metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C12⋊3M4(2), C42.206D6, C3⋊C8⋊18D4, C3⋊4(C8⋊6D4), (C6×D4).9C4, (C4×D4).6S3, C6.92(C4×D4), (D4×C12).7C2, C12⋊C8⋊22C2, C4.216(S3×D4), C4⋊C4.8Dic3, C2.9(D4×Dic3), C6.40(C8○D4), C12.375(C2×D4), C4⋊1(C4.Dic3), (C2×D4).6Dic3, C12.55D4⋊3C2, (C4×C12).83C22, (C22×C4).131D6, C6.41(C2×M4(2)), C22⋊C4.5Dic3, C12.308(C4○D4), (C2×C12).850C23, C2.7(D4.Dic3), C4.135(D4⋊2S3), C23.14(C2×Dic3), (C22×C12).100C22, C22.46(C22×Dic3), (C4×C3⋊C8)⋊7C2, (C3×C4⋊C4).12C4, (C3×C22⋊C4).6C4, (C2×C4.Dic3)⋊5C2, (C2×C12).164(C2×C4), C2.9(C2×C4.Dic3), (C2×C3⋊C8).201C22, (C22×C6).61(C2×C4), (C2×C4).20(C2×Dic3), (C2×C4).792(C22×S3), (C2×C6).187(C22×C4), SmallGroup(192,571)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C12⋊3M4(2)
G = < a,b,c | a12=b8=c2=1, bab-1=a-1, cac=a7, cbc=b5 >
Subgroups: 232 in 122 conjugacy classes, 61 normal (33 characteristic)
C1, C2, C2, C3, C4, C4, C4, C22, C22, C6, C6, C8, C2×C4, C2×C4, C2×C4, D4, C23, C12, C12, C12, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, C2×D4, C3⋊C8, C3⋊C8, C2×C12, C2×C12, C2×C12, C3×D4, C22×C6, C4×C8, C22⋊C8, C4⋊C8, C4×D4, C2×M4(2), C2×C3⋊C8, C2×C3⋊C8, C4.Dic3, C4×C12, C3×C22⋊C4, C3×C4⋊C4, C22×C12, C6×D4, C8⋊6D4, C4×C3⋊C8, C12⋊C8, C12.55D4, C2×C4.Dic3, D4×C12, C12⋊3M4(2)
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, Dic3, D6, M4(2), C22×C4, C2×D4, C4○D4, C2×Dic3, C22×S3, C4×D4, C2×M4(2), C8○D4, C4.Dic3, S3×D4, D4⋊2S3, C22×Dic3, C8⋊6D4, C2×C4.Dic3, D4×Dic3, D4.Dic3, C12⋊3M4(2)
►On 96 points
(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 57 24 67 92 39 81 27)(2 56 13 66 93 38 82 26)(3 55 14 65 94 37 83 25)(4 54 15 64 95 48 84 36)(5 53 16 63 96 47 73 35)(6 52 17 62 85 46 74 34)(7 51 18 61 86 45 75 33)(8 50 19 72 87 44 76 32)(9 49 20 71 88 43 77 31)(10 60 21 70 89 42 78 30)(11 59 22 69 90 41 79 29)(12 58 23 68 91 40 80 28)
(1 89)(2 96)(3 91)(4 86)(5 93)(6 88)(7 95)(8 90)(9 85)(10 92)(11 87)(12 94)(13 73)(14 80)(15 75)(16 82)(17 77)(18 84)(19 79)(20 74)(21 81)(22 76)(23 83)(24 78)(25 28)(26 35)(27 30)(29 32)(31 34)(33 36)(37 40)(38 47)(39 42)(41 44)(43 46)(45 48)(49 52)(50 59)(51 54)(53 56)(55 58)(57 60)(61 64)(62 71)(63 66)(65 68)(67 70)(69 72)
G:=sub<Sym(96)| (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), (1,57,24,67,92,39,81,27)(2,56,13,66,93,38,82,26)(3,55,14,65,94,37,83,25)(4,54,15,64,95,48,84,36)(5,53,16,63,96,47,73,35)(6,52,17,62,85,46,74,34)(7,51,18,61,86,45,75,33)(8,50,19,72,87,44,76,32)(9,49,20,71,88,43,77,31)(10,60,21,70,89,42,78,30)(11,59,22,69,90,41,79,29)(12,58,23,68,91,40,80,28), (1,89)(2,96)(3,91)(4,86)(5,93)(6,88)(7,95)(8,90)(9,85)(10,92)(11,87)(12,94)(13,73)(14,80)(15,75)(16,82)(17,77)(18,84)(19,79)(20,74)(21,81)(22,76)(23,83)(24,78)(25,28)(26,35)(27,30)(29,32)(31,34)(33,36)(37,40)(38,47)(39,42)(41,44)(43,46)(45,48)(49,52)(50,59)(51,54)(53,56)(55,58)(57,60)(61,64)(62,71)(63,66)(65,68)(67,70)(69,72)>;
