
G:=SmallGroup(192,1459);
// by ID
G=gap.SmallGroup(192,1459);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-2,672,701,6053,3036,124]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^6=c^8=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^3>;
// generators/relations