direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C3×C8○2M4(2), M4(2)⋊5C12, C12.31C42, (C4×C8)⋊14C6, (C2×C8)⋊9C12, (C2×C24)⋊19C4, (C4×C24)⋊30C2, C4.5(C4×C12), C8⋊C4⋊13C6, C4⋊C4.10C12, C24.88(C2×C4), C8.23(C2×C12), C6.44(C8○D4), C22⋊C4.6C12, C6.35(C2×C42), C42.59(C2×C6), (C22×C8).17C6, C22.5(C4×C12), (C2×C6).12C42, (C3×M4(2))⋊11C4, C23.21(C2×C12), (C22×C24).33C2, C4.34(C22×C12), (C2×C24).453C22, C42⋊C2.14C6, C12.192(C22×C4), (C2×C12).980C23, (C4×C12).300C22, (C2×M4(2)).17C6, (C6×M4(2)).36C2, C22.18(C22×C12), (C22×C12).579C22, C2.7(C2×C4×C12), C2.1(C3×C8○D4), (C3×C4⋊C4).22C4, (C3×C8⋊C4)⋊27C2, (C2×C8).100(C2×C6), (C2×C4).35(C2×C12), (C2×C12).332(C2×C4), (C3×C22⋊C4).13C4, (C22×C6).83(C2×C4), (C2×C6).230(C22×C4), (C2×C4).148(C22×C6), (C22×C4).116(C2×C6), (C3×C42⋊C2).28C2, SmallGroup(192,838)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C8○2M4(2)
G = < a,b,c,d | a3=b8=d2=1, c4=b4, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b4c >
Subgroups: 146 in 130 conjugacy classes, 114 normal (30 characteristic)
C1, C2, C2, C2, C3, C4, C4, C4, C22, C22, C22, C6, C6, C6, C8, C2×C4, C2×C4, C23, C12, C12, C12, C2×C6, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), C22×C4, C24, C2×C12, C2×C12, C22×C6, C4×C8, C8⋊C4, C42⋊C2, C22×C8, C2×M4(2), C4×C12, C3×C22⋊C4, C3×C4⋊C4, C2×C24, C2×C24, C3×M4(2), C22×C12, C8○2M4(2), C4×C24, C3×C8⋊C4, C3×C42⋊C2, C22×C24, C6×M4(2), C3×C8○2M4(2)
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, C23, C12, C2×C6, C42, C22×C4, C2×C12, C22×C6, C2×C42, C8○D4, C4×C12, C22×C12, C8○2M4(2), C2×C4×C12, C3×C8○D4, C3×C8○2M4(2)
►On 96 points
(1 20 61)(2 21 62)(3 22 63)(4 23 64)(5 24 57)(6 17 58)(7 18 59)(8 19 60)(9 29 70)(10 30 71)(11 31 72)(12 32 65)(13 25 66)(14 26 67)(15 27 68)(16 28 69)(33 74 82)(34 75 83)(35 76 84)(36 77 85)(37 78 86)(38 79 87)(39 80 88)(40 73 81)(41 55 90)(42 56 91)(43 49 92)(44 50 93)(45 51 94)(46 52 95)(47 53 96)(48 54 89)
(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 73 66 5 45 77 70)(2 42 74 67 6 46 78 71)(3 43 75 68 7 47 79 72)(4 44 76 69 8 48 80 65)(9 20 55 81 13 24 51 85)(10 21 56 82 14 17 52 86)(11 22 49 83 15 18 53 87)(12 23 50 84 16 19 54 88)(25 57 94 36 29 61 90 40)(26 58 95 37 30 62 91 33)(27 59 96 38 31 63 92 34)(28 60 89 39 32 64 93 35)
(1 75)(2 76)(3 77)(4 78)(5 79)(6 80)(7 73)(8 74)(9 53)(10 54)(11 55)(12 56)(13 49)(14 50)(15 51)(16 52)(17 88)(18 81)(19 82)(20 83)(21 84)(22 85)(23 86)(24 87)(25 92)(26 93)(27 94)(28 95)(29 96)(30 89)(31 90)(32 91)(33 60)(34 61)(35 62)(36 63)(37 64)(38 57)(39 58)(40 59)(41 72)(42 65)(43 66)(44 67)(45 68)(46 69)(47 70)(48 71)
