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## G = C2×D8⋊C22order 128 = 27

### Direct product of C2 and D8⋊C22

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C4 — C2×D8⋊C22
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C23×C4 — C22×C4○D4 — C2×D8⋊C22
 Lower central C1 — C2 — C4 — C2×D8⋊C22
 Upper central C1 — C2×C4 — C23×C4 — C2×D8⋊C22
 Jennings C1 — C2 — C2 — C4 — C2×D8⋊C22

Generators and relations for C2×D8⋊C22
G = < a,b,c,d,e | a2=b8=c2=d2=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, dbd=b5, be=eb, dcd=ece=b4c, de=ed >

Subgroups: 1148 in 742 conjugacy classes, 428 normal (14 characteristic)
C1, C2, C2 [×2], C2 [×14], C4 [×2], C4 [×6], C4 [×8], C22, C22 [×6], C22 [×42], C8 [×8], C2×C4 [×2], C2×C4 [×26], C2×C4 [×44], D4 [×8], D4 [×44], Q8 [×8], Q8 [×12], C23, C23 [×6], C23 [×22], C2×C8 [×12], M4(2) [×16], D8 [×16], SD16 [×32], Q16 [×16], C22×C4 [×2], C22×C4 [×12], C22×C4 [×26], C2×D4 [×12], C2×D4 [×30], C2×Q8 [×12], C2×Q8 [×6], C4○D4 [×32], C4○D4 [×48], C24, C24 [×2], C22×C8 [×2], C2×M4(2) [×12], C2×D8 [×4], C2×SD16 [×8], C2×Q16 [×4], C4○D8 [×32], C8⋊C22 [×32], C8.C22 [×32], C23×C4, C23×C4 [×2], C22×D4 [×2], C22×D4 [×2], C22×Q8 [×2], C2×C4○D4 [×24], C2×C4○D4 [×12], C22×M4(2), C2×C4○D8 [×4], C2×C8⋊C22 [×4], C2×C8.C22 [×4], D8⋊C22 [×16], C22×C4○D4 [×2], C2×D8⋊C22
Quotients: C1, C2 [×31], C22 [×155], D4 [×8], C23 [×155], C2×D4 [×28], C24 [×31], C22×D4 [×14], C25, D8⋊C22 [×2], D4×C23, C2×D8⋊C22

Smallest permutation representation of C2×D8⋊C22
On 32 points
Generators in S32
(1 29)(2 30)(3 31)(4 32)(5 25)(6 26)(7 27)(8 28)(9 23)(10 24)(11 17)(12 18)(13 19)(14 20)(15 21)(16 22)
(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 7)(4 6)(10 16)(11 15)(12 14)(17 21)(18 20)(22 24)(26 32)(27 31)(28 30)
(1 21)(2 18)(3 23)(4 20)(5 17)(6 22)(7 19)(8 24)(9 31)(10 28)(11 25)(12 30)(13 27)(14 32)(15 29)(16 26)
(1 15)(2 16)(3 9)(4 10)(5 11)(6 12)(7 13)(8 14)(17 25)(18 26)(19 27)(20 28)(21 29)(22 30)(23 31)(24 32)

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

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

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

44 conjugacy classes

 class 1 2A 2B 2C 2D ··· 2I 2J ··· 2Q 4A 4B 4C 4D 4E ··· 4J 4K ··· 4R 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 4 ··· 4 1 1 1 1 2 ··· 2 4 ··· 4 4 ··· 4

44 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 4 type + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 D4 D4 D8⋊C22 kernel C2×D8⋊C22 C22×M4(2) C2×C4○D8 C2×C8⋊C22 C2×C8.C22 D8⋊C22 C22×C4○D4 C22×C4 C24 C2 # reps 1 1 4 4 4 16 2 7 1 4

Matrix representation of C2×D8⋊C22 in GL6(𝔽17)

 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 1 0 0 0 0 0 0 1
,
 0 16 0 0 0 0 1 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 13 13 0 0 4 8 0 0 0 0 0 13 0 0
,
 1 0 0 0 0 0 0 16 0 0 0 0 0 0 1 0 0 0 0 0 16 16 0 0 0 0 0 0 16 15 0 0 0 0 0 1
,
 16 0 0 0 0 0 0 16 0 0 0 0 0 0 4 8 0 0 0 0 13 13 0 0 0 0 0 0 4 8 0 0 0 0 13 13
,
 16 0 0 0 0 0 0 16 0 0 0 0 0 0 4 8 0 0 0 0 13 13 0 0 0 0 0 0 13 9 0 0 0 0 4 4

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,16,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,8,13,0,0,4,13,0,0,0,0,0,13,0,0],[1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,16,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,15,1],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,4,13,0,0,0,0,8,13,0,0,0,0,0,0,4,13,0,0,0,0,8,13],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,4,13,0,0,0,0,8,13,0,0,0,0,0,0,13,4,0,0,0,0,9,4] >;

C2×D8⋊C22 in GAP, Magma, Sage, TeX

C_2\times D_8\rtimes C_2^2
% in TeX

G:=Group("C2xD8:C2^2");
// GroupNames label

G:=SmallGroup(128,2312);
// by ID

G=gap.SmallGroup(128,2312);
# by ID

G:=PCGroup([7,-2,2,2,2,2,-2,-2,477,1430,248,4037,2028,124]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^8=c^2=d^2=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,d*b*d=b^5,b*e=e*b,d*c*d=e*c*e=b^4*c,d*e=e*d>;
// generators/relations

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