G:=Group( (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), (1,57,24,67,92,39,81,27)(2,56,13,66,93,38,82,26)(3,55,14,65,94,37,83,25)(4,54,15,64,95,48,84,36)(5,53,16,63,96,47,73,35)(6,52,17,62,85,46,74,34)(7,51,18,61,86,45,75,33)(8,50,19,72,87,44,76,32)(9,49,20,71,88,43,77,31)(10,60,21,70,89,42,78,30)(11,59,22,69,90,41,79,29)(12,58,23,68,91,40,80,28), (1,89)(2,96)(3,91)(4,86)(5,93)(6,88)(7,95)(8,90)(9,85)(10,92)(11,87)(12,94)(13,73)(14,80)(15,75)(16,82)(17,77)(18,84)(19,79)(20,74)(21,81)(22,76)(23,83)(24,78)(25,28)(26,35)(27,30)(29,32)(31,34)(33,36)(37,40)(38,47)(39,42)(41,44)(43,46)(45,48)(49,52)(50,59)(51,54)(53,56)(55,58)(57,60)(61,64)(62,71)(63,66)(65,68)(67,70)(69,72) );
G=PermutationGroup([[(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)], [(1,57,24,67,92,39,81,27),(2,56,13,66,93,38,82,26),(3,55,14,65,94,37,83,25),(4,54,15,64,95,48,84,36),(5,53,16,63,96,47,73,35),(6,52,17,62,85,46,74,34),(7,51,18,61,86,45,75,33),(8,50,19,72,87,44,76,32),(9,49,20,71,88,43,77,31),(10,60,21,70,89,42,78,30),(11,59,22,69,90,41,79,29),(12,58,23,68,91,40,80,28)], [(1,89),(2,96),(3,91),(4,86),(5,93),(6,88),(7,95),(8,90),(9,85),(10,92),(11,87),(12,94),(13,73),(14,80),(15,75),(16,82),(17,77),(18,84),(19,79),(20,74),(21,81),(22,76),(23,83),(24,78),(25,28),(26,35),(27,30),(29,32),(31,34),(33,36),(37,40),(38,47),(39,42),(41,44),(43,46),(45,48),(49,52),(50,59),(51,54),(53,56),(55,58),(57,60),(61,64),(62,71),(63,66),(65,68),(67,70),(69,72)]])
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 8A | ··· | 8H | 8I | 8J | 8K | 8L | 12A | 12B | 12C | 12D | 12E | ··· | 12L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | ··· | 8 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | - | + | - | + | - | ||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | S3 | D4 | D6 | Dic3 | Dic3 | D6 | Dic3 | M4(2) | C4○D4 | C8○D4 | C4.Dic3 | S3×D4 | D4⋊2S3 | D4.Dic3 |
| kernel | C12⋊3M4(2) | C4×C3⋊C8 | C12⋊C8 | C12.55D4 | C2×C4.Dic3 | D4×C12 | C3×C22⋊C4 | C3×C4⋊C4 | C6×D4 | C4×D4 | C3⋊C8 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C12 | C12 | C6 | C4 | C4 | C4 | C2 |
| # reps | 1 | 1 | 1 | 2 | 2 | 1 | 4 | 2 | 2 | 1 | 2 | 1 | 2 | 1 | 2 | 1 | 4 | 2 | 4 | 8 | 1 | 1 | 2 |
Matrix representation of C12⋊3M4(2) ►in GL4(𝔽73) generated by
| 8 | 37 | 0 | 0 |
| 0 | 64 | 0 | 0 |
| 0 | 0 | 1 | 31 |
| 0 | 0 | 7 | 72 |
| 70 | 56 | 0 | 0 |
| 29 | 3 | 0 | 0 |
| 0 | 0 | 46 | 39 |
| 0 | 0 | 0 | 27 |
| 72 | 30 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 72 | 42 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(73))| [8,0,0,0,37,64,0,0,0,0,1,7,0,0,31,72],[70,29,0,0,56,3,0,0,0,0,46,0,0,0,39,27],[72,0,0,0,30,1,0,0,0,0,72,0,0,0,42,1] >;
C12⋊3M4(2) in GAP, Magma, Sage, TeX
C_{12}\rtimes_3M_4(2) % in TeX
G:=Group("C12:3M4(2)"); // GroupNames label
G:=SmallGroup(192,571);
// by ID
G=gap.SmallGroup(192,571);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,56,477,758,219,136,6278]);
// Polycyclic
G:=Group<a,b,c|a^12=b^8=c^2=1,b*a*b^-1=a^-1,c*a*c=a^7,c*b*c=b^5>;
// generators/relations