direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C22⋊C4×D11, D22.10D4, C23.13D22, (C2×C4)⋊5D22, D22⋊C4⋊9C2, D22⋊5(C2×C4), C2.1(D4×D11), (C2×C44)⋊6C22, C22.17(C2×D4), C22⋊3(C4×D11), (C22×D11)⋊2C4, C22.6(C22×C4), C23.D11⋊3C2, (C2×C22).21C23, (C23×D11).1C2, (C2×Dic11)⋊5C22, (C22×C22).10C22, C22.13(C22×D11), (C22×D11).33C22, (C2×C4×D11)⋊8C2, C2.8(C2×C4×D11), (C2×C22)⋊1(C2×C4), C11⋊1(C2×C22⋊C4), (C11×C22⋊C4)⋊8C2, SmallGroup(352,75)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C22⋊C4×D11
G = < a,b,c,d,e | a2=b2=c4=d11=e2=1, cac-1=ab=ba, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >
Subgroups: 898 in 132 conjugacy classes, 49 normal (15 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C2×C4, C2×C4, C23, C23, C11, C22⋊C4, C22⋊C4, C22×C4, C24, D11, D11, C22, C22, C22, C2×C22⋊C4, Dic11, C44, D22, D22, C2×C22, C2×C22, C2×C22, C4×D11, C2×Dic11, C2×C44, C22×D11, C22×D11, C22×D11, C22×C22, D22⋊C4, C23.D11, C11×C22⋊C4, C2×C4×D11, C23×D11, C22⋊C4×D11
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, D11, C2×C22⋊C4, D22, C4×D11, C22×D11, C2×C4×D11, D4×D11, C22⋊C4×D11
►On 88 points
Generators in S88
(1 32)(2 33)(3 23)(4 24)(5 25)(6 26)(7 27)(8 28)(9 29)(10 30)(11 31)(12 34)(13 35)(14 36)(15 37)(16 38)(17 39)(18 40)(19 41)(20 42)(21 43)(22 44)(45 78)(46 79)(47 80)(48 81)(49 82)(50 83)(51 84)(52 85)(53 86)(54 87)(55 88)(56 67)(57 68)(58 69)(59 70)(60 71)(61 72)(62 73)(63 74)(64 75)(65 76)(66 77)
(1 21)(2 22)(3 12)(4 13)(5 14)(6 15)(7 16)(8 17)(9 18)(10 19)(11 20)(23 34)(24 35)(25 36)(26 37)(27 38)(28 39)(29 40)(30 41)(31 42)(32 43)(33 44)(45 56)(46 57)(47 58)(48 59)(49 60)(50 61)(51 62)(52 63)(53 64)(54 65)(55 66)(67 78)(68 79)(69 80)(70 81)(71 82)(72 83)(73 84)(74 85)(75 86)(76 87)(77 88)
(1 76 32 54)(2 77 33 55)(3 67 23 45)(4 68 24 46)(5 69 25 47)(6 70 26 48)(7 71 27 49)(8 72 28 50)(9 73 29 51)(10 74 30 52)(11 75 31 53)(12 78 34 56)(13 79 35 57)(14 80 36 58)(15 81 37 59)(16 82 38 60)(17 83 39 61)(18 84 40 62)(19 85 41 63)(20 86 42 64)(21 87 43 65)(22 88 44 66)
(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)| (1,32)(2,33)(3,23)(4,24)(5,25)(6,26)(7,27)(8,28)(9,29)(10,30)(11,31)(12,34)(13,35)(14,36)(15,37)(16,38)(17,39)(18,40)(19,41)(20,42)(21,43)(22,44)(45,78)(46,79)(47,80)(48,81)(49,82)(50,83)(51,84)(52,85)(53,86)(54,87)(55,88)(56,67)(57,68)(58,69)(59,70)(60,71)(61,72)(62,73)(63,74)(64,75)(65,76)(66,77), (1,21)(2,22)(3,12)(4,13)(5,14)(6,15)(7,16)(8,17)(9,18)(10,19)(11,20)(23,34)(24,35)(25,36)(26,37)(27,38)(28,39)(29,40)(30,41)(31,42)(32,43)(33,44)(45,56)(46,57)(47,58)(48,59)(49,60)(50,61)(51,62)(52,63)(53,64)(54,65)(55,66)(67,78)(68,79)(69,80)(70,81)(71,82)(72,83)(73,84)(74,85)(75,86)(76,87)(77,88), (1,76,32,54)(2,77,33,55)(3,67,23,45)(4,68,24,46)(5,69,25,47)(6,70,26,48)(7,71,27,49)(8,72,28,50)(9,73,29,51)(10,74,30,52)(11,75,31,53)(12,78,34,56)(13,79,35,57)(14,80,36,58)(15,81,37,59)(16,82,38,60)(17,83,39,61)(18,84,40,62)(19,85,41,63)(20,86,42,64)(21,87,43,65)(22,88,44,66), (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( (1,32)(2,33)(3,23)(4,24)(5,25)(6,26)(7,27)(8,28)(9,29)(10,30)(11,31)(12,34)(13,35)(14,36)(15,37)(16,38)(17,39)(18,40)(19,41)(20,42)(21,43)(22,44)(45,78)(46,79)(47,80)(48,81)(49,82)(50,83)(51,84)(52,85)(53,86)(54,87)(55,88)(56,67)(57,68)(58,69)(59,70)(60,71)(61,72)(62,73)(63,74)(64,75)(65,76)(66,77), (1,21)(2,22)(3,12)(4,13)(5,14)(6,15)(7,16)(8,17)(9,18)(10,19)(11,20)(23,34)(24,35