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## G = C2×C42⋊6C4order 128 = 27

### Direct product of C2 and C42⋊6C4

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C4 — C2×C42⋊6C4
 Chief series C1 — C2 — C22 — C23 — C22×C4 — C23×C4 — C22×C42 — C2×C42⋊6C4
 Lower central C1 — C2 — C4 — C2×C42⋊6C4
 Upper central C1 — C22×C4 — C23×C4 — C2×C42⋊6C4
 Jennings C1 — C2 — C2 — C22×C4 — C2×C42⋊6C4

Generators and relations for C2×C426C4
G = < a,b,c,d | a2=b4=c4=d4=1, ab=ba, ac=ca, ad=da, dbd-1=bc=cb, dcd-1=c-1 >

Subgroups: 388 in 244 conjugacy classes, 116 normal (32 characteristic)
C1, C2, C2 [×6], C2 [×4], C4 [×4], C4 [×4], C4 [×12], C22 [×3], C22 [×8], C22 [×12], C8 [×4], C2×C4 [×8], C2×C4 [×20], C2×C4 [×36], C23 [×3], C23 [×4], C23 [×4], C42 [×4], C42 [×10], C22⋊C4 [×4], C4⋊C4 [×4], C4⋊C4 [×2], C2×C8 [×6], M4(2) [×4], M4(2) [×6], C22×C4 [×6], C22×C4 [×8], C22×C4 [×16], C24, C2×C42 [×6], C2×C42 [×4], C2×C22⋊C4, C2×C4⋊C4 [×2], C42⋊C2 [×4], C42⋊C2 [×2], C22×C8, C2×M4(2) [×6], C2×M4(2) [×3], C23×C4, C23×C4, C426C4 [×4], C22×C42, C2×C42⋊C2, C22×M4(2), C2×C426C4
Quotients: C1, C2 [×7], C4 [×12], C22 [×7], C2×C4 [×18], D4 [×6], Q8 [×2], C23, C42 [×4], C22⋊C4 [×12], C4⋊C4 [×12], C22×C4 [×3], C2×D4 [×3], C2×Q8, C2.C42 [×8], C4≀C2 [×4], C2×C42, C2×C22⋊C4 [×3], C2×C4⋊C4 [×3], C426C4 [×4], C2×C2.C42, C2×C4≀C2 [×2], C2×C426C4

Smallest permutation representation of C2×C426C4
On 32 points
Generators in S32
(1 14)(2 13)(3 12)(4 11)(5 16)(6 15)(7 10)(8 9)(17 21)(18 22)(19 23)(20 24)(25 32)(26 29)(27 30)(28 31)
(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 6 4 9)(2 5 3 10)(7 13 16 12)(8 14 15 11)(17 26 19 28)(18 27 20 25)(21 29 23 31)(22 30 24 32)
(1 30 2 21)(3 23 4 32)(5 31 6 22)(7 26 8 20)(9 24 10 29)(11 25 12 19)(13 17 14 27)(15 18 16 28)

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

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

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

56 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 4A ··· 4H 4I ··· 4AB 4AC ··· 4AJ 8A ··· 8H order 1 2 ··· 2 2 2 2 2 4 ··· 4 4 ··· 4 4 ··· 4 8 ··· 8 size 1 1 ··· 1 2 2 2 2 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 C4 C4 C4 C4 D4 Q8 D4 C4≀C2 kernel C2×C42⋊6C4 C42⋊6C4 C22×C42 C2×C42⋊C2 C22×M4(2) C2×C42 C2×C4⋊C4 C42⋊C2 C2×M4(2) C22×C4 C22×C4 C24 C22 # reps 1 4 1 1 1 8 4 4 8 5 2 1 16

Matrix representation of C2×C426C4 in GL4(𝔽17) generated by

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

C2×C426C4 in GAP, Magma, Sage, TeX

C_2\times C_4^2\rtimes_6C_4
% in TeX

G:=Group("C2xC4^2:6C4");
// GroupNames label

G:=SmallGroup(128,464);
// by ID

G=gap.SmallGroup(128,464);
# by ID

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

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

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