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## G = C11×C8⋊C22order 352 = 25·11

### Direct product of C11 and C8⋊C22

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C4 — C11×C8⋊C22
 Chief series C1 — C2 — C4 — C44 — D4×C11 — C11×D8 — C11×C8⋊C22
 Lower central C1 — C2 — C4 — C11×C8⋊C22
 Upper central C1 — C22 — C2×C44 — C11×C8⋊C22

Generators and relations for C11×C8⋊C22
G = < a,b,c,d | a11=b8=c2=d2=1, ab=ba, ac=ca, ad=da, cbc=b3, dbd=b5, cd=dc >

Subgroups: 116 in 68 conjugacy classes, 40 normal (24 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, D4, D4, D4, Q8, C23, C11, M4(2), D8, SD16, C2×D4, C4○D4, C22, C22, C8⋊C22, C44, C44, C2×C22, C2×C22, C88, C2×C44, C2×C44, D4×C11, D4×C11, D4×C11, Q8×C11, C22×C22, C11×M4(2), C11×D8, C11×SD16, D4×C22, C11×C4○D4, C11×C8⋊C22
Quotients: C1, C2, C22, D4, C23, C11, C2×D4, C22, C8⋊C22, C2×C22, D4×C11, C22×C22, D4×C22, C11×C8⋊C22

Smallest permutation representation of C11×C8⋊C22
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 75 15 61 40 54 30)(2 80 76 16 62 41 55 31)(3 81 77 17 63 42 45 32)(4 82 67 18 64 43 46 33)(5 83 68 19 65 44 47 23)(6 84 69 20 66 34 48 24)(7 85 70 21 56 35 49 25)(8 86 71 22 57 36 50 26)(9 87 72 12 58 37 51 27)(10 88 73 13 59 38 52 28)(11 78 74 14 60 39 53 29)
(12 87)(13 88)(14 78)(15 79)(16 80)(17 81)(18 82)(19 83)(20 84)(21 85)(22 86)(23 44)(24 34)(25 35)(26 36)(27 37)(28 38)(29 39)(30 40)(31 41)(32 42)(33 43)(45 77)(46 67)(47 68)(48 69)(49 70)(50 71)(51 72)(52 73)(53 74)(54 75)(55 76)
(12 27)(13 28)(14 29)(15 30)(16 31)(17 32)(18 33)(19 23)(20 24)(21 25)(22 26)(34 84)(35 85)(36 86)(37 87)(38 88)(39 78)(40 79)(41 80)(42 81)(43 82)(44 83)

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

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

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

121 conjugacy classes

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

121 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 4 4 type + + + + + + + + + image C1 C2 C2 C2 C2 C2 C11 C22 C22 C22 C22 C22 D4 D4 D4×C11 D4×C11 C8⋊C22 C11×C8⋊C22 kernel C11×C8⋊C22 C11×M4(2) C11×D8 C11×SD16 D4×C22 C11×C4○D4 C8⋊C22 M4(2) D8 SD16 C2×D4 C4○D4 C44 C2×C22 C4 C22 C11 C1 # reps 1 1 2 2 1 1 10 10 20 20 10 10 1 1 10 10 1 10

Matrix representation of C11×C8⋊C22 in GL4(𝔽89) generated by

 39 0 0 0 0 39 0 0 0 0 39 0 0 0 0 39
,
 0 0 1 0 0 0 0 88 0 1 0 0 1 0 0 0
,
 1 0 0 0 0 88 0 0 0 0 0 88 0 0 88 0
,
 1 0 0 0 0 1 0 0 0 0 88 0 0 0 0 88
G:=sub<GL(4,GF(89))| [39,0,0,0,0,39,0,0,0,0,39,0,0,0,0,39],[0,0,0,1,0,0,1,0,1,0,0,0,0,88,0,0],[1,0,0,0,0,88,0,0,0,0,0,88,0,0,88,0],[1,0,0,0,0,1,0,0,0,0,88,0,0,0,0,88] >;

C11×C8⋊C22 in GAP, Magma, Sage, TeX

C_{11}\times C_8\rtimes C_2^2
% in TeX

G:=Group("C11xC8:C2^2");
// GroupNames label

G:=SmallGroup(352,171);
// by ID

G=gap.SmallGroup(352,171);
# by ID

G:=PCGroup([6,-2,-2,-2,-11,-2,-2,1081,3242,7924,3970,88]);
// Polycyclic

G:=Group<a,b,c,d|a^11=b^8=c^2=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^3,d*b*d=b^5,c*d=d*c>;
// generators/relations

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