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## G = M4(2)×D11order 352 = 25·11

### Direct product of M4(2) and D11

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — M4(2)×D11
 Chief series C1 — C11 — C22 — C44 — C4×D11 — C2×C4×D11 — M4(2)×D11
 Lower central C11 — C22 — M4(2)×D11
 Upper central C1 — C4 — M4(2)

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

Smallest permutation representation of 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

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