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## G = C2×C16⋊C22order 128 = 27

### Direct product of C2 and C16⋊C22

direct product, p-group, metabelian, nilpotent (class 4), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C8 — C2×C16⋊C22
 Chief series C1 — C2 — C4 — C8 — C2×C8 — C22×C8 — C22×D8 — C2×C16⋊C22
 Lower central C1 — C2 — C4 — C8 — C2×C16⋊C22
 Upper central C1 — C22 — C22×C4 — C22×C8 — C2×C16⋊C22
 Jennings C1 — C2 — C2 — C2 — C2 — C4 — C4 — C8 — C2×C16⋊C22

Generators and relations for C2×C16⋊C22
G = < a,b,c,d | a2=b16=c2=d2=1, ab=ba, ac=ca, ad=da, cbc=b7, dbd=b9, cd=dc >

Subgroups: 532 in 200 conjugacy classes, 92 normal (20 characteristic)
C1, C2, C2 [×2], C2 [×8], C4 [×2], C4 [×2], C4 [×2], C22, C22 [×2], C22 [×22], C8 [×2], C8 [×2], C2×C4 [×2], C2×C4 [×4], C2×C4 [×5], D4 [×17], Q8 [×3], C23, C23 [×11], C16 [×4], C2×C8 [×2], C2×C8 [×4], D8 [×6], D8 [×7], SD16 [×4], Q16 [×2], Q16, C22×C4, C22×C4, C2×D4 [×11], C2×Q8, C4○D4 [×6], C24, C2×C16 [×2], M5(2) [×4], D16 [×8], SD32 [×8], C22×C8, C2×D8, C2×D8 [×6], C2×D8 [×3], C2×SD16, C2×Q16, C4○D8 [×4], C4○D8 [×2], C22×D4, C2×C4○D4, C2×M5(2), C2×D16 [×2], C2×SD32 [×2], C16⋊C22 [×8], C22×D8, C2×C4○D8, C2×C16⋊C22
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D8 [×4], C2×D4 [×6], C24, C2×D8 [×6], C22×D4, C16⋊C22 [×2], C22×D8, C2×C16⋊C22

Smallest permutation representation of C2×C16⋊C22
On 32 points
Generators in S32
(1 21)(2 22)(3 23)(4 24)(5 25)(6 26)(7 27)(8 28)(9 29)(10 30)(11 31)(12 32)(13 17)(14 18)(15 19)(16 20)
(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)
(2 8)(3 15)(4 6)(5 13)(7 11)(10 16)(12 14)(17 25)(18 32)(19 23)(20 30)(22 28)(24 26)(27 31)
(2 10)(4 12)(6 14)(8 16)(18 26)(20 28)(22 30)(24 32)

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

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

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

32 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F ··· 2K 4A 4B 4C 4D 4E 4F 8A 8B 8C 8D 8E 8F 16A ··· 16H order 1 2 2 2 2 2 2 ··· 2 4 4 4 4 4 4 8 8 8 8 8 8 16 ··· 16 size 1 1 1 1 2 2 8 ··· 8 2 2 2 2 8 8 2 2 2 2 4 4 4 ··· 4

32 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 D8 D8 C16⋊C22 kernel C2×C16⋊C22 C2×M5(2) C2×D16 C2×SD32 C16⋊C22 C22×D8 C2×C4○D8 C2×C8 C22×C4 C2×C4 C23 C2 # reps 1 1 2 2 8 1 1 3 1 6 2 4

Matrix representation of C2×C16⋊C22 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
,
 5 12 0 0 0 0 6 1 0 0 0 0 0 0 16 16 16 15 0 0 0 0 1 0 0 0 14 3 0 0 0 0 9 6 0 1
,
 1 0 0 0 0 0 11 16 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 3 3 6 6 0 0 5 5 14 11
,
 16 0 0 0 0 0 0 16 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 16 0 0 0 16 16 0 16

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],[5,6,0,0,0,0,12,1,0,0,0,0,0,0,16,0,14,9,0,0,16,0,3,6,0,0,16,1,0,0,0,0,15,0,0,1],[1,11,0,0,0,0,0,16,0,0,0,0,0,0,0,1,3,5,0,0,1,0,3,5,0,0,0,0,6,14,0,0,0,0,6,11],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,16,0,0,0,1,0,16,0,0,0,0,16,0,0,0,0,0,0,16] >;

C2×C16⋊C22 in GAP, Magma, Sage, TeX

C_2\times C_{16}\rtimes C_2^2
% in TeX

G:=Group("C2xC16:C2^2");
// GroupNames label

G:=SmallGroup(128,2144);
// by ID

G=gap.SmallGroup(128,2144);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,-2,-2,253,1430,1684,851,242,4037,2028,124]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^16=c^2=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^7,d*b*d=b^9,c*d=d*c>;
// generators/relations

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