metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C12⋊1M4(2), C42.203D6, Dic3⋊3M4(2), C4⋊C8⋊13S3, C4⋊3(C8⋊S3), (C4×S3).8Q8, C4.55(S3×Q8), D6.1(C4⋊C4), C12⋊C8⋊14C2, (C4×S3).33D4, (C2×C8).182D6, C4.207(S3×D4), Dic3⋊C8⋊28C2, C12.366(C2×D4), (S3×C42).4C2, C12.113(C2×Q8), (C4×Dic3).7C4, C6.8(C2×M4(2)), Dic3.2(C4⋊C4), C3⋊1(C4⋊M4(2)), (C4×C12).63C22, C2.18(S3×M4(2)), (C2×C12).834C23, (C2×C24).256C22, (C4×Dic3).277C22, C6.8(C2×C4⋊C4), C2.9(S3×C4⋊C4), (S3×C2×C4).7C4, (C3×C4⋊C8)⋊23C2, (C2×C8⋊S3).8C2, C2.13(C2×C8⋊S3), (C2×C4).147(C4×S3), (C2×C12).71(C2×C4), C22.112(S3×C2×C4), (C2×C3⋊C8).196C22, (S3×C2×C4).280C22, (C2×C6).89(C22×C4), (C22×S3).59(C2×C4), (C2×C4).776(C22×S3), (C2×Dic3).90(C2×C4), SmallGroup(192,396)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C12⋊M4(2)
G = < a,b,c | a12=b8=c2=1, bab-1=a-1, cac=a5, cbc=b5 >
Subgroups: 280 in 126 conjugacy classes, 61 normal (37 characteristic)
C1, C2, C2, C3, C4, C4, C4, C22, C22, S3, C6, C8, C2×C4, C2×C4, C23, Dic3, Dic3, C12, C12, C12, D6, D6, C2×C6, C42, C42, C2×C8, C2×C8, M4(2), C22×C4, C3⋊C8, C24, C4×S3, C4×S3, C2×Dic3, C2×C12, C22×S3, C4⋊C8, C4⋊C8, C2×C42, C2×M4(2), C8⋊S3, C2×C3⋊C8, C4×Dic3, C4×C12, C2×C24, S3×C2×C4, C4⋊M4(2), C12⋊C8, Dic3⋊C8, C3×C4⋊C8, S3×C42, C2×C8⋊S3, C12⋊M4(2)
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, C23, D6, C4⋊C4, M4(2), C22×C4, C2×D4, C2×Q8, C4×S3, C22×S3, C2×C4⋊C4, C2×M4(2), C8⋊S3, S3×C2×C4, S3×D4, S3×Q8, C4⋊M4(2), S3×C4⋊C4, C2×C8⋊S3, S3×M4(2), C12⋊M4(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 71 23 85 55 83 29 47)(2 70 24 96 56 82 30 46)(3 69 13 95 57 81 31 45)(4 68 14 94 58 80 32 44)(5 67 15 93 59 79 33 43)(6 66 16 92 60 78 34 42)(7 65 17 91 49 77 35 41)(8 64 18 90 50 76 36 40)(9 63 19 89 51 75 25 39)(10 62 20 88 52 74 26 38)(11 61 21 87 53 73 27 37)(12 72 22 86 54 84 28 48)
(2 6)(3 11)(5 9)(8 12)(13 21)(15 19)(16 24)(18 22)(25 33)(27 31)(28 36)(30 34)(37 95)(38 88)(39 93)(40 86)(41 91)(42 96)(43 89)(44 94)(45 87)(46 92)(47 85)(48 90)(50 54)(51 59)(53 57)(56 60)(61 81)(62 74)(63 79)(64 84)(65 77)(66 82)(67 75)(68 80)(69 73)(70 78)(71 83)(72 76)
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), (1,71,23,85,55,83,29,47)(2,70,24,96,56,82,30,46)(3,69,13,95,57,81,31,45)(4,68,14,94,58,80,32,44)(5,67,15,93,59,79,33,43)(6,66,16,92,60,78,34,42)(7,65,17,91,49,77,35,41)(8,64,18,90,50,76,36,40)(9,63,19,89,51,75,25,39)(10,62,20,88,52,74,26,38)(11,61,21,87,53,73,27,37)(12,72,22,86,54,84,28,48), (2,6)(3,11)(5,9)(8,12)(13,21)(15,19)(16,24)(18,22)(25,33)(27,31)(28,36)(30,34)(37,95)(38,88)(39,93)(40,86)(41,91)(42,96)(43,89)(44,94)(45,87)(46,92)(47,85)(48,90)(50,54)(51,59)(53,57)(56,60)(61,81)(62,74)(63,79)(64,84)(65,77)(66,82)(67,75)(68,80)(69,73)(70,78)(71,83)(72,76)>;
