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## G = C4×C22≀C2order 128 = 27

### Direct product of C4 and C22≀C2

direct product, p-group, metabelian, nilpotent (class 2), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22 — C4×C22≀C2
 Chief series C1 — C2 — C22 — C23 — C22×C4 — C23×C4 — C24×C4 — C4×C22≀C2
 Lower central C1 — C22 — C4×C22≀C2
 Upper central C1 — C22×C4 — C4×C22≀C2
 Jennings C1 — C23 — C4×C22≀C2

Generators and relations for C4×C22≀C2
G = < a,b,c,d,e,f | a4=b2=c2=d2=e2=f2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, fbf=bd=db, be=eb, cd=dc, fcf=ce=ec, de=ed, df=fd, ef=fe >

Subgroups: 1068 in 612 conjugacy classes, 184 normal (13 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C2×C4, C2×C4, D4, C23, C23, C23, C42, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C24, C24, C24, C2.C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C4×D4, C22≀C2, C23×C4, C23×C4, C23×C4, C22×D4, C25, C4×C22⋊C4, C243C4, C23.8Q8, C23.23D4, C2×C4×D4, C2×C22≀C2, C24×C4, C4×C22≀C2
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22×C4, C2×D4, C4○D4, C24, C4×D4, C22≀C2, C23×C4, C22×D4, C2×C4○D4, C2×C4×D4, C2×C22≀C2, C22.19C24, C4×C22≀C2

Smallest permutation representation of C4×C22≀C2
On 32 points
Generators in S32
(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 21)(2 22)(3 23)(4 24)(5 12)(6 9)(7 10)(8 11)(13 18)(14 19)(15 20)(16 17)(25 30)(26 31)(27 32)(28 29)
(1 11)(2 12)(3 9)(4 10)(5 22)(6 23)(7 24)(8 21)(13 18)(14 19)(15 20)(16 17)(25 30)(26 31)(27 32)(28 29)
(1 11)(2 12)(3 9)(4 10)(5 22)(6 23)(7 24)(8 21)(13 29)(14 30)(15 31)(16 32)(17 27)(18 28)(19 25)(20 26)
(1 21)(2 22)(3 23)(4 24)(5 12)(6 9)(7 10)(8 11)(13 28)(14 25)(15 26)(16 27)(17 32)(18 29)(19 30)(20 31)
(1 17)(2 18)(3 19)(4 20)(5 13)(6 14)(7 15)(8 16)(9 25)(10 26)(11 27)(12 28)(21 32)(22 29)(23 30)(24 31)

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

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

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

56 conjugacy classes

 class 1 2A ··· 2G 2H ··· 2S 2T 2U 4A ··· 4H 4I ··· 4T 4U ··· 4AH order 1 2 ··· 2 2 ··· 2 2 2 4 ··· 4 4 ··· 4 4 ··· 4 size 1 1 ··· 1 2 ··· 2 4 4 1 ··· 1 2 ··· 2 4 ··· 4

56 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 type + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C4 D4 C4○D4 kernel C4×C22≀C2 C4×C22⋊C4 C24⋊3C4 C23.8Q8 C23.23D4 C2×C4×D4 C2×C22≀C2 C24×C4 C22≀C2 C22×C4 C23 # reps 1 3 1 3 3 3 1 1 16 12 12

Matrix representation of C4×C22≀C2 in GL5(𝔽5)

 3 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 4 0 0 0 0 0 4
,
 4 0 0 0 0 0 1 0 0 0 0 0 4 0 0 0 0 0 4 0 0 0 0 0 1
,
 1 0 0 0 0 0 4 0 0 0 0 0 4 0 0 0 0 0 4 0 0 0 0 0 1
,
 1 0 0 0 0 0 4 0 0 0 0 0 4 0 0 0 0 0 4 0 0 0 0 0 4
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 4 0 0 0 0 0 4
,
 1 0 0 0 0 0 0 4 0 0 0 4 0 0 0 0 0 0 0 4 0 0 0 4 0

G:=sub<GL(5,GF(5))| [3,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,4,0,0,0,0,0,4],[4,0,0,0,0,0,1,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,1],[1,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,1],[1,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,4],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,4,0,0,0,0,0,4],[1,0,0,0,0,0,0,4,0,0,0,4,0,0,0,0,0,0,0,4,0,0,0,4,0] >;

C4×C22≀C2 in GAP, Magma, Sage, TeX

C_4\times C_2^2\wr C_2
% in TeX

G:=Group("C4xC2^2wrC2");
// GroupNames label

G:=SmallGroup(128,1031);
// by ID

G=gap.SmallGroup(128,1031);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,758,248]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^4=b^2=c^2=d^2=e^2=f^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,f*b*f=b*d=d*b,b*e=e*b,c*d=d*c,f*c*f=c*e=e*c,d*e=e*d,d*f=f*d,e*f=f*e>;
// generators/relations

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