direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C3×C4⋊M4(2), C12⋊9M4(2), C42.11C12, C4⋊C8⋊11C6, C4.75(C6×D4), C4.22(C6×Q8), (C4×C12).26C4, C4⋊2(C3×M4(2)), C12.60(C4⋊C4), (C2×C12).74Q8, (C2×C12).416D4, C12.480(C2×D4), C42.67(C2×C6), (C2×C42).17C6, C12.128(C2×Q8), C2.7(C6×M4(2)), C23.36(C2×C12), (C22×C4).18C12, (C22×C12).35C4, C6.50(C2×M4(2)), (C2×C24).324C22, (C4×C12).351C22, (C2×C12).985C23, (C2×M4(2)).13C6, (C6×M4(2)).31C2, C22.43(C22×C12), (C22×C12).585C22, C2.9(C6×C4⋊C4), (C3×C4⋊C8)⋊30C2, C4.11(C3×C4⋊C4), C6.65(C2×C4⋊C4), (C2×C4×C12).37C2, (C2×C8).49(C2×C6), (C2×C4).71(C3×D4), C22.7(C3×C4⋊C4), (C2×C4).16(C3×Q8), (C2×C6).24(C4⋊C4), (C2×C4).73(C2×C12), (C2×C12).289(C2×C4), (C22×C6).117(C2×C4), (C2×C4).153(C22×C6), (C2×C6).235(C22×C4), (C22×C4).121(C2×C6), SmallGroup(192,856)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C4⋊M4(2)
G = < a,b,c,d | a3=b4=c8=d2=1, ab=ba, ac=ca, ad=da, cbc-1=b-1, bd=db, dcd=c5 >
Subgroups: 162 in 126 conjugacy classes, 90 normal (26 characteristic)
C1, C2, C2, C2, C3, C4, C4, C4, C22, C22, C22, C6, C6, C6, C8, C2×C4, C2×C4, C2×C4, C23, C12, C12, C12, C2×C6, C2×C6, C2×C6, C42, C42, C2×C8, M4(2), C22×C4, C22×C4, C24, C2×C12, C2×C12, C2×C12, C22×C6, C4⋊C8, C2×C42, C2×M4(2), C4×C12, C4×C12, C2×C24, C3×M4(2), C22×C12, C22×C12, C4⋊M4(2), C3×C4⋊C8, C2×C4×C12, C6×M4(2), C3×C4⋊M4(2)
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D4, Q8, C23, C12, C2×C6, C4⋊C4, M4(2), C22×C4, C2×D4, C2×Q8, C2×C12, C3×D4, C3×Q8, C22×C6, C2×C4⋊C4, C2×M4(2), C3×C4⋊C4, C3×M4(2), C22×C12, C6×D4, C6×Q8, C4⋊M4(2), C6×C4⋊C4, C6×M4(2), C3×C4⋊M4(2)
►On 96 points
(1 83 39)(2 84 40)(3 85 33)(4 86 34)(5 87 35)(6 88 36)(7 81 37)(8 82 38)(9 17 63)(10 18 64)(11 19 57)(12 20 58)(13 21 59)(14 22 60)(15 23 61)(16 24 62)(25 75 72)(26 76 65)(27 77 66)(28 78 67)(29 79 68)(30 80 69)(31 73 70)(32 74 71)(41 52 93)(42 53 94)(43 54 95)(44 55 96)(45 56 89)(46 49 90)(47 50 91)(48 51 92)
(1 41 15 25)(2 26 16 42)(3 43 9 27)(4 28 10 44)(5 45 11 29)(6 30 12 46)(7 47 13 31)(8 32 14 48)(17 77 85 54)(18 55 86 78)(19 79 87 56)(20 49 88 80)(21 73 81 50)(22 51 82 74)(23 75 83 52)(24 53 84 76)(33 95 63 66)(34 67 64 96)(35 89 57 68)(36 69 58 90)(37 91 59 70)(38 71 60 92)(39 93 61 72)(40 65 62 94)
(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 11)(2 16)(3 13)(4 10)(5 15)(6 12)(7 9)(8 14)(17 81)(18 86)(19 83)(20 88)(21 85)(22 82)(23 87)(24 84)(25 45)(26 42)(27 47)(28 44)(29 41)(30 46)(31 43)(32 48)(33 59)(34 64)(35 61)(36 58)(37 63)(38 60)(39 57)(40 62)(49 80)(50 77)(51 74)(52 79)(53 76)(54 73)(55 78)(56 75)(65 94)(66 91)(67 96)(68 93)(69 90)(70 95)(71 92)(72 89)
