metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C88⋊4C2, C8⋊3D11, D22.C4, C4.13D22, Dic11.C4, C11⋊1M4(2), C44.13C22, C11⋊C8⋊4C2, C22.2(C2×C4), C2.3(C4×D11), (C4×D11).2C2, SmallGroup(176,4)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C88⋊C2
G = < a,b | a88=b2=1, bab=a21 >
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 | 2A | 2B | 4A | 4B | 4C | 8A | 8B | 8C | 8D | 11A | ··· | 11E | 22A | ··· | 22E | 44A | ··· | 44J | 88A | ··· | 88T |
| order | 1 | 2 | 2 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 11 | ··· | 11 | 22 | ··· | 22 | 44 | ··· | 44 | 88 | ··· | 88 |
| size | 1 | 1 | 22 | 1 | 1 | 22 | 2 | 2 | 22 | 22 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | M4(2) | D11 | D22 | C4×D11 | C88⋊C2 |
| kernel | C88⋊C2 | C11⋊C8 | C88 | C4×D11 | Dic11 | D22 | C11 | C8 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 5 | 5 | 10 | 20 |
Matrix representation of C88⋊C2 ►in GL2(𝔽89) generated by
| 84 | 83 |
| 6 | 14 |
| 1 | 0 |
| 71 | 88 |
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
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