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## G = C8×D11order 176 = 24·11

### Direct product of C8 and D11

Aliases: C8×D11, C883C2, D22.2C4, C4.12D22, C44.12C22, Dic11.2C4, C11⋊C86C2, C111(C2×C8), C22.1(C2×C4), C2.1(C4×D11), (C4×D11).3C2, SmallGroup(176,3)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C11 — C8×D11
 Chief series C1 — C11 — C22 — C44 — C4×D11 — C8×D11
 Lower central C11 — C8×D11
 Upper central C1 — C8

Generators and relations for C8×D11
G = < a,b,c | a8=b11=c2=1, ab=ba, ac=ca, cbc=b-1 >

Smallest permutation representation of C8×D11
On 88 points
Generators in S88
(1 87 43 65 21 76 32 54)(2 88 44 66 22 77 33 55)(3 78 34 56 12 67 23 45)(4 79 35 57 13 68 24 46)(5 80 36 58 14 69 25 47)(6 81 37 59 15 70 26 48)(7 82 38 60 16 71 27 49)(8 83 39 61 17 72 28 50)(9 84 40 62 18 73 29 51)(10 85 41 63 19 74 30 52)(11 86 42 64 20 75 31 53)
(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)| (1,87,43,65,21,76,32,54)(2,88,44,66,22,77,33,55)(3,78,34,56,12,67,23,45)(4,79,35,57,13,68,24,46)(5,80,36,58,14,69,25,47)(6,81,37,59,15,70,26,48)(7,82,38,60,16,71,27,49)(8,83,39,61,17,72,28,50)(9,84,40,62,18,73,29,51)(10,85,41,63,19,74,30,52)(11,86,42,64,20,75,31,53), (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( (1,87,43,65,21,76,32,54)(2,88,44,66,22,77,33,55)(3,78,34,56,12,67,23,45)(4,79,35,57,13,68,24,46)(5,80,36,58,14,69,25,47)(6,81,37,59,15,70,26,48)(7,82,38,60,16,71,27,49)(8,83,39,61,17,72,28,50)(9,84,40,62,18,73,29,51)(10,85,41,63,19,74,30,52)(11,86,42,64,20,75,31,53), (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([[(1,87,43,65,21,76,32,54),(2,88,44,66,22,77,33,55),(3,78,34,56,12,67,23,45),(4,79,35,57,13,68,24,46),(5,80,36,58,14,69,25,47),(6,81,37,59,15,70,26,48),(7,82,38,60,16,71,27,49),(8,83,39,61,17,72,28,50),(9,84,40,62,18,73,29,51),(10,85,41,63,19,74,30,52),(11,86,42,64,20,75,31,53)], [(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)]])

C8×D11 is a maximal subgroup of   D22.C8  D44.2C4  D44.C4  D83D11  Q8.D22  D885C2
C8×D11 is a maximal quotient of   D22.C8  Dic11⋊C8  D22⋊C8

56 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 8A 8B 8C 8D 8E 8F 8G 8H 11A ··· 11E 22A ··· 22E 44A ··· 44J 88A ··· 88T order 1 2 2 2 4 4 4 4 8 8 8 8 8 8 8 8 11 ··· 11 22 ··· 22 44 ··· 44 88 ··· 88 size 1 1 11 11 1 1 11 11 1 1 1 1 11 11 11 11 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2

56 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 type + + + + + + image C1 C2 C2 C2 C4 C4 C8 D11 D22 C4×D11 C8×D11 kernel C8×D11 C11⋊C8 C88 C4×D11 Dic11 D22 D11 C8 C4 C2 C1 # reps 1 1 1 1 2 2 8 5 5 10 20

Matrix representation of C8×D11 in GL3(𝔽89) generated by

 12 0 0 0 34 0 0 0 34
,
 1 0 0 0 0 1 0 88 55
,
 1 0 0 0 0 1 0 1 0
G:=sub<GL(3,GF(89))| [12,0,0,0,34,0,0,0,34],[1,0,0,0,0,88,0,1,55],[1,0,0,0,0,1,0,1,0] >;

C8×D11 in GAP, Magma, Sage, TeX

C_8\times D_{11}
% in TeX

G:=Group("C8xD11");
// GroupNames label

G:=SmallGroup(176,3);
// by ID

G=gap.SmallGroup(176,3);
# by ID

G:=PCGroup([5,-2,-2,-2,-2,-11,26,42,4004]);
// Polycyclic

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

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