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## G = C2×D42order 128 = 27

### Direct product of C2, D4 and D4

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22 — C2×D42
 Chief series C1 — C2 — C22 — C23 — C24 — C25 — D4×C23 — C2×D42
 Lower central C1 — C22 — C2×D42
 Upper central C1 — C23 — C2×D42
 Jennings C1 — C22 — C2×D42

Generators and relations for C2×D42
G = < a,b,c,d,e | a2=b4=c2=d4=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 2428 in 1268 conjugacy classes, 484 normal (8 characteristic)
C1, C2, C2 [×6], C2 [×24], C4 [×8], C4 [×10], C22, C22 [×22], C22 [×152], C2×C4 [×22], C2×C4 [×42], D4 [×32], D4 [×104], C23, C23 [×32], C23 [×184], C42 [×4], C22⋊C4 [×32], C4⋊C4 [×8], C22×C4, C22×C4 [×22], C22×C4 [×16], C2×D4 [×80], C2×D4 [×180], C24 [×24], C24 [×40], C2×C42, C2×C22⋊C4 [×8], C2×C4⋊C4 [×2], C4×D4 [×16], C22≀C2 [×32], C4⋊D4 [×32], C41D4 [×8], C23×C4 [×4], C22×D4 [×48], C22×D4 [×32], C25 [×4], C2×C4×D4 [×2], C2×C22≀C2 [×4], C2×C4⋊D4 [×4], C2×C41D4, D42 [×16], D4×C23 [×4], C2×D42
Quotients: C1, C2 [×31], C22 [×155], D4 [×16], C23 [×155], C2×D4 [×56], C24 [×31], C22×D4 [×28], 2+ 1+4 [×2], C25, D42 [×4], D4×C23 [×2], C2×2+ 1+4, C2×D42

Smallest permutation representation of C2×D42
On 32 points
Generators in S32
(1 27)(2 28)(3 25)(4 26)(5 10)(6 11)(7 12)(8 9)(13 22)(14 23)(15 24)(16 21)(17 30)(18 31)(19 32)(20 29)
(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 9)(2 12)(3 11)(4 10)(5 26)(6 25)(7 28)(8 27)(13 30)(14 29)(15 32)(16 31)(17 22)(18 21)(19 24)(20 23)
(1 13 10 31)(2 14 11 32)(3 15 12 29)(4 16 9 30)(5 18 27 22)(6 19 28 23)(7 20 25 24)(8 17 26 21)
(1 29)(2 30)(3 31)(4 32)(5 24)(6 21)(7 22)(8 23)(9 14)(10 15)(11 16)(12 13)(17 28)(18 25)(19 26)(20 27)

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

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

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

50 conjugacy classes

 class 1 2A ··· 2G 2H ··· 2W 2X ··· 2AE 4A ··· 4H 4I ··· 4R order 1 2 ··· 2 2 ··· 2 2 ··· 2 4 ··· 4 4 ··· 4 size 1 1 ··· 1 2 ··· 2 4 ··· 4 2 ··· 2 4 ··· 4

50 irreducible representations

 dim 1 1 1 1 1 1 1 2 4 type + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 D4 2+ 1+4 kernel C2×D42 C2×C4×D4 C2×C22≀C2 C2×C4⋊D4 C2×C4⋊1D4 D42 D4×C23 C2×D4 C22 # reps 1 2 4 4 1 16 4 16 2

Matrix representation of C2×D42 in GL5(ℤ)

 -1 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 1 -2 0 0 0 1 -1 0 0 0 0 0 -1 0 0 0 0 0 -1
,
 1 0 0 0 0 0 1 -2 0 0 0 0 -1 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 -1 0 0 0 0 0 -1 0 0 0 0 0 -1 -2 0 0 0 1 1
,
 -1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 -1 -2 0 0 0 0 1

G:=sub<GL(5,Integers())| [-1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,-1,0,0,0,0,0,-1],[1,0,0,0,0,0,1,1,0,0,0,-2,-1,0,0,0,0,0,-1,0,0,0,0,0,-1],[1,0,0,0,0,0,1,0,0,0,0,-2,-1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,-1,0,0,0,0,0,-1,0,0,0,0,0,-1,1,0,0,0,-2,1],[-1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,-1,0,0,0,0,-2,1] >;

C2×D42 in GAP, Magma, Sage, TeX

C_2\times D_4^2
% in TeX

G:=Group("C2xD4^2");
// GroupNames label

G:=SmallGroup(128,2194);
// by ID

G=gap.SmallGroup(128,2194);
# by ID

G:=PCGroup([7,-2,2,2,2,2,-2,2,477,1430,570]);
// Polycyclic

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

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