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## G = C24⋊7D4order 128 = 27

### 2nd semidirect product of C24 and D4 acting via D4/C2=C22

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C23 — C24⋊7D4
 Chief series C1 — C2 — C22 — C23 — C24 — C25 — D4×C23 — C24⋊7D4
 Lower central C1 — C23 — C24⋊7D4
 Upper central C1 — C23 — C24⋊7D4
 Jennings C1 — C23 — C24⋊7D4

Generators and relations for C247D4
G = < a,b,c,d,e,f | a2=b2=c2=d2=e4=f2=1, ab=ba, faf=ac=ca, ad=da, eae-1=acd, ebe-1=fbf=bc=cb, bd=db, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef=e-1 >

Subgroups: 1780 in 782 conjugacy classes, 140 normal (8 characteristic)
C1, C2, C2 [×6], C2 [×18], C4 [×12], C22, C22 [×14], C22 [×122], C2×C4 [×8], C2×C4 [×36], D4 [×80], C23, C23 [×22], C23 [×138], C22⋊C4 [×26], C22×C4 [×8], C22×C4 [×12], C2×D4 [×16], C2×D4 [×124], C24, C24 [×10], C24 [×34], C2.C42 [×4], C2×C22⋊C4 [×14], C22≀C2 [×16], C23×C4 [×2], C22×D4 [×12], C22×D4 [×24], C25, C25 [×2], C243C4, C23.23D4 [×4], C232D4 [×4], C2×C22≀C2 [×4], D4×C23 [×2], C247D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×20], C23 [×15], C2×D4 [×30], C24, C22≀C2 [×8], C22×D4 [×5], 2+ 1+4 [×2], C2×C22≀C2 [×2], C233D4, D42 [×4], C247D4

Smallest permutation representation of C247D4
On 32 points
Generators in S32
(1 29)(2 12)(3 31)(4 10)(5 18)(6 23)(7 20)(8 21)(9 16)(11 14)(13 32)(15 30)(17 28)(19 26)(22 25)(24 27)
(1 27)(2 13)(3 25)(4 15)(5 16)(6 28)(7 14)(8 26)(9 18)(10 30)(11 20)(12 32)(17 23)(19 21)(22 31)(24 29)
(1 5)(2 6)(3 7)(4 8)(9 24)(10 21)(11 22)(12 23)(13 28)(14 25)(15 26)(16 27)(17 32)(18 29)(19 30)(20 31)
(1 25)(2 26)(3 27)(4 28)(5 14)(6 15)(7 16)(8 13)(9 20)(10 17)(11 18)(12 19)(21 32)(22 29)(23 30)(24 31)
(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 2)(3 4)(5 6)(7 8)(9 17)(10 20)(11 19)(12 18)(13 16)(14 15)(21 31)(22 30)(23 29)(24 32)(25 26)(27 28)

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

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

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

38 conjugacy classes

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

38 irreducible representations

 dim 1 1 1 1 1 1 2 2 4 type + + + + + + + + + image C1 C2 C2 C2 C2 C2 D4 D4 2+ 1+4 kernel C24⋊7D4 C24⋊3C4 C23.23D4 C23⋊2D4 C2×C22≀C2 D4×C23 C2×D4 C24 C22 # reps 1 1 4 4 4 2 16 4 2

Matrix representation of C247D4 in GL6(𝔽5)

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

G:=sub<GL(6,GF(5))| [0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,4,0,0,0,0,1,0,0,0,0,0,0,0,3,0,0,0,0,0,2,2,0,0,0,0,0,0,0,1,0,0,0,0,4,0],[0,4,0,0,0,0,4,0,0,0,0,0,0,0,2,4,0,0,0,0,3,3,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

C247D4 in GAP, Magma, Sage, TeX

C_2^4\rtimes_7D_4
% in TeX

G:=Group("C2^4:7D4");
// GroupNames label

G:=SmallGroup(128,1135);
// by ID

G=gap.SmallGroup(128,1135);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,253,758,723,675]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=e^4=f^2=1,a*b=b*a,f*a*f=a*c=c*a,a*d=d*a,e*a*e^-1=a*c*d,e*b*e^-1=f*b*f=b*c=c*b,b*d=d*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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