G:=sub<Sym(96)| (1,83,39)(2,84,40)(3,85,33)(4,86,34)(5,87,35)(6,88,36)(7,81,37)(8,82,38)(9,17,63)(10,18,64)(11,19,57)(12,20,58)(13,21,59)(14,22,60)(15,23,61)(16,24,62)(25,75,72)(26,76,65)(27,77,66)(28,78,67)(29,79,68)(30,80,69)(31,73,70)(32,74,71)(41,52,93)(42,53,94)(43,54,95)(44,55,96)(45,56,89)(46,49,90)(47,50,91)(48,51,92), (1,41,15,25)(2,26,16,42)(3,43,9,27)(4,28,10,44)(5,45,11,29)(6,30,12,46)(7,47,13,31)(8,32,14,48)(17,77,85,54)(18,55,86,78)(19,79,87,56)(20,49,88,80)(21,73,81,50)(22,51,82,74)(23,75,83,52)(24,53,84,76)(33,95,63,66)(34,67,64,96)(35,89,57,68)(36,69,58,90)(37,91,59,70)(38,71,60,92)(39,93,61,72)(40,65,62,94), (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), (1,11)(2,16)(3,13)(4,10)(5,15)(6,12)(7,9)(8,14)(17,81)(18,86)(19,83)(20,88)(21,85)(22,82)(23,87)(24,84)(25,45)(26,42)(27,47)(28,44)(29,41)(30,46)(31,43)(32,48)(33,59)(34,64)(35,61)(36,58)(37,63)(38,60)(39,57)(40,62)(49,80)(50,77)(51,74)(52,79)(53,76)(54,73)(55,78)(56,75)(65,94)(66,91)(67,96)(68,93)(69,90)(70,95)(71,92)(72,89)>;
G:=Group( (1,83,39)(2,84,40)(3,85,33)(4,86,34)(5,87,35)(6,88,36)(7,81,37)(8,82,38)(9,17,63)(10,18,64)(11,19,57)(12,20,58)(13,21,59)(14,22,60)(15,23,61)(16,24,62)(25,75,72)(26,76,65)(27,77,66)(28,78,67)(29,79,68)(30,80,69)(31,73,70)(32,74,71)(41,52,93)(42,53,94)(43,54,95)(44,55,96)(45,56,89)(46,49,90)(47,50,91)(48,51,92), (1,41,15,25)(2,26,16,42)(3,43,9,27)(4,28,10,44)(5,45,11,29)(6,30,12,46)(7,47,13,31)(8,32,14,48)(17,77,85,54)(18,55,86,78)(19,79,87,56)(20,49,88,80)(21,73,81,50)(22,51,82,74)(23,75,83,52)(24,53,84,76)(33,95,63,66)(34,67,64,96)(35,89,57,68)(36,69,58,90)(37,91,59,70)(38,71,60,92)(39,93,61,72)(40,65,62,94), (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), (1,11)(2,16)(3,13)(4,10)(5,15)(6,12)(7,9)(8,14)(17,81)(18,86)(19,83)(20,88)(21,85)(22,82)(23,87)(24,84)(25,45)(26,42)(27,47)(28,44)(29,41)(30,46)(31,43)(32,48)(33,59)(34,64)(35,61)(36,58)(37,63)(38,60)(39,57)(40,62)(49,80)(50,77)(51,74)(52,79)(53,76)(54,73)(55,78)(56,75)(65,94)(66,91)(67,96)(68,93)(69,90)(70,95)(71,92)(72,89) );
