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G = C2×C42⋊C22  order 128 = 27

Direct product of C2 and C42⋊C22

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

Aliases: C2×C42⋊C22, C42⋊5C23, C24.99D4, M4(2)⋊12C23, C4≀C2⋊15C22, (C22×D4)⋊29C4, C4.11(C23×C4), (C22×Q8)⋊23C4, (C2×C4).181C24, (C2×C42)⋊34C22, C4○D4.20C23, D4.22(C22×C4), C23.641(C2×D4), (C22×C4).782D4, C4.181(C22×D4), Q8.22(C22×C4), C4○(C42⋊C22), C22.28(C22×D4), C42⋊C2⋊76C22, C23.88(C22⋊C4), (C2×M4(2))⋊73C22, (C22×M4(2))⋊22C2, (C23×C4).516C22, (C22×C4).1499C23, (C2×C4≀C2)⋊29C2, (C2×C4○D4)⋊22C4, C4○D4⋊16(C2×C4), (C2×D4)⋊51(C2×C4), (C2×Q8)⋊42(C2×C4), C4.77(C2×C22⋊C4), (C2×C4).1407(C2×D4), (C2×C42⋊C2)⋊43C2, (C2×C4).246(C22×C4), (C22×C4).326(C2×C4), (C22×C4○D4).22C2, C2.43(C22×C22⋊C4), C22.24(C2×C22⋊C4), (C2×C4).286(C22⋊C4), (C2×C4○D4).276C22, SmallGroup(128,1632)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series
C1 — C4 — C2×C42⋊C22
Chief series
C1 — C2 — C4 — C2×C4 — C22×C4 — C23×C4 — C22×C4○D4 — C2×C42⋊C22
Lower central
C1 — C2 — C4 — C2×C42⋊C22
Upper central
C1 — C2×C4 — C23×C4 — C2×C42⋊C22
Jennings
C1 — C2 — C2 — C2×C4 — C2×C42⋊C22

Subgroups: 668 in 386 conjugacy classes, 172 normal (34 characteristic)
C1, C2, C2 [×2], C2 [×10], C4 [×4], C4 [×4], C4 [×8], C22 [×3], C22 [×4], C22 [×26], C8 [×4], C2×C4 [×8], C2×C4 [×20], C2×C4 [×30], D4 [×4], D4 [×22], Q8 [×4], Q8 [×6], C23 [×3], C23 [×4], C23 [×12], C42 [×4], C42 [×2], C22⋊C4 [×4], C4⋊C4 [×4], C2×C8 [×6], M4(2) [×4], M4(2) [×6], C22×C4 [×6], C22×C4 [×8], C22×C4 [×15], C2×D4 [×6], C2×D4 [×15], C2×Q8 [×6], C2×Q8 [×3], C4○D4 [×16], C4○D4 [×24], C24, C24, C4≀C2 [×16], C2×C42 [×2], C2×C22⋊C4, C2×C4⋊C4, C42⋊C2 [×4], C42⋊C2 [×2], C22×C8, C2×M4(2) [×6], C2×M4(2) [×3], C23×C4, C23×C4, C22×D4, C22×D4, C22×Q8, C2×C4○D4 [×12], C2×C4○D4 [×6], C2×C4≀C2 [×4], C42⋊C22 [×8], C2×C42⋊C2, C22×M4(2), C22×C4○D4, C2×C42⋊C22

Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], D4 [×8], C23 [×15], C22⋊C4 [×16], C22×C4 [×14], C2×D4 [×12], C24, C2×C22⋊C4 [×12], C23×C4, C22×D4 [×2], C42⋊C22 [×2], C22×C22⋊C4, C2×C42⋊C22

Generators and relations
 G = < a,b,c,d,e | a2=b4=c4=d2=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd=b-1c, ebe=bc2, cd=dc, ce=ec, de=ed >

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

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

(2 16)(3 25)(4 7)(5 28)(8 15)(10 20)(11 30)(12 24)(14 26)(18 31)(19 21)(22 29)

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

Matrix representation ►G ⊆ GL6(𝔽17)

1600000
0160000
001000
000100
000010
000001
,
0130000
1300000
000008
0000134
008000
0091600
,
100000
010000
004000
000400
000040
000004
,
1600000
010000
001000
0011600
0000016
0000160
,
1600000
0160000
0016000
0001600
000010
000001

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],[0,13,0,0,0,0,13,0,0,0,0,0,0,0,0,0,8,9,0,0,0,0,0,16,0,0,0,13,0,0,0,0,8,4,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,0,16,0,0,0,0,0,0,0,16,0,0,0,0,16,0],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;
 

44 conjugacy classes

class 1 2A2B2C2D···2I2J2K2L2M4A4B4C4D4E···4J4K···4V8A···8H
order12222···2222244444···44···48···8
size11112···2444411112···24···44···4

44 irreducible representations

dim111111111224
type++++++++
imageC1C2C2C2C2C2C4C4C4D4D4C42⋊C22
kernelC2×C42⋊C22C2×C4≀C2C42⋊C22C2×C42⋊C2C22×M4(2)C22×C4○D4C22×D4C22×Q8C2×C4○D4C22×C4C24C2
# reps1481112212714

In GAP, Magma, Sage, TeX 

C_2\times C_4^2\rtimes C_2^2
 
% in TeX
 
G:=Group("C2xC4^2:C2^2");
 
// GroupNames label
 
G:=SmallGroup(128,1632);
 
// by ID
 
G=gap.SmallGroup(128,1632);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,1430,2804,1411,172,124]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^2=b^4=c^4=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,d*b*d=b^-1*c,e*b*e=b*c^2,c*d=d*c,c*e=e*c,d*e=e*d>;
 
// generators/relations
 

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