G:=sub<Sym(96)| (1,20,61)(2,21,62)(3,22,63)(4,23,64)(5,24,57)(6,17,58)(7,18,59)(8,19,60)(9,29,70)(10,30,71)(11,31,72)(12,32,65)(13,25,66)(14,26,67)(15,27,68)(16,28,69)(33,74,82)(34,75,83)(35,76,84)(36,77,85)(37,78,86)(38,79,87)(39,80,88)(40,73,81)(41,55,90)(42,56,91)(43,49,92)(44,50,93)(45,51,94)(46,52,95)(47,53,96)(48,54,89), (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), (1,41,73,66,5,45,77,70)(2,42,74,67,6,46,78,71)(3,43,75,68,7,47,79,72)(4,44,76,69,8,48,80,65)(9,20,55,81,13,24,51,85)(10,21,56,82,14,17,52,86)(11,22,49,83,15,18,53,87)(12,23,50,84,16,19,54,88)(25,57,94,36,29,61,90,40)(26,58,95,37,30,62,91,33)(27,59,96,38,31,63,92,34)(28,60,89,39,32,64,93,35), (1,75)(2,76)(3,77)(4,78)(5,79)(6,80)(7,73)(8,74)(9,53)(10,54)(11,55)(12,56)(13,49)(14,50)(15,51)(16,52)(17,88)(18,81)(19,82)(20,83)(21,84)(22,85)(23,86)(24,87)(25,92)(26,93)(27,94)(28,95)(29,96)(30,89)(31,90)(32,91)(33,60)(34,61)(35,62)(36,63)(37,64)(38,57)(39,58)(40,59)(41,72)(42,65)(43,66)(44,67)(45,68)(46,69)(47,70)(48,71)>;
G:=Group( (1,20,61)(2,21,62)(3,22,63)(4,23,64)(5,24,57)(6,17,58)(7,18,59)(8,19,60)(9,29,70)(10,30,71)(11,31,72)(12,32,65)(13,25,66)(14,26,67)(15,27,68)(16,28,69)(33,74,82)(34,75,83)(35,76,84)(36,77,85)(37,78,86)(38,79,87)(39,80,88)(40,73,81)(41,55,90)(42,56,91)(43,49,92)(44,50,93)(45,51,94)(46,52,95)(47,53,96)(48,54,89), (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), (1,41,73,66,5,45,77,70)(2,42,74,67,6,46,78,71)(3,43,75,68,7,47,79,72)(4,44,76,69,8,48,80,65)(9,20,55,81,13,24,51,85)(10,21,56,82,14,17,52,86)(11,22,49,83,15,18,53,87)(12,23,50,84,16,19,54,88)(25,57,94,36,29,61,90,40)(26,58,95,37,30,62,91,33)(27,59,96,38,31,63,92,34)(28,60,89,39,32,64,93,35), (1,75)(2,76)(3,77)(4,78)(5,79)(6,80)(7,73)(8,74)(9,53)(10,54)(11,55)(12,56)(13,49)(14,50)(15,51)(16,52)(17,88)(18,81)(19,82)(20,83)(21,84)(22,85)(23,86)(24,87)(25,92)(26,93)(27,94)(28,95)(29,96)(30,89)(31,90)(32,91)(33,60)(34,61)(35,62)(36,63)(37,64)(38,57)(39,58)(40,59)(41,72)(42,65)(43,66)(44,67)(45,68)(46,69)(47,70)(48,71) );
G=PermutationGroup([[(1,20,61),(2,21,62),(3,22,63),(4,23,64),(5,24,57),(6,17,58),(7,18,59),(8,19,60),(9,29,70),(10,30,71),(11,31,72),(12,32,65),(13,25,66),(14,26,67),(15,27,68),(16,28,69),(33,74,82),(34,75,83),(35,76,84),(36,77,85),(37,78,86),(38,79,87),(39,80,88),(40,73,81),(41,55,90),(42,56,91),(43,49,92),(44,50,93),(45,51,94),(46,52,95),(47,53,96),(48,54,89)], [(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)], [(1,41,73,66,5,45,77,70),(2,42,74,67,6,46,78,71),(3,43,75,68,7,47,79,72),