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G = C22×C4≀C2  order 128 = 27

Direct product of C22 and C4≀C2

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

Aliases: C22×C4≀C2, C42⋊19C23, C24.171D4, M4(2)⋊11C23, D4⋊6(C22×C4), Q8⋊6(C22×C4), (C22×D4)⋊28C4, C4.10(C23×C4), (C22×Q8)⋊22C4, (C2×C4).180C24, (C2×C42)⋊86C22, (C22×C42)⋊18C2, C4○D4.19C23, (C22×C4).820D4, C23.640(C2×D4), C4.180(C22×D4), C22.27(C22×D4), (C2×M4(2))⋊72C22, (C22×M4(2))⋊21C2, (C23×C4).691C22, C23.132(C22⋊C4), (C22×C4).1498C23, C4○(C2×C4≀C2), (C2×C4)○C4≀C2, (C2×C4○D4)⋊21C4, C4○D4⋊15(C2×C4), (C2×D4)⋊50(C2×C4), (C2×Q8)⋊41(C2×C4), C4.76(C2×C22⋊C4), (C2×C4).1564(C2×D4), (C2×C4).462(C22×C4), (C22×C4).416(C2×C4), (C22×C4○D4).21C2, C2.42(C22×C22⋊C4), C22.23(C2×C22⋊C4), (C2×C4).285(C22⋊C4), (C2×C4○D4).275C22, (C2×C4)○(C2×C4≀C2), SmallGroup(128,1631)

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

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

Subgroups: 716 in 428 conjugacy classes, 180 normal (17 characteristic)
C1, C2, C2 [×6], C2 [×8], C4 [×2], C4 [×6], C4 [×12], C22, C22 [×10], C22 [×28], C8 [×4], C2×C4, C2×C4 [×27], C2×C4 [×50], D4 [×4], D4 [×22], Q8 [×4], Q8 [×6], C23, C23 [×6], C23 [×14], C42 [×4], C42 [×6], C2×C8 [×6], M4(2) [×4], M4(2) [×6], C22×C4 [×2], C22×C4 [×12], C22×C4 [×27], 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 [×6], C2×C42 [×3], C22×C8, C2×M4(2) [×6], C2×M4(2) [×3], C23×C4, C23×C4 [×2], C22×D4, C22×D4, C22×Q8, C2×C4○D4 [×12], C2×C4○D4 [×6], C2×C4≀C2 [×12], C22×C42, C22×M4(2), C22×C4○D4, C22×C4≀C2

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, C4≀C2 [×4], C2×C22⋊C4 [×12], C23×C4, C22×D4 [×2], C2×C4≀C2 [×6], C22×C22⋊C4, C22×C4≀C2

Generators and relations
 G = < a,b,c,d,e | a2=b2=c4=d2=e4=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd=c-1, ce=ec, ede-1=c-1d >

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

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

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

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

Matrix representation ►G ⊆ GL5(𝔽17)

160000
016000
001600
000160
000016
,
160000
01000
00100
00010
00001
,
10000
016000
001600
000130
00004
,
10000
00100
01000
00004
000130
,
160000
01000
001600
000160
00004

G:=sub<GL(5,GF(17))| [16,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,16],[16,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,13,0,0,0,0,0,4],[1,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,13,0,0,0,4,0],[16,0,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,4] >;
 

56 conjugacy classes

class 1 2A···2G2H2I2J2K2L2M2N2O4A···4H4I···4AB4AC4AD4AE4AF8A···8H
order12···2222222224···44···444448···8
size11···1222244441···12···244444···4

56 irreducible representations

dim11111111222
type+++++++
imageC1C2C2C2C2C4C4C4D4D4C4≀C2
kernelC22×C4≀C2C2×C4≀C2C22×C42C22×M4(2)C22×C4○D4C22×D4C22×Q8C2×C4○D4C22×C4C24C22
# reps11211122127116

In GAP, Magma, Sage, TeX 

C_2^2\times C_4\wr C_2
 
% in TeX
 
G:=Group("C2^2xC4wrC2");
 
// GroupNames label
 
G:=SmallGroup(128,1631);
 
// by ID
 
G=gap.SmallGroup(128,1631);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,2804,1411,172,124]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^2=b^2=c^4=d^2=e^4=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^-1,c*e=e*c,e*d*e^-1=c^-1*d>;
 
// generators/relations
 

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