direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C6×C8○D4, C24.80C23, C12.93C24, (C22×C8)⋊17C6, C4○D4.7C12, D4.8(C2×C12), (C6×D4).24C4, (C6×Q8).20C4, (C2×C24)⋊54C22, (C22×C24)⋊27C2, (C2×D4).12C12, C8.17(C22×C6), C4.17(C23×C6), C6.63(C23×C4), (C2×Q8).13C12, Q8.14(C2×C12), (C2×M4(2))⋊17C6, (C6×M4(2))⋊35C2, M4(2)⋊11(C2×C6), C2.11(C23×C12), C23.25(C2×C12), C4.22(C22×C12), (C2×C12).969C23, C12.167(C22×C4), C22.4(C22×C12), (C3×M4(2))⋊40C22, (C22×C12).600C22, (C2×C8)⋊16(C2×C6), (C3×C4○D4).8C4, (C2×C4).53(C2×C12), C4○D4.21(C2×C6), (C2×C4○D4).20C6, (C6×C4○D4).28C2, (C3×D4).30(C2×C4), (C3×Q8).33(C2×C4), (C2×C12).274(C2×C4), (C2×C6).35(C22×C4), (C22×C6).86(C2×C4), (C2×C4).139(C22×C6), (C22×C4).134(C2×C6), (C3×C4○D4).59C22, SmallGroup(192,1456)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 290 in 266 conjugacy classes, 242 normal (20 characteristic)
C1, C2, C2 [×2], C2 [×6], C3, C4 [×2], C4 [×6], C22, C22 [×6], C22 [×6], C6, C6 [×2], C6 [×6], C8 [×8], C2×C4, C2×C4 [×15], D4 [×12], Q8 [×4], C23 [×3], C12 [×2], C12 [×6], C2×C6, C2×C6 [×6], C2×C6 [×6], C2×C8, C2×C8 [×15], M4(2) [×12], C22×C4 [×3], C2×D4 [×3], C2×Q8, C4○D4 [×8], C24 [×8], C2×C12, C2×C12 [×15], C3×D4 [×12], C3×Q8 [×4], C22×C6 [×3], C22×C8 [×3], C2×M4(2) [×3], C8○D4 [×8], C2×C4○D4, C2×C24, C2×C24 [×15], C3×M4(2) [×12], C22×C12 [×3], C6×D4 [×3], C6×Q8, C3×C4○D4 [×8], C2×C8○D4, C22×C24 [×3], C6×M4(2) [×3], C3×C8○D4 [×8], C6×C4○D4, C6×C8○D4
Quotients:
C1, C2 [×15], C3, C4 [×8], C22 [×35], C6 [×15], C2×C4 [×28], C23 [×15], C12 [×8], C2×C6 [×35], C22×C4 [×14], C24, C2×C12 [×28], C22×C6 [×15], C8○D4 [×2], C23×C4, C22×C12 [×14], C23×C6, C2×C8○D4, C3×C8○D4 [×2], C23×C12, C6×C8○D4
Generators and relations
G = < a,b,c,d | a6=b8=d2=1, c2=b4, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b4c >
►On 96 points
(1 75 87 41 23 50)(2 76 88 42 24 51)(3 77 81 43 17 52)(4 78 82 44 18 53)(5 79 83 45 19 54)(6 80 84 46 20 55)(7 73 85 47 21 56)(8 74 86 48 22 49)(9 26 94 38 62 65)(10 27 95 39 63 66)(11 28 96 40 64 67)(12 29 89 33 57 68)(13 30 90 34 58 69)(14 31 91 35 59 70)(15 32 92 36 60 71)(16 25 93 37 61 72)
(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 91 5 95)(2 92 6 96)(3 93 7 89)(4 94 8 90)(9 22 13 18)(10 23 14 19)(11 24 15 20)(12 17 16 21)(25 56 29 52)(26 49 30 53)(27 50 31 54)(28 51 32 55)(33 77 37 73)(34 78 38 74)(35 79 39 75)(36 80 40 76)(41 70 45 66)(42 71 46 67)(43 72 47 68)(44 65 48 69)(57 81 61 85)(58 82 62 86)(59 83 63 87)(60 84 64 88)
