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## G = C24⋊D11order 352 = 25·11

### 1st semidirect product of C24 and D11 acting via D11/C11=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C22 — C24⋊D11
 Chief series C1 — C11 — C22 — C2×C22 — C22×D11 — C2×C11⋊D4 — C24⋊D11
 Lower central C11 — C2×C22 — C24⋊D11
 Upper central C1 — C22 — C24

Generators and relations for C24⋊D11
G = < a,b,c,d,e,f | a2=b2=c2=d2=e11=f2=1, ab=ba, ac=ca, faf=ad=da, ae=ea, fbf=bc=cb, bd=db, be=eb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef=e-1 >

Subgroups: 602 in 130 conjugacy classes, 41 normal (8 characteristic)
C1, C2, C2, C4, C22, C22, C22, C2×C4, D4, C23, C23, C11, C22⋊C4, C2×D4, C24, D11, C22, C22, C22≀C2, Dic11, D22, C2×C22, C2×C22, C2×C22, C2×Dic11, C11⋊D4, C22×D11, C22×C22, C22×C22, C23.D11, C2×C11⋊D4, C23×C22, C24⋊D11
Quotients: C1, C2, C22, D4, C23, C2×D4, D11, C22≀C2, D22, C11⋊D4, C22×D11, C2×C11⋊D4, C24⋊D11