(4,44,76,69,8,48,80,65),(9,20,55,81,13,24,51,85),(10,21,56,82,14,17,52,86),(11,22,49,83,15,18,53,87),(12,23,50,84,16,19,54,88),(25,57,94,36,29,61,90,40),(26,58,95,37,30,62,91,33),(27,59,96,38,31,63,92,34),(28,60,89,39,32,64,93,35)], [(1,75),(2,76),(3,77),(4,78),(5,79),(6,80),(7,73),(8,74),(9,53),(10,54),(11,55),(12,56),(13,49),(14,50),(15,51),(16,52),(17,88),(18,81),(19,82),(20,83),(21,84),(22,85),(23,86),(24,87),(25,92),(26,93),(27,94),(28,95),(29,96),(30,89),(31,90),(32,91),(33,60),(34,61),(35,62),(36,63),(37,64),(38,57),(39,58),(40,59),(41,72),(42,65),(43,66),(44,67),(45,68),(46,69),(47,70),(48,71)]])
120 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 4A | 4B | 4C | 4D | 4E | ··· | 4N | 6A | ··· | 6F | 6G | 6H | 6I | 6J | 8A | ··· | 8H | 8I | ··· | 8T | 12A | ··· | 12H | 12I | ··· | 12AB | 24A | ··· | 24P | 24Q | ··· | 24AN |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 6 | ··· | 6 | 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 | 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 | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | + | + | + | ||||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C3 | C4 | C4 | C4 | C4 | C6 | C6 | C6 | C6 | C6 | C12 | C12 | C12 | C12 | C8○D4 | C3×C8○D4 |
| kernel | C3×C8○2M4(2) | C4×C24 | C3×C8⋊C4 | C3×C42⋊C2 | C22×C24 | C6×M4(2) | C8○2M4(2) | C3×C22⋊C4 | C3×C4⋊C4 | C2×C24 | C3×M4(2) | C4×C8 | C8⋊C4 | C42⋊C2 | C22×C8 | C2×M4(2) | C22⋊C4 | C4⋊C4 | C2×C8 | M4(2) | C6 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 2 | 4 | 4 | 8 | 8 | 4 | 4 | 2 | 2 | 2 | 8 | 8 | 16 | 16 | 8 | 16 |
Matrix representation of C3×C8○2M4(2) ►in GL4(𝔽73) generated by
| 1 | 0 | 0 | 0 |
| 0 | 64 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 72 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 51 | 0 |
| 0 | 0 | 0 | 51 |
| 27 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 51 | 0 |
| 0 | 0 | 53 | 22 |
| 1 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 72 | 46 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(73))| [1,0,0,0,0,64,0,0,0,0,1,0,0,0,0,1],[72,0,0,0,0,72,0,0,0,0,51,0,0,0,0,51],[27,0,0,0,0,72,0,0,0,0,51,53,0,0,0,22],[1,0,0,0,0,72,0,0,0,0,72,0,0,0,46,1] >;
C3×C8○2M4(2) in GAP, Magma, Sage, TeX
C_3\times C_8\circ_2M_4(2)
% in TeX
G:=Group("C3xC8o2M4(2)"); // GroupNames label
G:=SmallGroup(192,838);
// by ID
G=gap.SmallGroup(192,838);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,168,344,1059,172]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^8=d^2=1,c^4=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