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## G = C22×C8⋊C22order 128 = 27

### Direct product of C22 and C8⋊C22

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C4 — C22×C8⋊C22
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C23×C4 — D4×C23 — C22×C8⋊C22
 Lower central C1 — C2 — C4 — C22×C8⋊C22
 Upper central C1 — C23 — C23×C4 — C22×C8⋊C22
 Jennings C1 — C2 — C2 — C4 — C22×C8⋊C22

Generators and relations for C22×C8⋊C22
G = < a,b,c,d,e | a2=b2=c8=d2=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd=c3, ece=c5, de=ed >

Subgroups: 1580 in 860 conjugacy classes, 436 normal (14 characteristic)
C1, C2, C2 [×6], C2 [×16], C4, C4 [×7], C4 [×4], C22 [×11], C22 [×92], C8 [×8], C2×C4 [×28], C2×C4 [×22], D4 [×12], D4 [×50], Q8 [×4], Q8 [×6], C23, C23 [×6], C23 [×98], C2×C8 [×12], M4(2) [×16], D8 [×32], SD16 [×32], C22×C4 [×2], C22×C4 [×12], C22×C4 [×13], C2×D4 [×34], C2×D4 [×57], C2×Q8 [×6], C2×Q8 [×3], C4○D4 [×16], C4○D4 [×24], C24, C24 [×23], C22×C8 [×2], C2×M4(2) [×12], C2×D8 [×24], C2×SD16 [×24], C8⋊C22 [×64], C23×C4, C23×C4, C22×D4, C22×D4 [×14], C22×D4 [×8], C22×Q8, C2×C4○D4 [×12], C2×C4○D4 [×6], C25, C22×M4(2), C22×D8 [×2], C22×SD16 [×2], C2×C8⋊C22 [×24], D4×C23, C22×C4○D4, C22×C8⋊C22
Quotients: C1, C2 [×31], C22 [×155], D4 [×8], C23 [×155], C2×D4 [×28], C24 [×31], C8⋊C22 [×4], C22×D4 [×14], C25, C2×C8⋊C22 [×6], D4×C23, C22×C8⋊C22

Smallest permutation representation of C22×C8⋊C22
On 32 points
Generators in S32
(1 23)(2 24)(3 17)(4 18)(5 19)(6 20)(7 21)(8 22)(9 30)(10 31)(11 32)(12 25)(13 26)(14 27)(15 28)(16 29)
(1 31)(2 32)(3 25)(4 26)(5 27)(6 28)(7 29)(8 30)(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 31)(2 26)(3 29)(4 32)(5 27)(6 30)(7 25)(8 28)(9 20)(10 23)(11 18)(12 21)(13 24)(14 19)(15 22)(16 17)
(1 23)(2 20)(3 17)(4 22)(5 19)(6 24)(7 21)(8 18)(9 26)(10 31)(11 28)(12 25)(13 30)(14 27)(15 32)(16 29)

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

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

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

44 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 2L ··· 2W 4A ··· 4H 4I 4J 4K 4L 8A ··· 8H order 1 2 ··· 2 2 2 2 2 2 ··· 2 4 ··· 4 4 4 4 4 8 ··· 8 size 1 1 ··· 1 2 2 2 2 4 ··· 4 2 ··· 2 4 4 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 C8⋊C22 kernel C22×C8⋊C22 C22×M4(2) C22×D8 C22×SD16 C2×C8⋊C22 D4×C23 C22×C4○D4 C22×C4 C24 C22 # reps 1 1 2 2 24 1 1 7 1 4

Matrix representation of C22×C8⋊C22 in GL8(ℤ)

 -1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1
,
 -1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1
,
 0 1 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 -1 0 0 0 0 0 0 -1 0 0 0
,
 -1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1

G:=sub<GL(8,Integers())| [-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1],[-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1],[0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0],[-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1] >;

C22×C8⋊C22 in GAP, Magma, Sage, TeX

C_2^2\times C_8\rtimes C_2^2
% in TeX

G:=Group("C2^2xC8:C2^2");
// GroupNames label

G:=SmallGroup(128,2310);
// by ID

G=gap.SmallGroup(128,2310);
# by ID

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

G:=Group<a,b,c,d,e|a^2=b^2=c^8=d^2=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,d*c*d=c^3,e*c*e=c^5,d*e=e*d>;
// generators/relations

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