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## G = C22×C4≀C2order 128 = 27

### Direct product of C22 and C4≀C2

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

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

Generators and relations for C22×C4≀C2
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 >

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

Smallest permutation representation of C22×C4≀C2
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)])

56 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 2L 2M 2N 2O 4A ··· 4H 4I ··· 4AB 4AC 4AD 4AE 4AF 8A ··· 8H order 1 2 ··· 2 2 2 2 2 2 2 2 2 4 ··· 4 4 ··· 4 4 4 4 4 8 ··· 8 size 1 1 ··· 1 2 2 2 2 4 4 4 4 1 ··· 1 2 ··· 2 4 4 4 4 4 ··· 4

56 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 type + + + + + + + image C1 C2 C2 C2 C2 C4 C4 C4 D4 D4 C4≀C2 kernel C22×C4≀C2 C2×C4≀C2 C22×C42 C22×M4(2) C22×C4○D4 C22×D4 C22×Q8 C2×C4○D4 C22×C4 C24 C22 # reps 1 12 1 1 1 2 2 12 7 1 16

Matrix representation of C22×C4≀C2 in GL5(𝔽17)

 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 4 0 0 0 13 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

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] >;

C22×C4≀C2 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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