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## G = C23.D7order 112 = 24·7

### The non-split extension by C23 of D7 acting via D7/C7=C2

Aliases: C23.D7, C22⋊Dic7, C14.11D4, C22.7D14, (C2×C14)⋊2C4, C72(C22⋊C4), C14.9(C2×C4), (C2×Dic7)⋊2C2, C2.3(C7⋊D4), C2.5(C2×Dic7), (C22×C14).2C2, (C2×C14).7C22, SmallGroup(112,18)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C14 — C23.D7
 Chief series C1 — C7 — C14 — C2×C14 — C2×Dic7 — C23.D7
 Lower central C7 — C14 — C23.D7
 Upper central C1 — C22 — C23

Generators and relations for C23.D7
G = < a,b,c,d,e | a2=b2=c2=d7=1, e2=b, ab=ba, eae-1=ac=ca, ad=da, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >

Smallest permutation representation of C23.D7
On 56 points
Generators in S56
```(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 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 49 20 35)(2 48 21 34)(3 47 15 33)(4 46 16 32)(5 45 17 31)(6 44 18 30)(7 43 19 29)(8 54 22 40)(9 53 23 39)(10 52 24 38)(11 51 25 37)(12 50 26 36)(13 56 27 42)(14 55 28 41)```

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

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

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

34 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 7A 7B 7C 14A ··· 14U order 1 2 2 2 2 2 4 4 4 4 7 7 7 14 ··· 14 size 1 1 1 1 2 2 14 14 14 14 2 2 2 2 ··· 2

34 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 type + + + + + - + image C1 C2 C2 C4 D4 D7 Dic7 D14 C7⋊D4 kernel C23.D7 C2×Dic7 C22×C14 C2×C14 C14 C23 C22 C22 C2 # reps 1 2 1 4 2 3 6 3 12

Matrix representation of C23.D7 in GL3(𝔽29) generated by

 28 0 0 0 1 0 0 0 28
,
 28 0 0 0 28 0 0 0 28
,
 1 0 0 0 28 0 0 0 28
,
 1 0 0 0 16 0 0 0 20
,
 12 0 0 0 0 20 0 13 0
`G:=sub<GL(3,GF(29))| [28,0,0,0,1,0,0,0,28],[28,0,0,0,28,0,0,0,28],[1,0,0,0,28,0,0,0,28],[1,0,0,0,16,0,0,0,20],[12,0,0,0,0,13,0,20,0] >;`

C23.D7 in GAP, Magma, Sage, TeX

`C_2^3.D_7`
`% in TeX`

`G:=Group("C2^3.D7");`
`// GroupNames label`

`G:=SmallGroup(112,18);`
`// by ID`

`G=gap.SmallGroup(112,18);`
`# by ID`

`G:=PCGroup([5,-2,-2,-2,-2,-7,20,101,2404]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^7=1,e^2=b,a*b=b*a,e*a*e^-1=a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^-1>;`
`// generators/relations`

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