Smallest permutation representation of C24⋊D11
On 88 points
Generators in S88
```(1 43)(2 44)(3 34)(4 35)(5 36)(6 37)(7 38)(8 39)(9 40)(10 41)(11 42)(12 23)(13 24)(14 25)(15 26)(16 27)(17 28)(18 29)(19 30)(20 31)(21 32)(22 33)(45 67)(46 68)(47 69)(48 70)(49 71)(50 72)(51 73)(52 74)(53 75)(54 76)(55 77)(56 78)(57 79)(58 80)(59 81)(60 82)(61 83)(62 84)(63 85)(64 86)(65 87)(66 88)
(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 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 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 67)(46 68)(47 69)(48 70)(49 71)(50 72)(51 73)(52 74)(53 75)(54 76)(55 77)(56 78)(57 79)(58 80)(59 81)(60 82)(61 83)(62 84)(63 85)(64 86)(65 87)(66 88)
(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 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 55)(2 54)(3 53)(4 52)(5 51)(6 50)(7 49)(8 48)(9 47)(10 46)(11 45)(12 64)(13 63)(14 62)(15 61)(16 60)(17 59)(18 58)(19 57)(20 56)(21 66)(22 65)(23 75)(24 74)(25 73)(26 72)(27 71)(28 70)(29 69)(30 68)(31 67)(32 77)(33 76)(34 86)(35 85)(36 84)(37 83)(38 82)(39 81)(40 80)(41 79)(42 78)(43 88)(44 87)```

`G:=sub<Sym(88)| (1,43)(2,44)(3,34)(4,35)(5,36)(6,37)(7,38)(8,39)(9,40)(10,41)(11,42)(12,23)(13,24)(14,25)(15,26)(16,27)(17,28)(18,29)(19,30)(20,31)(21,32)(22,33)(45,67)(46,68)(47,69)(48,70)(49,71)(50,72)(51,73)(52,74)(53,75)(54,76)(55,77)(56,78)(57,79)(58,80)(59,81)(60,82)(61,83)(62,84)(63,85)(64,86)(65,87)(66,88), (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)(45,78)(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), (1,32)(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)(45,67)(46,68)(47,69)(48,70)(49,71)(50,72)(51,73)(52,74)(53,75)(54,76)(55,77)(56,78)(57,79)(58,80)(59,81)(60,82)(61,83)(62,84)(63,85)(64,86)(65,87)(66,88), (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), (1,2,3,4,5,6,7,8,9,10,11)(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), (1,55)(2,54)(3,53)(4,52)(5,51)(6,50)(7,49)(8,48)(9,47)(10,46)(11,45)(12,64)(13,63)(14,62)(15,61)(16,60)(17,59)(18,58)(19,57)(20,56)(21,66)(22,65)(23,75)(24,74)(25,73)(26,72)(27,71)(28,70)(29,69)(30,68)(31,67)(32,77)(33,76)(34,86)(35,85)(36,84)(37,83)(38,82)(39,81)(40,80)(41,79)(42,78)(43,88)(44,87)>;`

`G:=Group( (1,43)(2,44)(3,34)(4,35)(5,36)(6,37)(7,38)(8,39)(9,40)(10,41)(11,42)(12,23)(13,24)(14,25)(15,26)(16,27)(17,28)(18,29)(19,30)(20,31)(21,32)(22,33)(45,67)(46,68)(47,69)(48,70)(49,71)(50,72)(51,73)(52,74)(53,75)(54,76)(55,77)(56,78)(57,79)(58,80)(59,81)(60,82)(61,83)(62,84)(63,85)(64,86)(65,87)(66,88), (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)(45,78)(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), (1,32)(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)(45,67)(46,68)(47,69)(48,70)(49,71)(50,72)(51,73)(52,74)(53,75)(54,76)(55,77)(56,78)(57,79)(58,80)(59,81)(60,82)(61,83)(62,84)(63,85)(64,86)(65,87)(66,88), (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), (1,2,3,4,5,6,7,8,9,10,11)(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), (1,55)(2,54)(3,53)(4,52)(5,51)(6,50)(7,49)(8,48)(9,47)(10,46)(11,45)(12,64)(13,63)(14,62)(15,61)(16,60)(17,59)(18,58)(19,57)(20,56)(21,66)(22,65)(23,75)(24,74)(25,73)(26,72)(27,71)(28,70)(29,69)(30,68)(31,67)(32,77)(33,76)(34,86)(35,85)(36,84)(37,83)(38,82)(39,81)(40,80)(41,79)(42,78)(43,88)(44,87) );`

`G=PermutationGroup([[(1,43),(2,44),(3,34),(4,35),(5,36),(6,37),(7,38),(8,39),(9,40),(10,41),(11,42),(12,23),(13,24),(14,25),(15,26),(16,27),(17,28),(18,29),(19,30),(20,31),(21,32),(22,33),(45,67),(46,68),(47,69),(48,70),(49,71),(50,72),(51,73),(52,74),(53,75),(54,76),(55,77),(56,78),(57,79),(58,80),(59,81),(60,82),(61,83),(62,84),(63,85),(64,86),(65,87),(66,88)], [(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),(45,78),(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)], [(1,32),(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),(45,67),(46,68),(47,69),(48,70),(49,71),(50,72),(51,73),(52,74),(53,75),(54,76),(55,77),(56,78),(57,79),(58,80),(59,81),(60,82),(61,83),(62,84),(63,85),(64,86),(65,87),(66,88)], [(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)], [(1,2,3,4,5,6,7,8,9,10,11),(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)], [(1,55),(2,54),(3,53),(4,52),(5,51),(6,50),(7,49),(8,48),(9,47),(10,46),(11,45),(12,64),(13,63),(14,62),(15,61),(16,60),(17,59),(18,58),(19,57),(20,56),(21,66),(22,65),(23,75),(24,74),(25,73),(26,72),(27,71),(28,70),(29,69),(30,68),(31,67),(32,77),(33,76),(34,86),(35,85),(36,84),(37,83),(38,82),(39,81),(40,80),(41,79),(42,78),(43,88),(44,87)]])`

94 conjugacy classes

 class 1 2A 2B 2C 2D ··· 2I 2J 4A 4B 4C 11A ··· 11E 22A ··· 22BW order 1 2 2 2 2 ··· 2 2 4 4 4 11 ··· 11 22 ··· 22 size 1 1 1 1 2 ··· 2 44 44 44 44 2 ··· 2 2 ··· 2

94 irreducible representations

 dim 1 1 1 1 2 2 2 2 type + + + + + + + image C1 C2 C2 C2 D4 D11 D22 C11⋊D4 kernel C24⋊D11 C23.D11 C2×C11⋊D4 C23×C22 C2×C22 C24 C23 C22 # reps 1 3 3 1 6 5 15 60

Matrix representation of C24⋊D11 in GL4(𝔽89) generated by

 88 0 0 0 0 88 0 0 0 0 1 0 0 0 0 88
,
 1 0 0 0 0 88 0 0 0 0 88 0 0 0 0 1
,
 88 0 0 0 0 88 0 0 0 0 88 0 0 0 0 88
,
 1 0 0 0 0 1 0 0 0 0 88 0 0 0 0 88
,
 8 0 0 0 0 78 0 0 0 0 2 0 0 0 0 45
,
 0 78 0 0 8 0 0 0 0 0 0 45 0 0 2 0
`G:=sub<GL(4,GF(89))| [88,0,0,0,0,88,0,0,0,0,1,0,0,0,0,88],[1,0,0,0,0,88,0,0,0,0,88,0,0,0,0,1],[88,0,0,0,0,88,0,0,0,0,88,0,0,0,0,88],[1,0,0,0,0,1,0,0,0,0,88,0,0,0,0,88],[8,0,0,0,0,78,0,0,0,0,2,0,0,0,0,45],[0,8,0,0,78,0,0,0,0,0,0,2,0,0,45,0] >;`

C24⋊D11 in GAP, Magma, Sage, TeX

`C_2^4\rtimes D_{11}`
`% in TeX`

`G:=Group("C2^4:D11");`
`// GroupNames label`

`G:=SmallGroup(352,148);`
`// by ID`

`G=gap.SmallGroup(352,148);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-11,217,218,11525]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=e^11=f^2=1,a*b=b*a,a*c=c*a,f*a*f=a*d=d*a,a*e=e*a,f*b*f=b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f=e^-1>;`
`// generators/relations`

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