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## G = C11×D8order 176 = 24·11

### Direct product of C11 and D8

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

Aliases: C11×D8, D4⋊C22, C81C22, C885C2, C22.14D4, C44.17C22, (D4×C11)⋊4C2, C4.1(C2×C22), C2.3(D4×C11), SmallGroup(176,24)

Series: Derived Chief Lower central Upper central

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

Generators and relations for C11×D8
G = < a,b,c | a11=b8=c2=1, ab=ba, ac=ca, cbc=b-1 >

Smallest permutation representation of C11×D8
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 40 75 18 61 79 47 30)(2 41 76 19 62 80 48 31)(3 42 77 20 63 81 49 32)(4 43 67 21 64 82 50 33)(5 44 68 22 65 83 51 23)(6 34 69 12 66 84 52 24)(7 35 70 13 56 85 53 25)(8 36 71 14 57 86 54 26)(9 37 72 15 58 87 55 27)(10 38 73 16 59 88 45 28)(11 39 74 17 60 78 46 29)
(1 30)(2 31)(3 32)(4 33)(5 23)(6 24)(7 25)(8 26)(9 27)(10 28)(11 29)(12 66)(13 56)(14 57)(15 58)(16 59)(17 60)(18 61)(19 62)(20 63)(21 64)(22 65)(34 52)(35 53)(36 54)(37 55)(38 45)(39 46)(40 47)(41 48)(42 49)(43 50)(44 51)(67 82)(68 83)(69 84)(70 85)(71 86)(72 87)(73 88)(74 78)(75 79)(76 80)(77 81)

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

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

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

C11×D8 is a maximal subgroup of   C11⋊D16  D8.D11  D4⋊D22  D83D11

77 conjugacy classes

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

77 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 type + + + + + image C1 C2 C2 C11 C22 C22 D4 D8 D4×C11 C11×D8 kernel C11×D8 C88 D4×C11 D8 C8 D4 C22 C11 C2 C1 # reps 1 1 2 10 10 20 1 2 10 20

Matrix representation of C11×D8 in GL2(𝔽23) generated by

 2 0 0 2
,
 5 14 18 0
,
 0 9 18 0
G:=sub<GL(2,GF(23))| [2,0,0,2],[5,18,14,0],[0,18,9,0] >;

C11×D8 in GAP, Magma, Sage, TeX

C_{11}\times D_8
% in TeX

G:=Group("C11xD8");
// GroupNames label

G:=SmallGroup(176,24);
// by ID

G=gap.SmallGroup(176,24);
# by ID

G:=PCGroup([5,-2,-2,-11,-2,-2,461,2643,1328,58]);
// 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^-1>;
// generators/relations

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