direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: M4(2)×D11, C8⋊6D22, C88⋊6C22, C44.38C23, (C8×D11)⋊7C2, C88⋊C2⋊5C2, C11⋊C8⋊11C22, C44.12(C2×C4), (C4×D11).1C4, D22.6(C2×C4), C4.15(C4×D11), (C2×C4).45D22, C11⋊2(C2×M4(2)), C44.C4⋊5C2, C22.7(C4×D11), (C11×M4(2))⋊3C2, C22.15(C22×C4), (C2×C44).25C22, (C2×Dic11).6C4, Dic11.7(C2×C4), (C22×D11).4C4, C4.38(C22×D11), (C4×D11).18C22, (C2×C4×D11).4C2, C2.16(C2×C4×D11), (C2×C22).5(C2×C4), SmallGroup(352,101)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for M4(2)×D11
G = < a,b,c,d | a8=b2=c11=d2=1, bab=a5, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Subgroups: 346 in 68 conjugacy classes, 39 normal (27 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, C23, C11, C2×C8, M4(2), M4(2), C22×C4, D11, D11, C22, C22, C2×M4(2), Dic11, C44, D22, D22, C2×C22, C11⋊C8, C88, C4×D11, C2×Dic11, C2×C44, C22×D11, C8×D11, C88⋊C2, C44.C4, C11×M4(2), C2×C4×D11, M4(2)×D11
Quotients: C1, C2, C4, C22, C2×C4, C23, M4(2), C22×C4, D11, C2×M4(2), D22, C4×D11, C22×D11, C2×C4×D11, M4(2)×D11
►On 88 points
Generators in S88
(1 76 32 65 21 87 43 54)(2 77 33 66 22 88 44 55)(3 67 23 56 12 78 34 45)(4 68 24 57 13 79 35 46)(5 69 25 58 14 80 36 47)(6 70 26 59 15 81 37 48)(7 71 27 60 16 82 38 49)(8 72 28 61 17 83 39 50)(9 73 29 62 18 84 40 51)(10 74 30 63 19 85 41 52)(11 75 31 64 20 86 42 53)
(45 56)(46 57)(47 58)(48 59)(49 60)(50 61)(51 62)(52 63)(53 64)(54 65)(55 66)(67 78)(68 79)(69 80)(70 81)(71 82)(72 83)(73 84)(74 85)(75 86)(76 87)(77 88)
(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)
(1 20)(2 19)(3 18)(4 17)(5 16)(6 15)(7 14)(8 13)(9 12)(10 22)(11 21)(23 40)(24 39)(25 38)(26 37)(27 36)(28 35)(29 34)(30 44)(31 43)(32 42)(33 41)(45 62)(46 61)(47 60)(48 59)(49 58)(50 57)(51 56)(52 66)(53 65)(54 64)(55 63)(67 84)(68 83)(69 82)(70 81)(71 80)(72 79)(73 78)(74 88)(75 87)(76 86)(77 85)
G:=sub<Sym(88)| (1,76,32,65,21,87,43,54)(2,77,33,66,22,88,44,55)(3,67,23,56,12,78,34,45)(4,68,24,57,13,79,35,46)(5,69,25,58,14,80,36,47)(6,70,26,59,15,81,37,48)(7,71,27,60,16,82,38,49)(8,72,28,61,17,83,39,50)(9,73,29,62,18,84,40,51)(10,74,30,63,19,85,41,52)(11,75,31,64,20,86,42,53), (45,56)(46,57)(47,58)(48,59)(49,60)(50,61)(51,62)(52,63)(53,64)(54,65)(55,66)(67,78)(68,79)(69,80)(70,81)(71,82)(72,83)(73,84)(74,85)(75,86)(76,87)(77,88), (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), (1,20)(2,19)(3,18)(4,17)(5,16)(6,15)(7,14)(8,13)(9,12)(10,22)(11,21)(23,40)(24,39)(25,38)(26,37)(27,36)(28,35)(29,34)(30,44)(31,43)(32,42)(33,41)(45,62)(46,61)(47,60)(48,59)(49,58)(50,57)(51,56)(52,66)(53,65)(54,64)(55,63)(67,84)(68,83)(69,82)(70,81)(71,80)(72,79)(73,78)(74,88)(75,87)(76,86)(77,85)>;
