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G = C88⋊C2order 176 = 24·11

4th semidirect product of C88 and C2 acting faithfully

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: C884C2, C83D11, D22.C4, C4.13D22, Dic11.C4, C111M4(2), C44.13C22, C11⋊C84C2, C22.2(C2×C4), C2.3(C4×D11), (C4×D11).2C2, SmallGroup(176,4)

Series: Derived Chief Lower central Upper central

C1C22 — C88⋊C2
C1C11C22C44C4×D11 — C88⋊C2
C11C22 — C88⋊C2
C1C4C8

Generators and relations for C88⋊C2
 G = < a,b | a88=b2=1, bab=a21 >

22C2
11C22
11C4
2D11
11C2×C4
11C8
11M4(2)

Smallest permutation representation of C88⋊C2
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)
(2 22)(3 43)(4 64)(5 85)(6 18)(7 39)(8 60)(9 81)(10 14)(11 35)(12 56)(13 77)(15 31)(16 52)(17 73)(19 27)(20 48)(21 69)(24 44)(25 65)(26 86)(28 40)(29 61)(30 82)(32 36)(33 57)(34 78)(37 53)(38 74)(41 49)(42 70)(46 66)(47 87)(50 62)(51 83)(54 58)(55 79)(59 75)(63 71)(68 88)(72 84)(76 80)

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

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

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

C88⋊C2 is a maximal subgroup of   D44.2C4  M4(2)×D11  D44.C4  D4⋊D22  D88⋊C2  D4.D22  Q16⋊D11
C88⋊C2 is a maximal quotient of   Dic11⋊C8  C88⋊C4  D22⋊C8

50 conjugacy classes

class 1 2A2B4A4B4C8A8B8C8D11A···11E22A···22E44A···44J88A···88T
order122444888811···1122···2244···4488···88
size112211222222222···22···22···22···2

50 irreducible representations

dim11111122222
type++++++
imageC1C2C2C2C4C4M4(2)D11D22C4×D11C88⋊C2
kernelC88⋊C2C11⋊C8C88C4×D11Dic11D22C11C8C4C2C1
# reps1111222551020

Matrix representation of C88⋊C2 in GL2(𝔽89) generated by

8483
614
,
10
7188
G:=sub<GL(2,GF(89))| [84,6,83,14],[1,71,0,88] >;

C88⋊C2 in GAP, Magma, Sage, TeX

C_{88}\rtimes C_2
% in TeX

G:=Group("C88:C2");
// GroupNames label

G:=SmallGroup(176,4);
// by ID

G=gap.SmallGroup(176,4);
# by ID

G:=PCGroup([5,-2,-2,-2,-2,-11,101,26,42,4004]);
// Polycyclic

G:=Group<a,b|a^88=b^2=1,b*a*b=a^21>;
// generators/relations

Export

Subgroup lattice of C88⋊C2 in TeX

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