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## G = C23⋊4SD16order 128 = 27

### 2nd semidirect product of C23 and SD16 acting via SD16/C4=C22

p-group, metabelian, nilpotent (class 3), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — C23⋊4SD16
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C22×D4 — C2×C4⋊D4 — C23⋊4SD16
 Lower central C1 — C2 — C2×C4 — C23⋊4SD16
 Upper central C1 — C22 — C23×C4 — C23⋊4SD16
 Jennings C1 — C2 — C2 — C2×C4 — C23⋊4SD16

Generators and relations for C234SD16
G = < a,b,c,d,e | a2=b2=c2=d8=e2=1, ab=ba, eae=ac=ca, ad=da, dbd-1=bc=cb, be=eb, cd=dc, ce=ec, ede=d3 >

Subgroups: 532 in 234 conjugacy classes, 94 normal (28 characteristic)
C1, C2 [×3], C2 [×8], C4 [×2], C4 [×9], C22, C22 [×6], C22 [×20], C8 [×4], C2×C4 [×4], C2×C4 [×21], D4 [×14], Q8 [×6], C23 [×3], C23 [×4], C23 [×10], C22⋊C4 [×8], C4⋊C4 [×4], C4⋊C4 [×6], C2×C8 [×4], C2×C8 [×2], SD16 [×4], C22×C4 [×6], C22×C4 [×5], C2×D4 [×2], C2×D4 [×11], C2×Q8 [×2], C2×Q8 [×3], C24, C24, C22⋊C8 [×4], D4⋊C4 [×4], Q8⋊C4 [×4], C4.Q8 [×4], C2×C22⋊C4 [×2], C2×C4⋊C4 [×2], C2×C4⋊C4, C4⋊D4 [×4], C4⋊D4 [×2], C22⋊Q8 [×4], C22⋊Q8 [×2], C22×C8 [×2], C2×SD16 [×4], C23×C4, C22×D4, C22×D4, C22×Q8, C2×C22⋊C8, Q8⋊D4 [×2], C22⋊SD16 [×2], C88D4 [×4], C23.46D4 [×2], C23.47D4 [×2], C2×C4⋊D4, C2×C22⋊Q8, C234SD16
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], SD16 [×4], C2×D4 [×6], C24, C2×SD16 [×6], C22×D4, 2+ 1+4 [×2], C233D4, C22×SD16, D8⋊C22, C234SD16

Smallest permutation representation of C234SD16
On 32 points
Generators in S32
(1 19)(2 20)(3 21)(4 22)(5 23)(6 24)(7 17)(8 18)(9 26)(10 27)(11 28)(12 29)(13 30)(14 31)(15 32)(16 25)
(1 5)(2 26)(3 7)(4 28)(6 30)(8 32)(9 20)(10 14)(11 22)(12 16)(13 24)(15 18)(17 21)(19 23)(25 29)(27 31)
(1 29)(2 30)(3 31)(4 32)(5 25)(6 26)(7 27)(8 28)(9 24)(10 17)(11 18)(12 19)(13 20)(14 21)(15 22)(16 23)
(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 7)(3 5)(4 8)(9 24)(10 19)(11 22)(12 17)(13 20)(14 23)(15 18)(16 21)(25 31)(27 29)(28 32)

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

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

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

32 conjugacy classes

 class 1 2A 2B 2C 2D ··· 2I 2J 2K 4A 4B 4C 4D 4E 4F 4G ··· 4L 8A ··· 8H order 1 2 2 2 2 ··· 2 2 2 4 4 4 4 4 4 4 ··· 4 8 ··· 8 size 1 1 1 1 2 ··· 2 8 8 2 2 2 2 4 4 8 ··· 8 4 ··· 4

32 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 4 4 type + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 D4 D4 SD16 2+ 1+4 D8⋊C22 kernel C23⋊4SD16 C2×C22⋊C8 Q8⋊D4 C22⋊SD16 C8⋊8D4 C23.46D4 C23.47D4 C2×C4⋊D4 C2×C22⋊Q8 C22×C4 C24 C23 C4 C2 # reps 1 1 2 2 4 2 2 1 1 3 1 8 2 2

Matrix representation of C234SD16 in GL6(𝔽17)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 16 2 0 0 0 0 0 1 0 0 0 0 0 0 0 16 0 0 0 0 16 0
,
 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 16 0 0 0 0 0 0 16
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 0 0 16
,
 5 12 0 0 0 0 5 5 0 0 0 0 0 0 0 0 15 0 0 0 0 0 16 1 0 0 9 0 0 0 0 0 9 16 0 0
,
 0 16 0 0 0 0 16 0 0 0 0 0 0 0 16 0 0 0 0 0 16 1 0 0 0 0 0 0 1 0 0 0 0 0 0 16

G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,2,1,0,0,0,0,0,0,0,16,0,0,0,0,16,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[5,5,0,0,0,0,12,5,0,0,0,0,0,0,0,0,9,9,0,0,0,0,0,16,0,0,15,16,0,0,0,0,0,1,0,0],[0,16,0,0,0,0,16,0,0,0,0,0,0,0,16,16,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16] >;

C234SD16 in GAP, Magma, Sage, TeX

C_2^3\rtimes_4{\rm SD}_{16}
% in TeX

G:=Group("C2^3:4SD16");
// GroupNames label

G:=SmallGroup(128,1919);
// by ID

G=gap.SmallGroup(128,1919);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,758,219,675,4037,1027,124]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^8=e^2=1,a*b=b*a,e*a*e=a*c=c*a,a*d=d*a,d*b*d^-1=b*c=c*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^3>;
// generators/relations

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