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## G = C24⋊D7order 224 = 25·7

### 1st semidirect product of C24 and D7 acting via D7/C7=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C14 — C24⋊D7
 Chief series C1 — C7 — C14 — C2×C14 — C22×D7 — C2×C7⋊D4 — C24⋊D7
 Lower central C7 — C2×C14 — C24⋊D7
 Upper central C1 — C22 — C24

Generators and relations for C24⋊D7
G = < a,b,c,d,e,f | a2=b2=c2=d2=e7=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: 446 in 130 conjugacy classes, 41 normal (8 characteristic)
C1, C2, C2, C4, C22, C22, C22, C7, C2×C4, D4, C23, C23, D7, C14, C14, C22⋊C4, C2×D4, C24, Dic7, D14, C2×C14, C2×C14, C2×C14, C22≀C2, C2×Dic7, C7⋊D4, C22×D7, C22×C14, C22×C14, C23.D7, C2×C7⋊D4, C23×C14, C24⋊D7
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, D14, C22≀C2, C7⋊D4, C22×D7, C2×C7⋊D4, C24⋊D7

Smallest permutation representation of C24⋊D7
On 56 points
Generators in S56
```(1 27)(2 28)(3 22)(4 23)(5 24)(6 25)(7 26)(8 15)(9 16)(10 17)(11 18)(12 19)(13 20)(14 21)(29 43)(30 44)(31 45)(32 46)(33 47)(34 48)(35 49)(36 50)(37 51)(38 52)(39 53)(40 54)(41 55)(42 56)
(1 13)(2 14)(3 8)(4 9)(5 10)(6 11)(7 12)(15 22)(16 23)(17 24)(18 25)(19 26)(20 27)(21 28)(29 50)(30 51)(31 52)(32 53)(33 54)(34 55)(35 56)(36 43)(37 44)(38 45)(39 46)(40 47)(41 48)(42 49)
(1 20)(2 21)(3 15)(4 16)(5 17)(6 18)(7 19)(8 22)(9 23)(10 24)(11 25)(12 26)(13 27)(14 28)(29 43)(30 44)(31 45)(32 46)(33 47)(34 48)(35 49)(36 50)(37 51)(38 52)(39 53)(40 54)(41 55)(42 56)
(1 13)(2 14)(3 8)(4 9)(5 10)(6 11)(7 12)(15 22)(16 23)(17 24)(18 25)(19 26)(20 27)(21 28)(29 36)(30 37)(31 38)(32 39)(33 40)(34 41)(35 42)(43 50)(44 51)(45 52)(46 53)(47 54)(48 55)(49 56)
(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)
(1 35)(2 34)(3 33)(4 32)(5 31)(6 30)(7 29)(8 40)(9 39)(10 38)(11 37)(12 36)(13 42)(14 41)(15 47)(16 46)(17 45)(18 44)(19 43)(20 49)(21 48)(22 54)(23 53)(24 52)(25 51)(26 50)(27 56)(28 55)```

`G:=sub<Sym(56)| (1,27)(2,28)(3,22)(4,23)(5,24)(6,25)(7,26)(8,15)(9,16)(10,17)(11,18)(12,19)(13,20)(14,21)(29,43)(30,44)(31,45)(32,46)(33,47)(34,48)(35,49)(36,50)(37,51)(38,52)(39,53)(40,54)(41,55)(42,56), (1,13)(2,14)(3,8)(4,9)(5,10)(6,11)(7,12)(15,22)(16,23)(17,24)(18,25)(19,26)(20,27)(21,28)(29,50)(30,51)(31,52)(32,53)(33,54)(34,55)(35,56)(36,43)(37,44)(38,45)(39,46)(40,47)(41,48)(42,49), (1,20)(2,21)(3,15)(4,16)(5,17)(6,18)(7,19)(8,22)(9,23)(10,24)(11,25)(12,26)(13,27)(14,28)(29,43)(30,44)(31,45)(32,46)(33,47)(34,48)(35,49)(36,50)(37,51)(38,52)(39,53)(40,54)(41,55)(42,56), (1,13)(2,14)(3,8)(4,9)(5,10)(6,11)(7,12)(15,22)(16,23)(17,24)(18,25)(19,26)(20,27)(21,28)(29,36)(30,37)(31,38)(32,39)(33,40)(34,41)(35,42)(43,50)(44,51)(45,52)(46,53)(47,54)(48,55)(49,56), (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), (1,35)(2,34)(3,33)(4,32)(5,31)(6,30)(7,29)(8,40)(9,39)(10,38)(11,37)(12,36)(13,42)(14,41)(15,47)(16,46)(17,45)(18,44)(19,43)(20,49)(21,48)(22,54)(23,53)(24,52)(25,51)(26,50)(27,56)(28,55)>;`