G:=Group( (1,76,32,65,21,87,43,54)(2,77,33,66,22,88,44,55)(3,67,23,56,12,78,34,45)(4,68,24,57,13,79,35,46)(5,69,25,58,14,80,36,47)(6,70,26,59,15,81,37,48)(7,71,27,60,16,82,38,49)(8,72,28,61,17,83,39,50)(9,73,29,62,18,84,40,51)(10,74,30,63,19,85,41,52)(11,75,31,64,20,86,42,53), (45,56)(46,57)(47,58)(48,59)(49,60)(50,61)(51,62)(52,63)(53,64)(54,65)(55,66)(67,78)(68,79)(69,80)(70,81)(71,82)(72,83)(73,84)(74,85)(75,86)(76,87)(77,88), (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), (1,20)(2,19)(3,18)(4,17)(5,16)(6,15)(7,14)(8,13)(9,12)(10,22)(11,21)(23,40)(24,39)(25,38)(26,37)(27,36)(28,35)(29,34)(30,44)(31,43)(32,42)(33,41)(45,62)(46,61)(47,60)(48,59)(49,58)(50,57)(51,56)(52,66)(53,65)(54,64)(55,63)(67,84)(68,83)(69,82)(70,81)(71,80)(72,79)(73,78)(74,88)(75,87)(76,86)(77,85) );
G=PermutationGroup([[(1,76,32,65,21,87,43,54),(2,77,33,66,22,88,44,55),(3,67,23,56,12,78,34,45),(4,68,24,57,13,79,35,46),(5,69,25,58,14,80,36,47),(6,70,26,59,15,81,37,48),(7,71,27,60,16,82,38,49),(8,72,28,61,17,83,39,50),(9,73,29,62,18,84,40,51),(10,74,30,63,19,85,41,52),(11,75,31,64,20,86,42,53)], [(45,56),(46,57),(47,58),(48,59),(49,60),(50,61),(51,62),(52,63),(53,64),(54,65),(55,66),(67,78),(68,79),(69,80),(70,81),(71,82),(72,83),(73,84),(74,85),(75,86),(76,87),(77,88)], [(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)], [(1,20),(2,19),(3,18),(4,17),(5,16),(6,15),(7,14),(8,13),(9,12),(10,22),(11,21),(23,40),(24,39),(25,38),(26,37),(27,36),(28,35),(29,34),(30,44),(31,43),(32,42),(33,41),(45,62),(46,61),(47,60),(48,59),(49,58),(50,57),(51,56),(52,66),(53,65),(54,64),(55,63),(67,84),(68,83),(69,82),(70,81),(71,80),(72,79),(73,78),(74,88),(75,87),(76,86),(77,85)]])
70 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 11A | ··· | 11E | 22A | ··· | 22E | 22F | ··· | 22J | 44A | ··· | 44J | 44K | ··· | 44O | 88A | ··· | 88T |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 11 | ··· | 11 | 22 | ··· | 22 | 22 | ··· | 22 | 44 | ··· | 44 | 44 | ··· | 44 | 88 | ··· | 88 |
| size | 1 | 1 | 2 | 11 | 11 | 22 | 1 | 1 | 2 | 11 | 11 | 22 | 2 | 2 | 2 | 2 | 22 | 22 | 22 | 22 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
70 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | M4(2) | D11 | D22 | D22 | C4×D11 | C4×D11 | M4(2)×D11 |
| kernel | M4(2)×D11 | C8×D11 | C88⋊C2 | C44.C4 | C11×M4(2) | C2×C4×D11 | C4×D11 | C2×Dic11 | C22×D11 | D11 | M4(2) | C8 | C2×C4 | C4 | C22 | C1 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 4 | 2 | 2 | 4 | 5 | 10 | 5 | 10 | 10 | 10 |
Matrix representation of M4(2)×D11 ►in GL4(𝔽89) generated by
| 88 | 0 | 0 | 0 |
| 0 | 88 | 0 | 0 |
| 0 | 0 | 46 | 2 |
| 0 | 0 | 82 | 43 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 43 | 88 |
| 0 | 1 | 0 | 0 |
| 88 | 7 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 88 | 0 |
| 0 | 0 | 0 | 88 |
G:=sub<GL(4,GF(89))| [88,0,0,0,0,88,0,0,0,0,46,82,0,0,2,43],[1,0,0,0,0,1,0,0,0,0,1,43,0,0,0,88],[0,88,0,0,1,7,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,88,0,0,0,0,88] >;
M4(2)×D11 in GAP, Magma, Sage, TeX
M_4(2)\times D_{11} % in TeX
G:=Group("M4(2)xD11"); // GroupNames label
G:=SmallGroup(352,101);
// by ID
G=gap.SmallGroup(352,101);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-11,188,50,69,11525]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=c^11=d^2=1,b*a*b=a^5,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations