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G = C22×C8⋊C22  order 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, D8⋊3C23, D4⋊2C24, C4.5C25, Q8⋊2C24, SD16⋊1C23, C24.185D4, M4(2)⋊4C23, (C2×C8)⋊4C23, C4○D4⋊5C23, (C2×D8)⋊54C22, (D4×C23)⋊18C2, (C22×D8)⋊22C2, (C2×D4)⋊21C23, (C2×Q8)⋊20C23, C2.40(D4×C23), C4.31(C22×D4), (C2×C4).611C24, (C22×C8)⋊26C22, (C22×SD16)⋊8C2, C23.710(C2×D4), (C22×C4).536D4, (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

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

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

Generators and relations
 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 >

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

(1 31)(2 32)(3 25)(4 26)(5 27)(6 28)(7 29)(8 30)(9 21)(10 22)(11 23)(12 24)(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)

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

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

Matrix representation ►G ⊆ 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] >;
 

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

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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