(1 70)(2 71)(3 72)(4 65)(5 66)(6 67)(7 68)(8 69)(9 78)(10 79)(11 80)(12 73)(13 74)(14 75)(15 76)(16 77)(17 37)(18 38)(19 39)(20 40)(21 33)(22 34)(23 35)(24 36)(25 81)(26 82)(27 83)(28 84)(29 85)(30 86)(31 87)(32 88)(41 91)(42 92)(43 93)(44 94)(45 95)(46 96)(47 89)(48 90)(49 58)(50 59)(51 60)(52 61)(53 62)(54 63)(55 64)(56 57)
G:=sub<Sym(96)| (1,75,87,41,23,50)(2,76,88,42,24,51)(3,77,81,43,17,52)(4,78,82,44,18,53)(5,79,83,45,19,54)(6,80,84,46,20,55)(7,73,85,47,21,56)(8,74,86,48,22,49)(9,26,94,38,62,65)(10,27,95,39,63,66)(11,28,96,40,64,67)(12,29,89,33,57,68)(13,30,90,34,58,69)(14,31,91,35,59,70)(15,32,92,36,60,71)(16,25,93,37,61,72), (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), (1,91,5,95)(2,92,6,96)(3,93,7,89)(4,94,8,90)(9,22,13,18)(10,23,14,19)(11,24,15,20)(12,17,16,21)(25,56,29,52)(26,49,30,53)(27,50,31,54)(28,51,32,55)(33,77,37,73)(34,78,38,74)(35,79,39,75)(36,80,40,76)(41,70,45,66)(42,71,46,67)(43,72,47,68)(44,65,48,69)(57,81,61,85)(58,82,62,86)(59,83,63,87)(60,84,64,88), (1,70)(2,71)(3,72)(4,65)(5,66)(6,67)(7,68)(8,69)(9,78)(10,79)(11,80)(12,73)(13,74)(14,75)(15,76)(16,77)(17,37)(18,38)(19,39)(20,40)(21,33)(22,34)(23,35)(24,36)(25,81)(26,82)(27,83)(28,84)(29,85)(30,86)(31,87)(32,88)(41,91)(42,92)(43,93)(44,94)(45,95)(46,96)(47,89)(48,90)(49,58)(50,59)(51,60)(52,61)(53,62)(54,63)(55,64)(56,57)>;
G:=Group( (1,75,87,41,23,50)(2,76,88,42,24,51)(3,77,81,43,17,52)(4,78,82,44,18,53)(5,79,83,45,19,54)(6,80,84,46,20,55)(7,73,85,47,21,56)(8,74,86,48,22,49)(9,26,94,38,62,65)(10,27,95,39,63,66)(11,28,96,40,64,67)(12,29,89,33,57,68)(13,30,90,34,58,69)(14,31,91,35,59,70)(15,32,92,36,60,71)(16,25,93,37,61,72), (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), (1,91,5,95)(2,92,6,96)(3,93,7,89)(4,94,8,90)(9,22,13,18)(10,23,14,19)(11,24,15,20)(12,17,16,21)(25,56,29,52)(26,49,30,53)(27,50,31,54)(28,51,32,55)(33,77,37,73)(34,78,38,74)(35,79,39,75)(36,80,40,76)(41,70,45,66)(42,71,46,67)(43,72,47,68)(44,65,48,69)(57,81,61,85)(58,82,62,86)(59,83,63,87)(60,84,64,88), (1,70)(2,71)(3,72)(4,65)(5,66)(6,67)(7,68)(8,69)(9,78)(10,79)(11,80)(12,73)(13,74)(14,75)(15,76)(16,77)(17,37)(18,38)(19,39)(20,40)(21,33)(22,34)(23,35)(24,36)(25,81)(26,82)(27,83)(28,84)(29,85)(30,86)(31,87)(32,88)(41,91)(42,92)(43,93)(44,94)(45,95)(46,96)(47,89)(48,90)(49,58)(50,59)(51,60)(52,61)(53,62)(54,63)(55,64)(56,57) );
