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## G = C2×C4.10C42order 128 = 27

### Direct product of C2 and C4.10C42

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C4 — C2×C4.10C42
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C23×C4 — C22×M4(2) — C2×C4.10C42
 Lower central C1 — C4 — C2×C4.10C42
 Upper central C1 — C2×C4 — C2×C4.10C42
 Jennings C1 — C2 — C2 — C22×C4 — C2×C4.10C42

Generators and relations for C2×C4.10C42
G = < a,b,c,d | a2=b4=1, c4=d4=b2, ab=ba, ac=ca, ad=da, dcd-1=bc=cb, bd=db >

Subgroups: 276 in 186 conjugacy classes, 108 normal (12 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×2], C4 [×6], C22, C22 [×6], C22 [×10], C8 [×12], C2×C4, C2×C4 [×27], C23, C23 [×6], C23 [×2], C2×C8 [×12], C2×C8 [×12], M4(2) [×24], C22×C4 [×2], C22×C4 [×12], C24, C22×C8 [×6], C2×M4(2) [×12], C2×M4(2) [×12], C23×C4, C4.10C42 [×4], C22×M4(2) [×3], C2×C4.10C42
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], C2×C42, C2×C22⋊C4 [×3], C2×C4⋊C4 [×3], C4.10C42 [×2], C2×C2.C42, C2×C4.10C42

Smallest permutation representation of C2×C4.10C42
On 32 points
Generators in S32
(1 25)(2 26)(3 27)(4 28)(5 29)(6 30)(7 31)(8 32)(9 18)(10 19)(11 20)(12 21)(13 22)(14 23)(15 24)(16 17)
(1 31 5 27)(2 32 6 28)(3 25 7 29)(4 26 8 30)(9 20 13 24)(10 21 14 17)(11 22 15 18)(12 23 16 19)
(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 16 7 10 5 12 3 14)(2 24 4 22 6 20 8 18)(9 26 15 28 13 30 11 32)(17 31 19 29 21 27 23 25)

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

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

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

44 conjugacy classes

 class 1 2A 2B 2C 2D ··· 2I 4A 4B 4C 4D 4E ··· 4J 8A ··· 8X order 1 2 2 2 2 ··· 2 4 4 4 4 4 ··· 4 8 ··· 8 size 1 1 1 1 2 ··· 2 1 1 1 1 2 ··· 2 4 ··· 4

44 irreducible representations

 dim 1 1 1 1 1 2 2 2 4 type + + + + - - image C1 C2 C2 C4 C4 D4 Q8 Q8 C4.10C42 kernel C2×C4.10C42 C4.10C42 C22×M4(2) C22×C8 C2×M4(2) C22×C4 C22×C4 C24 C2 # reps 1 4 3 12 12 6 1 1 4

Matrix representation of C2×C4.10C42 in GL6(𝔽17)

 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 16 0 0 0 0 0 0 16
,
 16 0 0 0 0 0 0 16 0 0 0 0 0 0 13 0 0 0 0 0 0 13 0 0 0 0 0 0 13 0 0 0 0 0 0 13
,
 10 16 0 0 0 0 16 7 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 13 0 0 0 0 0 0 4 0 0
,
 0 13 0 0 0 0 4 0 0 0 0 0 0 0 0 1 0 0 0 0 4 0 0 0 0 0 0 0 0 13 0 0 0 0 1 0

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,13],[10,16,0,0,0,0,16,7,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,4,0,0,1,0,0,0,0,0,0,1,0,0],[0,4,0,0,0,0,13,0,0,0,0,0,0,0,0,4,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,13,0] >;

C2×C4.10C42 in GAP, Magma, Sage, TeX

C_2\times C_4._{10}C_4^2
% in TeX

G:=Group("C2xC4.10C4^2");
// GroupNames label

G:=SmallGroup(128,463);
// by ID

G=gap.SmallGroup(128,463);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,112,141,232,248,1411,172,4037,124]);
// Polycyclic

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

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