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## G = C2×C42⋊C22order 128 = 27

### Direct product of C2 and C42⋊C22

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

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

Generators and relations for C2×C42⋊C22
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 >

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

Smallest permutation representation of C2×C42⋊C22
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)])

44 conjugacy classes

 class 1 2A 2B 2C 2D ··· 2I 2J 2K 2L 2M 4A 4B 4C 4D 4E ··· 4J 4K ··· 4V 8A ··· 8H order 1 2 2 2 2 ··· 2 2 2 2 2 4 4 4 4 4 ··· 4 4 ··· 4 8 ··· 8 size 1 1 1 1 2 ··· 2 4 4 4 4 1 1 1 1 2 ··· 2 4 ··· 4 4 ··· 4

44 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 4 type + + + + + + + + image C1 C2 C2 C2 C2 C2 C4 C4 C4 D4 D4 C42⋊C22 kernel C2×C42⋊C22 C2×C4≀C2 C42⋊C22 C2×C42⋊C2 C22×M4(2) C22×C4○D4 C22×D4 C22×Q8 C2×C4○D4 C22×C4 C24 C2 # reps 1 4 8 1 1 1 2 2 12 7 1 4

Matrix representation of C2×C42⋊C22 in GL6(𝔽17)

 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 0 8 0 0 0 0 13 4 0 0 8 0 0 0 0 0 9 16 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 0 0 0 0 0 1 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

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] >;

C2×C42⋊C22 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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