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## G = C11×SD16order 176 = 24·11

### Direct product of C11 and SD16

direct product, metacyclic, nilpotent (class 3), monomial, 2-elementary

Aliases: C11×SD16, Q8⋊C22, C82C22, C886C2, D4.C22, C22.15D4, C44.18C22, C4.2(C2×C22), (Q8×C11)⋊4C2, C2.4(D4×C11), (D4×C11).2C2, SmallGroup(176,25)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C4 — C11×SD16
 Chief series C1 — C2 — C4 — C44 — Q8×C11 — C11×SD16
 Lower central C1 — C2 — C4 — C11×SD16
 Upper central C1 — C22 — C44 — C11×SD16

Generators and relations for C11×SD16
G = < a,b,c | a11=b8=c2=1, ab=ba, ac=ca, cbc=b3 >

Smallest permutation representation of C11×SD16
On 88 points
Generators in S88
(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 79 47 30 73 40 66 21)(2 80 48 31 74 41 56 22)(3 81 49 32 75 42 57 12)(4 82 50 33 76 43 58 13)(5 83 51 23 77 44 59 14)(6 84 52 24 67 34 60 15)(7 85 53 25 68 35 61 16)(8 86 54 26 69 36 62 17)(9 87 55 27 70 37 63 18)(10 88 45 28 71 38 64 19)(11 78 46 29 72 39 65 20)
(12 42)(13 43)(14 44)(15 34)(16 35)(17 36)(18 37)(19 38)(20 39)(21 40)(22 41)(23 83)(24 84)(25 85)(26 86)(27 87)(28 88)(29 78)(30 79)(31 80)(32 81)(33 82)(45 64)(46 65)(47 66)(48 56)(49 57)(50 58)(51 59)(52 60)(53 61)(54 62)(55 63)

G:=sub<Sym(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), (1,79,47,30,73,40,66,21)(2,80,48,31,74,41,56,22)(3,81,49,32,75,42,57,12)(4,82,50,33,76,43,58,13)(5,83,51,23,77,44,59,14)(6,84,52,24,67,34,60,15)(7,85,53,25,68,35,61,16)(8,86,54,26,69,36,62,17)(9,87,55,27,70,37,63,18)(10,88,45,28,71,38,64,19)(11,78,46,29,72,39,65,20), (12,42)(13,43)(14,44)(15,34)(16,35)(17,36)(18,37)(19,38)(20,39)(21,40)(22,41)(23,83)(24,84)(25,85)(26,86)(27,87)(28,88)(29,78)(30,79)(31,80)(32,81)(33,82)(45,64)(46,65)(47,66)(48,56)(49,57)(50,58)(51,59)(52,60)(53,61)(54,62)(55,63)>;

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), (1,79,47,30,73,40,66,21)(2,80,48,31,74,41,56,22)(3,81,49,32,75,42,57,12)(4,82,50,33,76,43,58,13)(5,83,51,23,77,44,59,14)(6,84,52,24,67,34,60,15)(7,85,53,25,68,35,61,16)(8,86,54,26,69,36,62,17)(9,87,55,27,70,37,63,18)(10,88,45,28,71,38,64,19)(11,78,46,29,72,39,65,20), (12,42)(13,43)(14,44)(15,34)(16,35)(17,36)(18,37)(19,38)(20,39)(21,40)(22,41)(23,83)(24,84)(25,85)(26,86)(27,87)(28,88)(29,78)(30,79)(31,80)(32,81)(33,82)(45,64)(46,65)(47,66)(48,56)(49,57)(50,58)(51,59)(52,60)(53,61)(54,62)(55,63) );

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)], [(1,79,47,30,73,40,66,21),(2,80,48,31,74,41,56,22),(3,81,49,32,75,42,57,12),(4,82,50,33,76,43,58,13),(5,83,51,23,77,44,59,14),(6,84,52,24,67,34,60,15),(7,85,53,25,68,35,61,16),(8,86,54,26,69,36,62,17),(9,87,55,27,70,37,63,18),(10,88,45,28,71,38,64,19),(11,78,46,29,72,39,65,20)], [(12,42),(13,43),(14,44),(15,34),(16,35),(17,36),(18,37),(19,38),(20,39),(21,40),(22,41),(23,83),(24,84),(25,85),(26,86),(27,87),(28,88),(29,78),(30,79),(31,80),(32,81),(33,82),(45,64),(46,65),(47,66),(48,56),(49,57),(50,58),(51,59),(52,60),(53,61),(54,62),(55,63)])

C11×SD16 is a maximal subgroup of   D88⋊C2  D4.D22  Q8.D22

77 conjugacy classes

 class 1 2A 2B 4A 4B 8A 8B 11A ··· 11J 22A ··· 22J 22K ··· 22T 44A ··· 44J 44K ··· 44T 88A ··· 88T order 1 2 2 4 4 8 8 11 ··· 11 22 ··· 22 22 ··· 22 44 ··· 44 44 ··· 44 88 ··· 88 size 1 1 4 2 4 2 2 1 ··· 1 1 ··· 1 4 ··· 4 2 ··· 2 4 ··· 4 2 ··· 2

77 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 type + + + + + image C1 C2 C2 C2 C11 C22 C22 C22 D4 SD16 D4×C11 C11×SD16 kernel C11×SD16 C88 D4×C11 Q8×C11 SD16 C8 D4 Q8 C22 C11 C2 C1 # reps 1 1 1 1 10 10 10 10 1 2 10 20

Matrix representation of C11×SD16 in GL2(𝔽89) generated by

 78 0 0 78
,
 49 69 40 0
,
 1 1 0 88
G:=sub<GL(2,GF(89))| [78,0,0,78],[49,40,69,0],[1,0,1,88] >;

C11×SD16 in GAP, Magma, Sage, TeX

C_{11}\times {\rm SD}_{16}
% in TeX

G:=Group("C11xSD16");
// GroupNames label

G:=SmallGroup(176,25);
// by ID

G=gap.SmallGroup(176,25);
# by ID

G:=PCGroup([5,-2,-2,-11,-2,-2,440,461,2643,1328,58]);
// Polycyclic

G:=Group<a,b,c|a^11=b^8=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^3>;
// generators/relations

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