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G = C24.97D4order 128 = 27

52nd non-split extension by C24 of D4 acting via D4/C2=C22

p-group, metabelian, nilpotent (class 2), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C23 — C24.97D4
 Chief series C1 — C2 — C22 — C23 — C24 — C22×D4 — C23.23D4 — C24.97D4
 Lower central C1 — C23 — C24.97D4
 Upper central C1 — C23 — C24.97D4
 Jennings C1 — C23 — C24.97D4

Generators and relations for C24.97D4
G = < a,b,c,d,e,f | a2=b2=c2=d2=e4=1, f2=d, ab=ba, faf-1=ac=ca, ad=da, eae-1=acd, ebe-1=bc=cb, bd=db, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=de-1 >

Subgroups: 836 in 351 conjugacy classes, 100 normal (14 characteristic)
C1, C2, C2 [×6], C2 [×9], C4 [×11], C22, C22 [×10], C22 [×55], C2×C4 [×41], D4 [×12], C23, C23 [×10], C23 [×55], C22⋊C4 [×26], C4⋊C4 [×4], C22×C4, C22×C4 [×10], C22×C4 [×8], C2×D4 [×14], C24 [×2], C24 [×6], C24 [×8], C2.C42 [×8], C2×C22⋊C4, C2×C22⋊C4 [×16], C2×C22⋊C4 [×4], C2×C4⋊C4 [×4], C22≀C2 [×4], C23×C4 [×2], C22×D4, C22×D4 [×2], C25, C243C4, C23.8Q8 [×2], C23.23D4 [×2], C23.10D4 [×4], C23.11D4 [×4], C22×C22⋊C4, C2×C22≀C2, C24.97D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×4], C24, C22.D4 [×4], C22×D4, C2×C4○D4 [×2], 2+ 1+4 [×4], C2×C22.D4, C233D4 [×2], C22.32C24 [×4], C24.97D4

Smallest permutation representation of C24.97D4
On 32 points
Generators in S32
```(2 30)(4 32)(5 18)(6 22)(7 20)(8 24)(10 28)(12 26)(13 23)(14 17)(15 21)(16 19)
(1 9)(2 30)(3 11)(4 32)(5 15)(6 19)(7 13)(8 17)(10 28)(12 26)(14 24)(16 22)(18 21)(20 23)(25 31)(27 29)
(1 27)(2 28)(3 25)(4 26)(5 21)(6 22)(7 23)(8 24)(9 29)(10 30)(11 31)(12 32)(13 20)(14 17)(15 18)(16 19)
(1 9)(2 10)(3 11)(4 12)(5 18)(6 19)(7 20)(8 17)(13 23)(14 24)(15 21)(16 22)(25 31)(26 32)(27 29)(28 30)
(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)
(1 16 9 22)(2 21 10 15)(3 14 11 24)(4 23 12 13)(5 30 18 28)(6 27 19 29)(7 32 20 26)(8 25 17 31)```

`G:=sub<Sym(32)| (2,30)(4,32)(5,18)(6,22)(7,20)(8,24)(10,28)(12,26)(13,23)(14,17)(15,21)(16,19), (1,9)(2,30)(3,11)(4,32)(5,15)(6,19)(7,13)(8,17)(10,28)(12,26)(14,24)(16,22)(18,21)(20,23)(25,31)(27,29), (1,27)(2,28)(3,25)(4,26)(5,21)(6,22)(7,23)(8,24)(9,29)(10,30)(11,31)(12,32)(13,20)(14,17)(15,18)(16,19), (1,9)(2,10)(3,11)(4,12)(5,18)(6,19)(7,20)(8,17)(13,23)(14,24)(15,21)(16,22)(25,31)(26,32)(27,29)(28,30), (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), (1,16,9,22)(2,21,10,15)(3,14,11,24)(4,23,12,13)(5,30,18,28)(6,27,19,29)(7,32,20,26)(8,25,17,31)>;`

`G:=Group( (2,30)(4,32)(5,18)(6,22)(7,20)(8,24)(10,28)(12,26)(13,23)(14,17)(15,21)(16,19), (1,9)(2,30)(3,11)(4,32)(5,15)(6,19)(7,13)(8,17)(10,28)(12,26)(14,24)(16,22)(18,21)(20,23)(25,31)(27,29), (1,27)(2,28)(3,25)(4,26)(5,21)(6,22)(7,23)(8,24)(9,29)(10,30)(11,31)(12,32)(13,20)(14,17)(15,18)(16,19), (1,9)(2,10)(3,11)(4,12)(5,18)(6,19)(7,20)(8,17)(13,23)(14,24)(15,21)(16,22)(25,31)(26,32)(27,29)(28,30), (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), (1,16,9,22)(2,21,10,15)(3,14,11,24)(4,23,12,13)(5,30,18,28)(6,27,19,29)(7,32,20,26)(8,25,17,31) );`

`G=PermutationGroup([(2,30),(4,32),(5,18),(6,22),(7,20),(8,24),(10,28),(12,26),(13,23),(14,17),(15,21),(16,19)], [(1,9),(2,30),(3,11),(4,32),(5,15),(6,19),(7,13),(8,17),(10,28),(12,26),(14,24),(16,22),(18,21),(20,23),(25,31),(27,29)], [(1,27),(2,28),(3,25),(4,26),(5,21),(6,22),(7,23),(8,24),(9,29),(10,30),(11,31),(12,32),(13,20),(14,17),(15,18),(16,19)], [(1,9),(2,10),(3,11),(4,12),(5,18),(6,19),(7,20),(8,17),(13,23),(14,24),(15,21),(16,22),(25,31),(26,32),(27,29),(28,30)], [(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)], [(1,16,9,22),(2,21,10,15),(3,14,11,24),(4,23,12,13),(5,30,18,28),(6,27,19,29),(7,32,20,26),(8,25,17,31)])`

32 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 2L 2M 2N 2O 2P 4A ··· 4H 4I ··· 4O order 1 2 ··· 2 2 2 2 2 2 2 2 2 2 4 ··· 4 4 ··· 4 size 1 1 ··· 1 2 2 2 2 4 4 4 4 8 4 ··· 4 8 ··· 8

32 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 4 type + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 D4 C4○D4 2+ 1+4 kernel C24.97D4 C24⋊3C4 C23.8Q8 C23.23D4 C23.10D4 C23.11D4 C22×C22⋊C4 C2×C22≀C2 C24 C23 C22 # reps 1 1 2 2 4 4 1 1 4 8 4

Matrix representation of C24.97D4 in GL8(𝔽5)

 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 4 0 0 0 0 0 1 0 0 4
,
 4 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 4 1 0 0 0 0 0 1 0 0 4
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 4
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 0 1 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 1 0 0 3 0 0 0 0 0 1 3 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 4
,
 4 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 1 3 0 0 0 0 0 1 0 0 3 0 0 0 0 0 0 0 4 0 0 0 0 0 0 4 0

`G:=sub<GL(8,GF(5))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,1,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,4,4,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,3,4,0,0,0,0,0,3,0,0,4],[4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,3,0,0,4,0,0,0,0,0,3,4,0] >;`

C24.97D4 in GAP, Magma, Sage, TeX

`C_2^4._{97}D_4`
`% in TeX`

`G:=Group("C2^4.97D4");`
`// GroupNames label`

`G:=SmallGroup(128,1354);`
`// by ID`

`G=gap.SmallGroup(128,1354);`
`# by ID`

`G:=PCGroup([7,-2,2,2,2,-2,2,2,253,232,758,723,185]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=e^4=1,f^2=d,a*b=b*a,f*a*f^-1=a*c=c*a,a*d=d*a,e*a*e^-1=a*c*d,e*b*e^-1=b*c=c*b,b*d=d*b,b*f=f*b,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=d*e^-1>;`
`// generators/relations`

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