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G = C8×D11order 176 = 24·11

Direct product of C8 and D11

direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary

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

C1C11 — C8×D11
C1C11C22C44C4×D11 — C8×D11
C11 — C8×D11
C1C8

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

11C2
11C2
11C22
11C4
11C2×C4
11C8
11C2×C8

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 2A2B2C4A4B4C4D8A8B8C8D8E8F8G8H11A···11E22A···22E44A···44J88A···88T
order122244448888888811···1122···2244···4488···88
size1111111111111111111111112···22···22···22···2

56 irreducible representations

dim11111112222
type++++++
imageC1C2C2C2C4C4C8D11D22C4×D11C8×D11
kernelC8×D11C11⋊C8C88C4×D11Dic11D22D11C8C4C2C1
# reps1111228551020

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

1200
0340
0034
,
100
001
08855
,
100
001
010
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

Export

Subgroup lattice of C8×D11 in TeX

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