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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, C2, C4, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, C23, C2×C8, M4(2), D8, SD16, Q16, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C4○D4, C4○D4, C24, C24, C22×C8, C2×M4(2), C2×D8, C2×SD16, C2×Q16, C4○D8, C8⋊C22, C8.C22, C23×C4, C23×C4, C22×D4, C22×D4, C22×Q8, C2×C4○D4, C2×C4○D4, C22×M4(2), C2×C4○D8, C2×C8⋊C22, C2×C8.C22, D8⋊C22, C22×C4○D4, C2×D8⋊C22
Quotients: C1, C2, C22, D4, C23, C2×D4, C24, C22×D4, C25, D8⋊C22, D4×C23, C2×D8⋊C22

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

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

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

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

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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