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), (1,71,23,85,55,83,29,47)(2,70,24,96,56,82,30,46)(3,69,13,95,57,81,31,45)(4,68,14,94,58,80,32,44)(5,67,15,93,59,79,33,43)(6,66,16,92,60,78,34,42)(7,65,17,91,49,77,35,41)(8,64,18,90,50,76,36,40)(9,63,19,89,51,75,25,39)(10,62,20,88,52,74,26,38)(11,61,21,87,53,73,27,37)(12,72,22,86,54,84,28,48), (2,6)(3,11)(5,9)(8,12)(13,21)(15,19)(16,24)(18,22)(25,33)(27,31)(28,36)(30,34)(37,95)(38,88)(39,93)(40,86)(41,91)(42,96)(43,89)(44,94)(45,87)(46,92)(47,85)(48,90)(50,54)(51,59)(53,57)(56,60)(61,81)(62,74)(63,79)(64,84)(65,77)(66,82)(67,75)(68,80)(69,73)(70,78)(71,83)(72,76) );
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)], [(1,71,23,85,55,83,29,47),(2,70,24,96,56,82,30,46),(3,69,13,95,57,81,31,45),(4,68,14,94,58,80,32,44),(5,67,15,93,59,79,33,43),(6,66,16,92,60,78,34,42),(7,65,17,91,49,77,35,41),(8,64,18,90,50,76,36,40),(9,63,19,89,51,75,25,39),(10,62,20,88,52,74,26,38),(11,61,21,87,53,73,27,37),(12,72,22,86,54,84,28,48)], [(2,6),(3,11),(5,9),(8,12),(13,21),(15,19),(16,24),(18,22),(25,33),(27,31),(28,36),(30,34),(37,95),(38,88),(39,93),(40,86),(41,91),(42,96),(43,89),(44,94),(45,87),(46,92),(47,85),(48,90),(50,54),(51,59),(53,57),(56,60),(61,81),(62,74),(63,79),(64,84),(65,77),(66,82),(67,75),(68,80),(69,73),(70,78),(71,83),(72,76)]])
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4N | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 6 | 6 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 6 | ··· | 6 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | + | + | + | - | |||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | S3 | D4 | Q8 | D6 | D6 | M4(2) | M4(2) | C4×S3 | C8⋊S3 | S3×D4 | S3×Q8 | S3×M4(2) |
| kernel | C12⋊M4(2) | C12⋊C8 | Dic3⋊C8 | C3×C4⋊C8 | S3×C42 | C2×C8⋊S3 | C4×Dic3 | S3×C2×C4 | C4⋊C8 | C4×S3 | C4×S3 | C42 | C2×C8 | Dic3 | C12 | C2×C4 | C4 | C4 | C4 | C2 |
| # reps | 1 | 1 | 2 | 1 | 1 | 2 | 4 | 4 | 1 | 2 | 2 | 1 | 2 | 4 | 4 | 4 | 8 | 1 | 1 | 2 |
Matrix representation of C12⋊M4(2) ►in GL6(𝔽73)
| 0 | 72 | 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 71 | 0 | 0 |
| 0 | 0 | 60 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 72 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 72 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(73))| [0,1,0,0,0,0,72,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,1,0],[72,1,0,0,0,0,0,1,0,0,0,0,0,0,72,60,0,0,0,0,71,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[1,72,0,0,0,0,0,72,0,0,0,0,0,0,1,72,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;
C12⋊M4(2) in GAP, Magma, Sage, TeX
C_{12}\rtimes M_4(2) % in TeX
G:=Group("C12:M4(2)"); // GroupNames label
G:=SmallGroup(192,396);
// by ID
G=gap.SmallGroup(192,396);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,120,758,219,58,136,6278]);
// Polycyclic
G:=Group<a,b,c|a^12=b^8=c^2=1,b*a*b^-1=a^-1,c*a*c=a^5,c*b*c=b^5>;
// generators/relations