direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C2×C4×D11, C44⋊3C22, C22.2C23, C22.9D22, D22.4C22, Dic11⋊3C22, C22⋊1(C2×C4), (C2×C44)⋊5C2, C11⋊1(C22×C4), (C2×Dic11)⋊5C2, (C2×C22).9C22, C2.1(C22×D11), (C22×D11).2C2, SmallGroup(176,28)
Series: Derived ►Chief ►Lower central ►Upper central
| C11 — C2×C4×D11 |
Generators and relations for C2×C4×D11
G = < a,b,c,d | a2=b4=c11=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Subgroups: 244 in 54 conjugacy classes, 35 normal (11 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C2×C4, C2×C4, C23, C11, C22×C4, D11, C22, C22, Dic11, C44, D22, C2×C22, C4×D11, C2×Dic11, C2×C44, C22×D11, C2×C4×D11
Quotients: C1, C2, C4, C22, C2×C4, C23, C22×C4, D11, D22, C4×D11, C22×D11, C2×C4×D11
►On 88 points
Generators in S88
(1 54)(2 55)(3 45)(4 46)(5 47)(6 48)(7 49)(8 50)(9 51)(10 52)(11 53)(12 56)(13 57)(14 58)(15 59)(16 60)(17 61)(18 62)(19 63)(20 64)(21 65)(22 66)(23 67)(24 68)(25 69)(26 70)(27 71)(28 72)(29 73)(30 74)(31 75)(32 76)(33 77)(34 78)(35 79)(36 80)(37 81)(38 82)(39 83)(40 84)(41 85)(42 86)(43 87)(44 88)
(1 43 21 32)(2 44 22 33)(3 34 12 23)(4 35 13 24)(5 36 14 25)(6 37 15 26)(7 38 16 27)(8 39 17 28)(9 40 18 29)(10 41 19 30)(11 42 20 31)(45 78 56 67)(46 79 57 68)(47 80 58 69)(48 81 59 70)(49 82 60 71)(50 83 61 72)(51 84 62 73)(52 85 63 74)(53 86 64 75)(54 87 65 76)(55 88 66 77)
(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 11)(2 10)(3 9)(4 8)(5 7)(12 18)(13 17)(14 16)(19 22)(20 21)(23 29)(24 28)(25 27)(30 33)(31 32)(34 40)(35 39)(36 38)(41 44)(42 43)(45 51)(46 50)(47 49)(52 55)(53 54)(56 62)(57 61)(58 60)(63 66)(64 65)(67 73)(68 72)(69 71)(74 77)(75 76)(78 84)(79 83)(80 82)(85 88)(86 87)
G:=sub<Sym(88)| (1,54)(2,55)(3,45)(4,46)(5,47)(6,48)(7,49)(8,50)(9,51)(10,52)(11,53)(12,56)(13,57)(14,58)(15,59)(16,60)(17,61)(18,62)(19,63)(20,64)(21,65)(22,66)(23,67)(24,68)(25,69)(26,70)(27,71)(28,72)(29,73)(30,74)(31,75)(32,76)(33,77)(34,78)(35,79)(36,80)(37,81)(38,82)(39,83)(40,84)(41,85)(42,86)(43,87)(44,88), (1,43,21,32)(2,44,22,33)(3,34,12,23)(4,35,13,24)(5,36,14,25)(6,37,15,26)(7,38,16,27)(8,39,17,28)(9,40,18,29)(10,41,19,30)(11,42,20,31)(45,78,56,67)(46,79,57,68)(47,80,58,69)(48,81,59,70)(49,82,60,71)(50,83,61,72)(51,84,62,73)(52,85,63,74)(53,86,64,75)(54,87,65,76)(55,88,66,77), (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), (1,11)(2,10)(3,9)(4,8)(5,7)(12,18)(13,17)(14,16)(19,22)(20,21)(23,29)(24,28)(25,27)(30,33)(31,32)(34,40)(35,39)(36,38)(41,44)(42,43)(45,51)(46,50)(47,49)(52,55)(53,54)(56,62)(57,61)(58,60)(63,66)(64,65)(67,73)(68,72)(69,71)(74,77)(75,76)(78,84)(79,83)(80,82)(85,88)(86,87)>;
G:=Group( (1,54)(2,55)(3,45)(4,46)(5,47)(6,48)(7,49)(8,50)(9,51)(10,52)(11,53)(12,56)(13,57)(14,58)(15,59)(16,60)(17,61)(18,62)(19,63)(20,64)(21,65)(22,66)(23,67)(24,68)(25,69)(26,70)(27,71)(28,72)(29,73)(30,74)(31,75)(32,76)(33,77)(34,78)(35,79)(36,80)(37,81)(38,82)(39,83)(40,84)(41,85)(42,86)(43,87)(44,88), (1,43,21,32)(2,44,22,33)(3,34,12,23)(4,35,13,24)(5,36,14,25)(6,37,15,26)(7,38,16,27)(8,39,17,28)(9,40,18,29)(10,41,19,30)(11,42,20,31)(45,78,56,67)(46,79,57,68)(47,80,58,69)(48,81,59,70)(49,82,60,71)(50,83,61,72)(51,84,62,73)(52,85,63,74)(53,86,64,75)(54,87,65,76)(55,88,66,77), (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), (1,11)(2,10)(3,9)(4,8)(5,7)(12,18)(13,17)(14,16)(19,22)(20,21)(23,29)(24,28)(25,27)(30,33)(31,32)(34,40)(35,39)(36,38)(41,44)(42,43)(45,51)(46,50)(47,49)(52,55)(53,54)(56,62)(57,61)(58,60)(63,66)(64,65)(67,73)(68,72)(69,71)(74,77)(75,76)(78,84)(79,83)(80,82)(85,88)(86,87) );
G=PermutationGroup([[(1,54),(2,55),(3,45),(4,46),(5,47),(6,48),(7,49),(8,50),(9,51),(10,52),(11,53),(12,56),(13,57),(14,58),(15,59),(16,60),(17,61),(18,62),(19,63),(20,64),(21,65),(22,66),(23,67),(24,68),(25,69),(26,70),(27,71),(28,72),(29,73),(30,74),(31,75),(32,76),(33,77),(34,78),(35,79),(36,80),(37,81),(38,82),(39,83),(40,84),(41,85),(42,86),(43,87),(44,88)], [(1,43,21,32),(2,44,22,33),(3,34,12,23),(4,35,13,24),(5,36,14,25),(6,37,15,26),(7,38,16,27),(8,39,17,28),(9,40,18,29),(10,41,19,30),(11,42,20,31),(45,78,56,67),(46,79,57,68),(47,80,58,69),(48,81,59,70),(49,82,60,71),(50,83,61,72),(51,84,62,73),(52,85,63,74),(53,86,64,75),(54,87,65,76),(55,88,66,77)], [(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)], [(1,11),(2,10),(3,9),(4,8),(5,7),(12,18),(13,17),(14,16),(19,22),(20,21),(23,29),(24,28),(25,27),(30,33),(31,32),(34,40),(35,39),(36,38),(41,44),(42,43),(45,51),(46,50),(47,49),(52,55),(53,54),(56,62),(57,61),(58,60),(63,66),(64,65),(67,73),(68,72),(69,71),(74,77),(75,76),(78,84),(79,83),(80,82),(85,88),(86,87)]])
C2×C4×D11 is a maximal subgroup of
D22⋊C8 C42⋊D11 Dic11⋊4D4 D22.D4 D22⋊D4 C4⋊C4⋊7D11 D44⋊C4 D22.5D4 C4⋊2D44 D22⋊Q8 D22⋊2Q8 C44⋊2D4 D22⋊3Q8
C2×C4×D11 is a maximal quotient of
C42⋊D11 C23.11D22 Dic11⋊4D4 Dic22⋊C4 C4⋊C4⋊7D11 D44⋊C4 D44.2C4 D44.C4
56 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 11A | ··· | 11E | 22A | ··· | 22O | 44A | ··· | 44T |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 11 | ··· | 11 | 22 | ··· | 22 | 44 | ··· | 44 |
| size | 1 | 1 | 1 | 1 | 11 | 11 | 11 | 11 | 1 | 1 | 1 | 1 | 11 | 11 | 11 | 11 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C4 | D11 | D22 | D22 | C4×D11 |
| kernel | C2×C4×D11 | C4×D11 | C2×Dic11 | C2×C44 | C22×D11 | D22 | C2×C4 | C4 | C22 | C2 |
| # reps | 1 | 4 | 1 | 1 | 1 | 8 | 5 | 10 | 5 | 20 |
Matrix representation of C2×C4×D11 ►in GL3(𝔽89) generated by
| 88 | 0 | 0 |
| 0 | 88 | 0 |
| 0 | 0 | 88 |
| 55 | 0 | 0 |
| 0 | 88 | 0 |
| 0 | 0 | 88 |
| 1 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 88 | 47 |
| 88 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 1 | 0 |
G:=sub<GL(3,GF(89))| [88,0,0,0,88,0,0,0,88],[55,0,0,0,88,0,0,0,88],[1,0,0,0,0,88,0,1,47],[88,0,0,0,0,1,0,1,0] >;
C2×C4×D11 in GAP, Magma, Sage, TeX
C_2\times C_4\times D_{11} % in TeX
G:=Group("C2xC4xD11"); // GroupNames label
G:=SmallGroup(176,28);
// by ID
G=gap.SmallGroup(176,28);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-11,42,4004]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^4=c^11=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations