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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

Aliases: C22×C8⋊C22, C8⋊C24, D83C23, D42C24, C4.5C25, Q82C24, SD161C23, C24.185D4, M4(2)⋊4C23, (C2×C8)⋊4C23, C4○D45C23, (C2×D4)⋊21C23, (C22×D8)⋊22C2, (D4×C23)⋊18C2, (C2×D8)⋊54C22, (C2×Q8)⋊20C23, C4.31(C22×D4), C2.40(D4×C23), (C2×C4).611C24, (C22×C8)⋊26C22, (C22×SD16)⋊8C2, (C22×C4).536D4, C23.710(C2×D4), (C2×SD16)⋊59C22, (C22×D4)⋊65C22, (C22×M4(2))⋊6C2, (C22×Q8)⋊68C22, C22.52(C22×D4), (C2×M4(2))⋊56C22, (C23×C4).622C22, (C22×C4).1222C23, (C2×C4).667(C2×D4), (C22×C4○D4)⋊26C2, (C2×C4○D4)⋊77C22, SmallGroup(128,2310)

Series: Derived Chief Lower central Upper central Jennings

C1C4 — C22×C8⋊C22
C1C2C4C2×C4C22×C4C23×C4D4×C23 — C22×C8⋊C22
C1C2C4 — C22×C8⋊C22
C1C23C23×C4 — C22×C8⋊C22
C1C2C2C4 — 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···2G2H2I2J2K2L···2W4A···4H4I4J4K4L8A···8H
order12···222222···24···444448···8
size11···122224···42···244444···4

44 irreducible representations

dim1111111224
type++++++++++
imageC1C2C2C2C2C2C2D4D4C8⋊C22
kernelC22×C8⋊C22C22×M4(2)C22×D8C22×SD16C2×C8⋊C22D4×C23C22×C4○D4C22×C4C24C22
# reps11222411714

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

-10000000
0-1000000
00-100000
000-10000
0000-1000
00000-100
000000-10
0000000-1
,
-10000000
0-1000000
00100000
00010000
0000-1000
00000-100
000000-10
0000000-1
,
01000000
-10000000
00010000
00-100000
000000-10
00000001
00000-100
0000-1000
,
-10000000
01000000
00100000
000-10000
0000-1000
00000100
00000001
00000010
,
10000000
01000000
00100000
00010000
0000-1000
00000-100
00000010
00000001

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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