)(25,36)(26,37)(27,38)(28,39)(29,40)(30,41)(31,42)(32,43)(33,44)(45,56)(46,57)(47,58)(48,59)(49,60)(50,61)(51,62)(52,63)(53,64)(54,65)(55,66)(67,78)(68,79)(69,80)(70,81)(71,82)(72,83)(73,84)(74,85)(75,86)(76,87)(77,88), (1,76,32,54)(2,77,33,55)(3,67,23,45)(4,68,24,46)(5,69,25,47)(6,70,26,48)(7,71,27,49)(8,72,28,50)(9,73,29,51)(10,74,30,52)(11,75,31,53)(12,78,34,56)(13,79,35,57)(14,80,36,58)(15,81,37,59)(16,82,38,60)(17,83,39,61)(18,84,40,62)(19,85,41,63)(20,86,42,64)(21,87,43,65)(22,88,44,66), (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([[(1,32),(2,33),(3,23),(4,24),(5,25),(6,26),(7,27),(8,28),(9,29),(10,30),(11,31),(12,34),(13,35),(14,36),(15,37),(16,38),(17,39),(18,40),(19,41),(20,42),(21,43),(22,44),(45,78),(46,79),(47,80),(48,81),(49,82),(50,83),(51,84),(52,85),(53,86),(54,87),(55,88),(56,67),(57,68),(58,69),(59,70),(60,71),(61,72),(62,73),(63,74),(64,75),(65,76),(66,77)], [(1,21),(2,22),(3,12),(4,13),(5,14),(6,15),(7,16),(8,17),(9,18),(10,19),(11,20),(23,34),(24,35),(25,36),(26,37),(27,38),(28,39),(29,40),(30,41),(31,42),(32,43),(33,44),(45,56),(46,57),(47,58),(48,59),(49,60),(50,61),(51,62),(52,63),(53,64),(54,65),(55,66),(67,78),(68,79),(69,80),(70,81),(71,82),(72,83),(73,84),(74,85),(75,86),(76,87),(77,88)], [(1,76,32,54),(2,77,33,55),(3,67,23,45),(4,68,24,46),(5,69,25,47),(6,70,26,48),(7,71,27,49),(8,72,28,50),(9,73,29,51),(10,74,30,52),(11,75,31,53),(12,78,34,56),(13,79,35,57),(14,80,36,58),(15,81,37,59),(16,82,38,60),(17,83,39,61),(18,84,40,62),(19,85,41,63),(20,86,42,64),(21,87,43,65),(22,88,44,66)], [(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)]])
70 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 11A | ··· | 11E | 22A | ··· | 22O | 22P | ··· | 22Y | 44A | ··· | 44T |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 11 | ··· | 11 | 22 | ··· | 22 | 22 | ··· | 22 | 44 | ··· | 44 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 11 | 11 | 11 | 11 | 22 | 22 | 2 | 2 | 2 | 2 | 22 | 22 | 22 | 22 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
70 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | D11 | D22 | D22 | C4×D11 | D4×D11 |
| kernel | C22⋊C4×D11 | D22⋊C4 | C23.D11 | C11×C22⋊C4 | C2×C4×D11 | C23×D11 | C22×D11 | D22 | C22⋊C4 | C2×C4 | C23 | C22 | C2 |
| # reps | 1 | 2 | 1 | 1 | 2 | 1 | 8 | 4 | 5 | 10 | 5 | 20 | 10 |
Matrix representation of C22⋊C4×D11 ►in GL4(𝔽89) generated by
| 88 | 0 | 0 | 0 |
| 0 | 88 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 21 | 88 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 88 | 0 |
| 0 | 0 | 0 | 88 |
| 34 | 0 | 0 | 0 |
| 0 | 34 | 0 | 0 |
| 0 | 0 | 18 | 11 |
| 0 | 0 | 3 | 71 |
| 42 | 1 | 0 | 0 |
| 67 | 44 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 44 | 88 | 0 | 0 |
| 66 | 45 | 0 | 0 |
| 0 | 0 | 88 | 0 |
| 0 | 0 | 0 | 88 |
G:=sub<GL(4,GF(89))| [88,0,0,0,0,88,0,0,0,0,1,21,0,0,0,88],[1,0,0,0,0,1,0,0,0,0,88,0,0,0,0,88],[34,0,0,0,0,34,0,0,0,0,18,3,0,0,11,71],[42,67,0,0,1,44,0,0,0,0,1,0,0,0,0,1],[44,66,0,0,88,45,0,0,0,0,88,0,0,0,0,88] >;
C22⋊C4×D11 in GAP, Magma, Sage, TeX
C_2^2\rtimes C_4\times D_{11} % in TeX
G:=Group("C2^2:C4xD11"); // GroupNames label
G:=SmallGroup(352,75);
// by ID
G=gap.SmallGroup(352,75);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-11,188,50,11525]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^4=d^11=e^2=1,c*a*c^-1=a*b=b*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations