direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C11⋊D4, C22⋊2D4, C23⋊D11, C22⋊2D22, D22⋊3C22, C22.10C23, Dic11⋊2C22, C11⋊3(C2×D4), (C2×C22)⋊3C22, (C22×C22)⋊2C2, (C2×Dic11)⋊4C2, (C22×D11)⋊3C2, C2.10(C22×D11), SmallGroup(176,36)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C11⋊D4
G = < a,b,c,d | a2=b11=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >
Subgroups: 260 in 54 conjugacy classes, 27 normal (11 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C2×C4, D4, C23, C23, C11, C2×D4, D11, C22, C22, C22, Dic11, D22, D22, C2×C22, C2×C22, C2×C22, C2×Dic11, C11⋊D4, C22×D11, C22×C22, C2×C11⋊D4
Quotients: C1, C2, C22, D4, C23, C2×D4, D11, D22, C11⋊D4, C22×D11, C2×C11⋊D4
►On 88 points
Generators in S88
(1 56)(2 57)(3 58)(4 59)(5 60)(6 61)(7 62)(8 63)(9 64)(10 65)(11 66)(12 45)(13 46)(14 47)(15 48)(16 49)(17 50)(18 51)(19 52)(20 53)(21 54)(22 55)(23 78)(24 79)(25 80)(26 81)(27 82)(28 83)(29 84)(30 85)(31 86)(32 87)(33 88)(34 67)(35 68)(36 69)(37 70)(38 71)(39 72)(40 73)(41 74)(42 75)(43 76)(44 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 23 12 34)(2 33 13 44)(3 32 14 43)(4 31 15 42)(5 30 16 41)(6 29 17 40)(7 28 18 39)(8 27 19 38)(9 26 20 37)(10 25 21 36)(11 24 22 35)(45 67 56 78)(46 77 57 88)(47 76 58 87)(48 75 59 86)(49 74 60 85)(50 73 61 84)(51 72 62 83)(52 71 63 82)(53 70 64 81)(54 69 65 80)(55 68 66 79)
(2 11)(3 10)(4 9)(5 8)(6 7)(13 22)(14 21)(15 20)(16 19)(17 18)(23 34)(24 44)(25 43)(26 42)(27 41)(28 40)(29 39)(30 38)(31 37)(32 36)(33 35)(46 55)(47 54)(48 53)(49 52)(50 51)(57 66)(58 65)(59 64)(60 63)(61 62)(67 78)(68 88)(69 87)(70 86)(71 85)(72 84)(73 83)(74 82)(75 81)(76 80)(77 79)
G:=sub<Sym(88)| (1,56)(2,57)(3,58)(4,59)(5,60)(6,61)(7,62)(8,63)(9,64)(10,65)(11,66)(12,45)(13,46)(14,47)(15,48)(16,49)(17,50)(18,51)(19,52)(20,53)(21,54)(22,55)(23,78)(24,79)(25,80)(26,81)(27,82)(28,83)(29,84)(30,85)(31,86)(32,87)(33,88)(34,67)(35,68)(36,69)(37,70)(38,71)(39,72)(40,73)(41,74)(42,75)(43,76)(44,77), (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), (1,23,12,34)(2,33,13,44)(3,32,14,43)(4,31,15,42)(5,30,16,41)(6,29,17,40)(7,28,18,39)(8,27,19,38)(9,26,20,37)(10,25,21,36)(11,24,22,35)(45,67,56,78)(46,77,57,88)(47,76,58,87)(48,75,59,86)(49,74,60,85)(50,73,61,84)(51,72,62,83)(52,71,63,82)(53,70,64,81)(54,69,65,80)(55,68,66,79), (2,11)(3,10)(4,9)(5,8)(6,7)(13,22)(14,21)(15,20)(16,19)(17,18)(23,34)(24,44)(25,43)(26,42)(27,41)(28,40)(29,39)(30,38)(31,37)(32,36)(33,35)(46,55)(47,54)(48,53)(49,52)(50,51)(57,66)(58,65)(59,64)(60,63)(61,62)(67,78)(68,88)(69,87)(70,86)(71,85)(72,84)(73,83)(74,82)(75,81)(76,80)(77,79)>;
G:=Group( (1,56)(2,57)(3,58)(4,59)(5,60)(6,61)(7,62)(8,63)(9,64)(10,65)(11,66)(12,45)(13,46)(14,47)(15,48)(16,49)(17,50)(18,51)(19,52)(20,53)(21,54)(22,55)(23,78)(24,79)(25,80)(26,81)(27,82)(28,83)(29,84)(30,85)(31,86)(32,87)(33,88)(34,67)(35,68)(36,69)(37,70)(38,71)(39,72)(40,73)(41,74)(42,75)(43,76)(44,77), (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), (1,23,12,34)(2,33,13,44)(3,32,14,43)(4,31,15,42)(5,30,16,41)(6,29,17,40)(7,28,18,39)(8,27,19,38)(9,26,20,37)(10,25,21,36)(11,24,22,35)(45,67,56,78)(46,77,57,88)(47,76,58,87)(48,75,59,86)(49,74,60,85)(50,73,61,84)(51,72,62,83)(52,71,63,82)(53,70,64,81)(54,69,65,80)(55,68,66,79), (2,11)(3,10)(4,9)(5,8)(6,7)(13,22)(14,21)(15,20)(16,19)(17,18)(23,34)(24,44)(25,43)(26,42)(27,41)(28,40)(29,39)(30,38)(31,37)(32,36)(33,35)(46,55)(47,54)(48,53)(49,52)(50,51)(57,66)(58,65)(59,64)(60,63)(61,62)(67,78)(68,88)(69,87)(70,86)(71,85)(72,84)(73,83)(74,82)(75,81)(76,80)(77,79) );
G=PermutationGroup([[(1,56),(2,57),(3,58),(4,59),(5,60),(6,61),(7,62),(8,63),(9,64),(10,65),(11,66),(12,45),(13,46),(14,47),(15,48),(16,49),(17,50),(18,51),(19,52),(20,53),(21,54),(22,55),(23,78),(24,79),(25,80),(26,81),(27,82),(28,83),(29,84),(30,85),(31,86),(32,87),(33,88),(34,67),(35,68),(36,69),(37,70),(38,71),(39,72),(40,73),(41,74),(42,75),(43,76),(44,77)], [(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)], [(1,23,12,34),(2,33,13,44),(3,32,14,43),(4,31,15,42),(5,30,16,41),(6,29,17,40),(7,28,18,39),(8,27,19,38),(9,26,20,37),(10,25,21,36),(11,24,22,35),(45,67,56,78),(46,77,57,88),(47,76,58,87),(48,75,59,86),(49,74,60,85),(50,73,61,84),(51,72,62,83),(52,71,63,82),(53,70,64,81),(54,69,65,80),(55,68,66,79)], [(2,11),(3,10),(4,9),(5,8),(6,7),(13,22),(14,21),(15,20),(16,19),(17,18),(23,34),(24,44),(25,43),(26,42),(27,41),(28,40),(29,39),(30,38),(31,37),(32,36),(33,35),(46,55),(47,54),(48,53),(49,52),(50,51),(57,66),(58,65),(59,64),(60,63),(61,62),(67,78),(68,88),(69,87),(70,86),(71,85),(72,84),(73,83),(74,82),(75,81),(76,80),(77,79)]])
C2×C11⋊D4 is a maximal subgroup of
C22.2D44 Dic11⋊4D4 C22⋊D44 D22.D4 D22⋊D4 Dic11.D4 C22.D44 C23.23D22 C44⋊7D4 C23⋊D22 C44⋊2D4 Dic11⋊D4 C44⋊D4 C24⋊D11 C2×D4×D11 D4⋊6D22
C2×C11⋊D4 is a maximal quotient of
C44.48D4 C23.23D22 C44⋊7D4 D44⋊6C22 C23.18D22 C44.17D4 C23⋊D22 C44⋊2D4 Dic11⋊D4 C44⋊D4 C44.C23 Dic11⋊Q8 D22⋊3Q8 C44.23D4 Q8⋊D22 D4.8D22 D4.9D22 C24⋊D11
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 11A | ··· | 11E | 22A | ··· | 22AI |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 11 | ··· | 11 | 22 | ··· | 22 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 22 | 22 | 22 | 22 | 2 | ··· | 2 | 2 | ··· | 2 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | D4 | D11 | D22 | C11⋊D4 |
| kernel | C2×C11⋊D4 | C2×Dic11 | C11⋊D4 | C22×D11 | C22×C22 | C22 | C23 | C22 | C2 |
| # reps | 1 | 1 | 4 | 1 | 1 | 2 | 5 | 15 | 20 |
Matrix representation of C2×C11⋊D4 ►in GL4(𝔽89) generated by
| 88 | 0 | 0 | 0 |
| 0 | 88 | 0 | 0 |
| 0 | 0 | 88 | 0 |
| 0 | 0 | 0 | 88 |
| 59 | 88 | 0 | 0 |
| 10 | 27 | 0 | 0 |
| 0 | 0 | 0 | 88 |
| 0 | 0 | 1 | 71 |
| 1 | 68 | 0 | 0 |
| 0 | 88 | 0 | 0 |
| 0 | 0 | 79 | 74 |
| 0 | 0 | 72 | 10 |
| 88 | 21 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 33 | 18 |
| 0 | 0 | 78 | 56 |
G:=sub<GL(4,GF(89))| [88,0,0,0,0,88,0,0,0,0,88,0,0,0,0,88],[59,10,0,0,88,27,0,0,0,0,0,1,0,0,88,71],[1,0,0,0,68,88,0,0,0,0,79,72,0,0,74,10],[88,0,0,0,21,1,0,0,0,0,33,78,0,0,18,56] >;
C2×C11⋊D4 in GAP, Magma, Sage, TeX
C_2\times C_{11}\rtimes D_4 % in TeX
G:=Group("C2xC11:D4"); // GroupNames label
G:=SmallGroup(176,36);
// by ID
G=gap.SmallGroup(176,36);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-11,182,4004]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^11=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations