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## G = C2×D4○SD16order 128 = 27

### Direct product of C2 and D4○SD16

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C4 — C2×D4○SD16
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C2×C4○D4 — C2×2+ 1+4 — C2×D4○SD16
 Lower central C1 — C2 — C4 — C2×D4○SD16
 Upper central C1 — C22 — C2×C4○D4 — C2×D4○SD16
 Jennings C1 — C2 — C2 — C4 — C2×D4○SD16

Generators and relations for C2×D4○SD16
G = < a,b,c,d,e | a2=b4=c2=e2=1, d4=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=d3 >

Subgroups: 1172 in 730 conjugacy classes, 428 normal (16 characteristic)
C1, C2, C2 [×2], C2 [×14], C4 [×2], C4 [×6], C4 [×8], C22, C22 [×6], C22 [×34], C8 [×8], C2×C4, C2×C4 [×15], C2×C4 [×40], D4 [×20], D4 [×40], Q8 [×12], Q8 [×16], C23 [×3], C23 [×22], C2×C8, C2×C8 [×15], M4(2) [×12], D8 [×12], SD16 [×40], Q16 [×12], C22×C4 [×3], C22×C4 [×9], C2×D4, C2×D4 [×18], C2×D4 [×39], C2×Q8 [×2], C2×Q8 [×15], C2×Q8 [×17], C4○D4 [×32], C4○D4 [×44], C24 [×3], C22×C8 [×3], C2×M4(2) [×3], C8○D4 [×8], C2×D8 [×3], C2×SD16, C2×SD16 [×33], C2×Q16 [×3], C4○D8 [×24], C8⋊C22 [×24], C8.C22 [×24], C22×D4 [×3], C22×D4 [×3], C22×Q8 [×3], C22×Q8, C2×C4○D4, C2×C4○D4 [×6], C2×C4○D4 [×4], 2+ 1+4 [×8], 2+ 1+4 [×4], 2- 1+4 [×8], 2- 1+4 [×4], C2×C8○D4, C22×SD16 [×3], C2×C4○D8 [×3], C2×C8⋊C22 [×3], C2×C8.C22 [×3], D4○SD16 [×16], C2×2+ 1+4, C2×2- 1+4, C2×D4○SD16
Quotients: C1, C2 [×31], C22 [×155], D4 [×8], C23 [×155], C2×D4 [×28], C24 [×31], C22×D4 [×14], C25, D4○SD16 [×2], D4×C23, C2×D4○SD16

Smallest permutation representation of C2×D4○SD16
On 32 points
Generators in S32
(1 26)(2 27)(3 28)(4 29)(5 30)(6 31)(7 32)(8 25)(9 23)(10 24)(11 17)(12 18)(13 19)(14 20)(15 21)(16 22)
(1 7 5 3)(2 8 6 4)(9 11 13 15)(10 12 14 16)(17 19 21 23)(18 20 22 24)(25 31 29 27)(26 32 30 28)
(1 15)(2 16)(3 9)(4 10)(5 11)(6 12)(7 13)(8 14)(17 30)(18 31)(19 32)(20 25)(21 26)(22 27)(23 28)(24 29)
(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 19)(2 22)(3 17)(4 20)(5 23)(6 18)(7 21)(8 24)(9 30)(10 25)(11 28)(12 31)(13 26)(14 29)(15 32)(16 27)

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

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

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

44 conjugacy classes

 class 1 2A 2B 2C 2D ··· 2I 2J ··· 2Q 4A ··· 4H 4I ··· 4P 8A 8B 8C 8D 8E ··· 8J order 1 2 2 2 2 ··· 2 2 ··· 2 4 ··· 4 4 ··· 4 8 8 8 8 8 ··· 8 size 1 1 1 1 2 ··· 2 4 ··· 4 2 ··· 2 4 ··· 4 2 2 2 2 4 ··· 4

44 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 4 type + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 D4 D4 D4 D4○SD16 kernel C2×D4○SD16 C2×C8○D4 C22×SD16 C2×C4○D8 C2×C8⋊C22 C2×C8.C22 D4○SD16 C2×2+ 1+4 C2×2- 1+4 C2×D4 C2×Q8 C4○D4 C2 # reps 1 1 3 3 3 3 16 1 1 3 1 4 4

Matrix representation of C2×D4○SD16 in GL6(𝔽17)

 16 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 0 0 0 0 0 0 16
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 16 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 16 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 16 0 0 0 0 16 0 0 0 0 1 0 0 0
,
 11 4 0 0 0 0 12 6 0 0 0 0 0 0 5 12 0 0 0 0 5 5 0 0 0 0 0 0 5 12 0 0 0 0 5 5
,
 16 0 0 0 0 0 14 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 16 0 0 1 0 0 0 0 0 0 16 0 0

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,16,0,0,0,0,16,0,0,0,0,1,0,0,0],[11,12,0,0,0,0,4,6,0,0,0,0,0,0,5,5,0,0,0,0,12,5,0,0,0,0,0,0,5,5,0,0,0,0,12,5],[16,14,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,1,0,0,0,0,0,0,16,0,0] >;

C2×D4○SD16 in GAP, Magma, Sage, TeX

C_2\times D_4\circ {\rm SD}_{16}
% in TeX

G:=Group("C2xD4oSD16");
// GroupNames label

G:=SmallGroup(128,2314);
// by ID

G=gap.SmallGroup(128,2314);
# by ID

G:=PCGroup([7,-2,2,2,2,2,-2,-2,448,477,521,4037,2028,124]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^4=c^2=e^2=1,d^4=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^3>;
// generators/relations

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