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## G = C4×C4≀C2order 128 = 27

### Direct product of C4 and C4≀C2

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C4 — C4×C4≀C2
 Chief series C1 — C2 — C22 — C23 — C22×C4 — C2×C42 — C43 — C4×C4≀C2
 Lower central C1 — C2 — C4 — C4×C4≀C2
 Upper central C1 — C42 — C2×C42 — C4×C4≀C2
 Jennings C1 — C2 — C2 — C22×C4 — C4×C4≀C2

Generators and relations for C4×C4≀C2
G = < a,b,c,d | a4=b4=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=b-1, bd=db, dcd-1=b-1c >

Subgroups: 300 in 180 conjugacy classes, 80 normal (20 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×2], C4 [×6], C4 [×14], C22, C22 [×2], C22 [×6], C8 [×4], C2×C4 [×2], C2×C4 [×8], C2×C4 [×27], D4 [×2], D4 [×5], Q8 [×2], Q8, C23, C23, C42 [×2], C42 [×6], C42 [×13], C22⋊C4 [×3], C4⋊C4 [×2], C4⋊C4 [×2], C2×C8 [×2], M4(2) [×4], M4(2) [×2], C22×C4, C22×C4 [×2], C22×C4 [×5], C2×D4, C2×D4, C2×Q8, C4○D4 [×4], C4○D4 [×2], C4×C8, C8⋊C4, C4≀C2 [×8], C2×C42, C2×C42 [×2], C2×C42 [×3], C42⋊C2, C42⋊C2, C4×D4 [×2], C4×D4 [×2], C4×Q8 [×2], C2×M4(2) [×2], C2×C4○D4, C426C4 [×2], C43, C4×M4(2), C2×C4≀C2 [×2], C4×C4○D4, C4×C4≀C2
Quotients: C1, C2 [×7], C4 [×12], C22 [×7], C2×C4 [×18], D4 [×4], C23, C42 [×4], C22⋊C4 [×4], C22×C4 [×3], C2×D4 [×2], C4○D4 [×2], C4≀C2 [×4], C2×C42, C2×C22⋊C4, C42⋊C2, C4×D4 [×4], C4×C22⋊C4, C2×C4≀C2 [×2], C4×C4≀C2

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

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

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

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

56 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A ··· 4L 4M ··· 4AH 4AI ··· 4AN 8A ··· 8H order 1 2 2 2 2 2 2 2 4 ··· 4 4 ··· 4 4 ··· 4 8 ··· 8 size 1 1 1 1 2 2 4 4 1 ··· 1 2 ··· 2 4 ··· 4 4 ··· 4

56 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 type + + + + + + + + image C1 C2 C2 C2 C2 C2 C4 C4 C4 D4 D4 C4○D4 C4≀C2 kernel C4×C4≀C2 C42⋊6C4 C43 C4×M4(2) C2×C4≀C2 C4×C4○D4 C4≀C2 C4×D4 C4×Q8 C42 C22×C4 C2×C4 C4 # reps 1 2 1 1 2 1 16 4 4 2 2 4 16

Matrix representation of C4×C4≀C2 in GL3(𝔽17) generated by

 4 0 0 0 4 0 0 0 4
,
 1 0 0 0 4 0 0 0 13
,
 16 0 0 0 0 1 0 1 0
,
 16 0 0 0 4 0 0 0 16
G:=sub<GL(3,GF(17))| [4,0,0,0,4,0,0,0,4],[1,0,0,0,4,0,0,0,13],[16,0,0,0,0,1,0,1,0],[16,0,0,0,4,0,0,0,16] >;

C4×C4≀C2 in GAP, Magma, Sage, TeX

C_4\times C_4\wr C_2
% in TeX

G:=Group("C4xC4wrC2");
// GroupNames label

G:=SmallGroup(128,490);
// by ID

G=gap.SmallGroup(128,490);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,112,141,232,100,2019,248,4037]);
// Polycyclic

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

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