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## G = C2×C22⋊SD16order 128 = 27

### Direct product of C2 and C22⋊SD16

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

Series: Derived Chief Lower central Upper central Jennings

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

Generators and relations for C2×C22⋊SD16
G = < a,b,c,d,e | a2=b2=c2=d8=e2=1, ab=ba, ac=ca, ad=da, ae=ea, dbd-1=bc=cb, be=eb, cd=dc, ce=ec, ede=d3 >

Subgroups: 1148 in 498 conjugacy classes, 124 normal (18 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, D4, Q8, C23, C23, C23, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, SD16, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C24, C24, C22⋊C8, D4⋊C4, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C22⋊Q8, C22⋊Q8, C22×C8, C2×SD16, C2×SD16, C23×C4, C22×D4, C22×D4, C22×Q8, C25, C2×C22⋊C8, C2×D4⋊C4, C22⋊SD16, C2×C22⋊Q8, C22×SD16, D4×C23, C2×C22⋊SD16
Quotients: C1, C2, C22, D4, C23, SD16, C2×D4, C24, C22≀C2, C2×SD16, C8⋊C22, C22×D4, C22⋊SD16, C2×C22≀C2, C22×SD16, C2×C8⋊C22, C2×C22⋊SD16

Smallest permutation representation of C2×C22⋊SD16
On 32 points
Generators in S32
(1 12)(2 13)(3 14)(4 15)(5 16)(6 9)(7 10)(8 11)(17 29)(18 30)(19 31)(20 32)(21 25)(22 26)(23 27)(24 28)
(2 30)(4 32)(6 26)(8 28)(9 22)(11 24)(13 18)(15 20)
(1 29)(2 30)(3 31)(4 32)(5 25)(6 26)(7 27)(8 28)(9 22)(10 23)(11 24)(12 17)(13 18)(14 19)(15 20)(16 21)
(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 27)(2 30)(3 25)(4 28)(5 31)(6 26)(7 29)(8 32)(9 22)(10 17)(11 20)(12 23)(13 18)(14 21)(15 24)(16 19)

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

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

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

38 conjugacy classes

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

38 irreducible representations

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

Matrix representation of C2×C22⋊SD16 in GL5(𝔽17)

 16 0 0 0 0 0 16 0 0 0 0 0 16 0 0 0 0 0 16 0 0 0 0 0 16
,
 16 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 7 16
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 16 0 0 0 0 0 16
,
 1 0 0 0 0 0 5 12 0 0 0 5 5 0 0 0 0 0 7 15 0 0 0 7 10
,
 16 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 16 0 0 0 0 0 16

G:=sub<GL(5,GF(17))| [16,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,16],[16,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,7,0,0,0,0,16],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,16],[1,0,0,0,0,0,5,5,0,0,0,12,5,0,0,0,0,0,7,7,0,0,0,15,10],[16,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,16] >;

C2×C22⋊SD16 in GAP, Magma, Sage, TeX

C_2\times C_2^2\rtimes {\rm SD}_{16}
% in TeX

G:=Group("C2xC2^2:SD16");
// GroupNames label

G:=SmallGroup(128,1729);
// by ID

G=gap.SmallGroup(128,1729);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,758,2804,1411,172]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^8=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*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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