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## G = C11×C4.D4order 352 = 25·11

### Direct product of C11 and C4.D4

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C11×C4.D4
 Chief series C1 — C2 — C4 — C2×C4 — C2×C44 — C11×M4(2) — C11×C4.D4
 Lower central C1 — C2 — C22 — C11×C4.D4
 Upper central C1 — C22 — C2×C44 — C11×C4.D4

Generators and relations for C11×C4.D4
G = < a,b,c,d | a11=b4=1, c4=b2, d2=b, ab=ba, ac=ca, ad=da, cbc-1=b-1, bd=db, dcd-1=b-1c3 >

Smallest permutation representation of C11×C4.D4
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 66 73 47)(2 56 74 48)(3 57 75 49)(4 58 76 50)(5 59 77 51)(6 60 67 52)(7 61 68 53)(8 62 69 54)(9 63 70 55)(10 64 71 45)(11 65 72 46)(12 42 32 81)(13 43 33 82)(14 44 23 83)(15 34 24 84)(16 35 25 85)(17 36 26 86)(18 37 27 87)(19 38 28 88)(20 39 29 78)(21 40 30 79)(22 41 31 80)
(1 30 66 40 73 21 47 79)(2 31 56 41 74 22 48 80)(3 32 57 42 75 12 49 81)(4 33 58 43 76 13 50 82)(5 23 59 44 77 14 51 83)(6 24 60 34 67 15 52 84)(7 25 61 35 68 16 53 85)(8 26 62 36 69 17 54 86)(9 27 63 37 70 18 55 87)(10 28 64 38 71 19 45 88)(11 29 65 39 72 20 46 78)
(1 40 66 30 73 79 47 21)(2 41 56 31 74 80 48 22)(3 42 57 32 75 81 49 12)(4 43 58 33 76 82 50 13)(5 44 59 23 77 83 51 14)(6 34 60 24 67 84 52 15)(7 35 61 25 68 85 53 16)(8 36 62 26 69 86 54 17)(9 37 63 27 70 87 55 18)(10 38 64 28 71 88 45 19)(11 39 65 29 72 78 46 20)

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

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

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

121 conjugacy classes

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

121 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 4 4 type + + + + + image C1 C2 C2 C4 C11 C22 C22 C44 D4 D4×C11 C4.D4 C11×C4.D4 kernel C11×C4.D4 C11×M4(2) D4×C22 C22×C22 C4.D4 M4(2) C2×D4 C23 C44 C4 C11 C1 # reps 1 2 1 4 10 20 10 40 2 20 1 10

Matrix representation of C11×C4.D4 in GL4(𝔽89) generated by

 4 0 0 0 0 4 0 0 0 0 4 0 0 0 0 4
,
 0 1 0 0 88 0 0 0 0 0 0 1 0 0 88 0
,
 0 0 0 1 0 0 1 0 88 0 0 0 0 1 0 0
,
 0 0 1 0 0 0 0 1 0 1 0 0 88 0 0 0
G:=sub<GL(4,GF(89))| [4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[0,88,0,0,1,0,0,0,0,0,0,88,0,0,1,0],[0,0,88,0,0,0,0,1,0,1,0,0,1,0,0,0],[0,0,0,88,0,0,1,0,1,0,0,0,0,1,0,0] >;

C11×C4.D4 in GAP, Magma, Sage, TeX

C_{11}\times C_4.D_4
% in TeX

G:=Group("C11xC4.D4");
// GroupNames label

G:=SmallGroup(352,49);
// by ID

G=gap.SmallGroup(352,49);
# by ID

G:=PCGroup([6,-2,-2,-11,-2,-2,-2,528,553,5283,3970,88]);
// Polycyclic

G:=Group<a,b,c,d|a^11=b^4=1,c^4=b^2,d^2=b,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^-1=b^-1*c^3>;
// generators/relations

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