G=PermutationGroup([[(1,83,39),(2,84,40),(3,85,33),(4,86,34),(5,87,35),(6,88,36),(7,81,37),(8,82,38),(9,17,63),(10,18,64),(11,19,57),(12,20,58),(13,21,59),(14,22,60),(15,23,61),(16,24,62),(25,75,72),(26,76,65),(27,77,66),(28,78,67),(29,79,68),(30,80,69),(31,73,70),(32,74,71),(41,52,93),(42,53,94),(43,54,95),(44,55,96),(45,56,89),(46,49,90),(47,50,91),(48,51,92)], [(1,41,15,25),(2,26,16,42),(3,43,9,27),(4,28,10,44),(5,45,11,29),(6,30,12,46),(7,47,13,31),(8,32,14,48),(17,77,85,54),(18,55,86,78),(19,79,87,56),(20,49,88,80),(21,73,81,50),(22,51,82,74),(23,75,83,52),(24,53,84,76),(33,95,63,66),(34,67,64,96),(35,89,57,68),(36,69,58,90),(37,91,59,70),(38,71,60,92),(39,93,61,72),(40,65,62,94)], [(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)], [(1,11),(2,16),(3,13),(4,10),(5,15),(6,12),(7,9),(8,14),(17,81),(18,86),(19,83),(20,88),(21,85),(22,82),(23,87),(24,84),(25,45),(26,42),(27,47),(28,44),(29,41),(30,46),(31,43),(32,48),(33,59),(34,64),(35,61),(36,58),(37,63),(38,60),(39,57),(40,62),(49,80),(50,77),(51,74),(52,79),(53,76),(54,73),(55,78),(56,75),(65,94),(66,91),(67,96),(68,93),(69,90),(70,95),(71,92),(72,89)]])
84 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 | 12A | ··· | 12H | 12I | ··· | 12AB | 24A | ··· | 24P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 8 | ··· | 8 | 12 | ··· | 12 | 12 | ··· | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 |
84 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | ||||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C4 | C4 | C6 | C6 | C6 | C12 | C12 | D4 | Q8 | M4(2) | C3×D4 | C3×Q8 | C3×M4(2) |
| kernel | C3×C4⋊M4(2) | C3×C4⋊C8 | C2×C4×C12 | C6×M4(2) | C4⋊M4(2) | C4×C12 | C22×C12 | C4⋊C8 | C2×C42 | C2×M4(2) | C42 | C22×C4 | C2×C12 | C2×C12 | C12 | C2×C4 | C2×C4 | C4 |
| # reps | 1 | 4 | 1 | 2 | 2 | 4 | 4 | 8 | 2 | 4 | 8 | 8 | 2 | 2 | 8 | 4 | 4 | 16 |
Matrix representation of C3×C4⋊M4(2) ►in GL4(𝔽73) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 64 | 0 |
| 0 | 0 | 0 | 64 |
| 0 | 72 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 72 |
| 47 | 37 | 0 | 0 |
| 37 | 26 | 0 | 0 |
| 0 | 0 | 0 | 2 |
| 0 | 0 | 50 | 0 |
| 72 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(73))| [1,0,0,0,0,1,0,0,0,0,64,0,0,0,0,64],[0,1,0,0,72,0,0,0,0,0,72,0,0,0,0,72],[47,37,0,0,37,26,0,0,0,0,0,50,0,0,2,0],[72,0,0,0,0,72,0,0,0,0,72,0,0,0,0,1] >;
C3×C4⋊M4(2) in GAP, Magma, Sage, TeX
C_3\times C_4\rtimes M_4(2)
% in TeX
G:=Group("C3xC4:M4(2)"); // GroupNames label
G:=SmallGroup(192,856);
// by ID
G=gap.SmallGroup(192,856);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,336,365,176,2102,124]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^4=c^8=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,b*d=d*b,d*c*d=c^5>;
// generators/relations