G=PermutationGroup([(1,75,87,41,23,50),(2,76,88,42,24,51),(3,77,81,43,17,52),(4,78,82,44,18,53),(5,79,83,45,19,54),(6,80,84,46,20,55),(7,73,85,47,21,56),(8,74,86,48,22,49),(9,26,94,38,62,65),(10,27,95,39,63,66),(11,28,96,40,64,67),(12,29,89,33,57,68),(13,30,90,34,58,69),(14,31,91,35,59,70),(15,32,92,36,60,71),(16,25,93,37,61,72)], [(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)], [(1,91,5,95),(2,92,6,96),(3,93,7,89),(4,94,8,90),(9,22,13,18),(10,23,14,19),(11,24,15,20),(12,17,16,21),(25,56,29,52),(26,49,30,53),(27,50,31,54),(28,51,32,55),(33,77,37,73),(34,78,38,74),(35,79,39,75),(36,80,40,76),(41,70,45,66),(42,71,46,67),(43,72,47,68),(44,65,48,69),(57,81,61,85),(58,82,62,86),(59,83,63,87),(60,84,64,88)], [(1,70),(2,71),(3,72),(4,65),(5,66),(6,67),(7,68),(8,69),(9,78),(10,79),(11,80),(12,73),(13,74),(14,75),(15,76),(16,77),(17,37),(18,38),(19,39),(20,40),(21,33),(22,34),(23,35),(24,36),(25,81),(26,82),(27,83),(28,84),(29,85),(30,86),(31,87),(32,88),(41,91),(42,92),(43,93),(44,94),(45,95),(46,96),(47,89),(48,90),(49,58),(50,59),(51,60),(52,61),(53,62),(54,63),(55,64),(56,57)])
Matrix representation ►G ⊆ GL3(𝔽73) generated by
| 72 | 0 | 0 |
| 0 | 8 | 0 |
| 0 | 0 | 8 |
| 27 | 0 | 0 |
| 0 | 22 | 0 |
| 0 | 0 | 22 |
| 1 | 0 | 0 |
| 0 | 46 | 0 |
| 0 | 2 | 27 |
| 1 | 0 | 0 |
| 0 | 46 | 1 |
| 0 | 2 | 27 |
G:=sub<GL(3,GF(73))| [72,0,0,0,8,0,0,0,8],[27,0,0,0,22,0,0,0,22],[1,0,0,0,46,2,0,0,27],[1,0,0,0,46,2,0,1,27] >;
120 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | ··· | 2I | 3A | 3B | 4A | 4B | 4C | 4D | 4E | ··· | 4J | 6A | ··· | 6F | 6G | ··· | 6R | 8A | ··· | 8H | 8I | ··· | 8T | 12A | ··· | 12H | 12I | ··· | 12T | 24A | ··· | 24P | 24Q | ··· | 24AN |
| order | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 8 | ··· | 8 | 8 | ··· | 8 | 12 | ··· | 12 | 12 | ··· | 12 | 24 | ··· | 24 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 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 |
| type | + | + | + | + | + | |||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C3 | C4 | C4 | C4 | C6 | C6 | C6 | C6 | C12 | C12 | C12 | C8○D4 | C3×C8○D4 |
| kernel | C6×C8○D4 | C22×C24 | C6×M4(2) | C3×C8○D4 | C6×C4○D4 | C2×C8○D4 | C6×D4 | C6×Q8 | C3×C4○D4 | C22×C8 | C2×M4(2) | C8○D4 | C2×C4○D4 | C2×D4 | C2×Q8 | C4○D4 | C6 | C2 |
| # reps | 1 | 3 | 3 | 8 | 1 | 2 | 6 | 2 | 8 | 6 | 6 | 16 | 2 | 12 | 4 | 16 | 8 | 16 |
In GAP, Magma, Sage, TeX
C_6\times C_8\circ D_4
% in TeX
G:=Group("C6xC8oD4"); // GroupNames label
G:=SmallGroup(192,1456);
// by ID
G=gap.SmallGroup(192,1456);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-2,336,1059,124]);
// Polycyclic
G:=Group<a,b,c,d|a^6=b^8=d^2=1,c^2=b^4,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=b^4*c>;
// generators/relations