`G:=Group( (1,27)(2,28)(3,22)(4,23)(5,24)(6,25)(7,26)(8,15)(9,16)(10,17)(11,18)(12,19)(13,20)(14,21)(29,43)(30,44)(31,45)(32,46)(33,47)(34,48)(35,49)(36,50)(37,51)(38,52)(39,53)(40,54)(41,55)(42,56), (1,13)(2,14)(3,8)(4,9)(5,10)(6,11)(7,12)(15,22)(16,23)(17,24)(18,25)(19,26)(20,27)(21,28)(29,50)(30,51)(31,52)(32,53)(33,54)(34,55)(35,56)(36,43)(37,44)(38,45)(39,46)(40,47)(41,48)(42,49), (1,20)(2,21)(3,15)(4,16)(5,17)(6,18)(7,19)(8,22)(9,23)(10,24)(11,25)(12,26)(13,27)(14,28)(29,43)(30,44)(31,45)(32,46)(33,47)(34,48)(35,49)(36,50)(37,51)(38,52)(39,53)(40,54)(41,55)(42,56), (1,13)(2,14)(3,8)(4,9)(5,10)(6,11)(7,12)(15,22)(16,23)(17,24)(18,25)(19,26)(20,27)(21,28)(29,36)(30,37)(31,38)(32,39)(33,40)(34,41)(35,42)(43,50)(44,51)(45,52)(46,53)(47,54)(48,55)(49,56), (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), (1,35)(2,34)(3,33)(4,32)(5,31)(6,30)(7,29)(8,40)(9,39)(10,38)(11,37)(12,36)(13,42)(14,41)(15,47)(16,46)(17,45)(18,44)(19,43)(20,49)(21,48)(22,54)(23,53)(24,52)(25,51)(26,50)(27,56)(28,55) );`

`G=PermutationGroup([[(1,27),(2,28),(3,22),(4,23),(5,24),(6,25),(7,26),(8,15),(9,16),(10,17),(11,18),(12,19),(13,20),(14,21),(29,43),(30,44),(31,45),(32,46),(33,47),(34,48),(35,49),(36,50),(37,51),(38,52),(39,53),(40,54),(41,55),(42,56)], [(1,13),(2,14),(3,8),(4,9),(5,10),(6,11),(7,12),(15,22),(16,23),(17,24),(18,25),(19,26),(20,27),(21,28),(29,50),(30,51),(31,52),(32,53),(33,54),(34,55),(35,56),(36,43),(37,44),(38,45),(39,46),(40,47),(41,48),(42,49)], [(1,20),(2,21),(3,15),(4,16),(5,17),(6,18),(7,19),(8,22),(9,23),(10,24),(11,25),(12,26),(13,27),(14,28),(29,43),(30,44),(31,45),(32,46),(33,47),(34,48),(35,49),(36,50),(37,51),(38,52),(39,53),(40,54),(41,55),(42,56)], [(1,13),(2,14),(3,8),(4,9),(5,10),(6,11),(7,12),(15,22),(16,23),(17,24),(18,25),(19,26),(20,27),(21,28),(29,36),(30,37),(31,38),(32,39),(33,40),(34,41),(35,42),(43,50),(44,51),(45,52),(46,53),(47,54),(48,55),(49,56)], [(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)], [(1,35),(2,34),(3,33),(4,32),(5,31),(6,30),(7,29),(8,40),(9,39),(10,38),(11,37),(12,36),(13,42),(14,41),(15,47),(16,46),(17,45),(18,44),(19,43),(20,49),(21,48),(22,54),(23,53),(24,52),(25,51),(26,50),(27,56),(28,55)]])`

62 conjugacy classes

 class 1 2A 2B 2C 2D ··· 2I 2J 4A 4B 4C 7A 7B 7C 14A ··· 14AS order 1 2 2 2 2 ··· 2 2 4 4 4 7 7 7 14 ··· 14 size 1 1 1 1 2 ··· 2 28 28 28 28 2 2 2 2 ··· 2

62 irreducible representations

 dim 1 1 1 1 2 2 2 2 type + + + + + + + image C1 C2 C2 C2 D4 D7 D14 C7⋊D4 kernel C24⋊D7 C23.D7 C2×C7⋊D4 C23×C14 C2×C14 C24 C23 C22 # reps 1 3 3 1 6 3 9 36

Matrix representation of C24⋊D7 in GL4(𝔽29) generated by

 28 0 0 0 0 1 0 0 0 0 1 0 0 0 0 28
,
 28 0 0 0 0 28 0 0 0 0 28 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 28 0 0 0 0 28
,
 28 0 0 0 0 28 0 0 0 0 28 0 0 0 0 28
,
 23 0 0 0 0 24 0 0 0 0 20 0 0 0 0 16
,
 0 24 0 0 23 0 0 0 0 0 0 16 0 0 20 0
`G:=sub<GL(4,GF(29))| [28,0,0,0,0,1,0,0,0,0,1,0,0,0,0,28],[28,0,0,0,0,28,0,0,0,0,28,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,28,0,0,0,0,28],[28,0,0,0,0,28,0,0,0,0,28,0,0,0,0,28],[23,0,0,0,0,24,0,0,0,0,20,0,0,0,0,16],[0,23,0,0,24,0,0,0,0,0,0,20,0,0,16,0] >;`

C24⋊D7 in GAP, Magma, Sage, TeX

`C_2^4\rtimes D_7`
`% in TeX`

`G:=Group("C2^4:D7");`
`// GroupNames label`

`G:=SmallGroup(224,148);`
`// by ID`

`G=gap.SmallGroup(224,148);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-7,217,218,6917]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=e^7=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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