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## G = C2×C23.31D4order 128 = 27

### Direct product of C2 and C23.31D4

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — C2×C23.31D4
 Chief series C1 — C2 — C22 — C23 — C22×C4 — C23×C4 — C2×C22⋊Q8 — C2×C23.31D4
 Lower central C1 — C22 — C2×C4 — C2×C23.31D4
 Upper central C1 — C23 — C23×C4 — C2×C23.31D4
 Jennings C1 — C2 — C22 — C22×C4 — C2×C23.31D4

Generators and relations for C2×C23.31D4
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e4=d, f2=b, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, ebe-1=bcd, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=bcde3 >

Subgroups: 364 in 168 conjugacy classes, 60 normal (28 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×12], C22 [×3], C22 [×8], C22 [×12], C8 [×2], C2×C4 [×4], C2×C4 [×32], Q8 [×4], C23 [×3], C23 [×4], C23 [×4], C22⋊C4 [×4], C4⋊C4 [×2], C4⋊C4 [×5], C2×C8 [×4], C22×C4 [×6], C22×C4 [×14], C2×Q8 [×2], C2×Q8 [×3], C24, C2.C42 [×2], C2.C42, C22⋊C8 [×2], C22⋊C8, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C22⋊Q8 [×4], C22⋊Q8 [×2], C22×C8, C23×C4, C23×C4, C22×Q8, C23.31D4 [×4], C2×C2.C42, C2×C22⋊C8, C2×C22⋊Q8, C2×C23.31D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], SD16 [×2], Q16 [×2], C22×C4, C2×D4 [×2], C23⋊C4 [×2], Q8⋊C4 [×4], C4≀C2 [×2], C2×C22⋊C4, C2×SD16, C2×Q16, C23.31D4 [×4], C2×C23⋊C4, C2×Q8⋊C4, C2×C4≀C2, C2×C23.31D4

Smallest permutation representation of C2×C23.31D4
On 32 points
Generators in S32
(1 12)(2 13)(3 14)(4 15)(5 16)(6 9)(7 10)(8 11)(17 29)(18 30)(19 31)(20 32)(21 25)(22 26)(23 27)(24 28)
(2 26)(4 28)(6 30)(8 32)(9 18)(11 20)(13 22)(15 24)
(1 29)(2 30)(3 31)(4 32)(5 25)(6 26)(7 27)(8 28)(9 22)(10 23)(11 24)(12 17)(13 18)(14 19)(15 20)(16 21)
(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)(17 21)(18 22)(19 23)(20 24)(25 29)(26 30)(27 31)(28 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 5)(2 8 26 32)(3 27)(4 30 28 6)(7 31)(9 15 18 24)(10 19)(11 22 20 13)(12 16)(14 23)(17 21)(25 29)

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

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

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

38 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 4A 4B 4C 4D 4E ··· 4N 4O 4P 4Q 4R 8A ··· 8H order 1 2 ··· 2 2 2 2 2 4 4 4 4 4 ··· 4 4 4 4 4 8 ··· 8 size 1 1 ··· 1 2 2 2 2 2 2 2 2 4 ··· 4 8 8 8 8 4 ··· 4

38 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 4 type + + + + + + + - + image C1 C2 C2 C2 C2 C4 C4 C4 D4 D4 SD16 Q16 C4≀C2 C23⋊C4 kernel C2×C23.31D4 C23.31D4 C2×C2.C42 C2×C22⋊C8 C2×C22⋊Q8 C2×C4⋊C4 C22⋊Q8 C22×Q8 C22×C4 C24 C23 C23 C22 C22 # reps 1 4 1 1 1 2 4 2 3 1 4 4 8 2

Matrix representation of C2×C23.31D4 in GL6(𝔽17)

 1 0 0 0 0 0 0 1 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
,
 1 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 0 0 0 0 0 16
,
 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
,
 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
,
 0 10 0 0 0 0 12 10 0 0 0 0 0 0 12 5 0 0 0 0 12 12 0 0 0 0 0 0 0 4 0 0 0 0 16 0
,
 16 0 0 0 0 0 16 1 0 0 0 0 0 0 16 0 0 0 0 0 0 1 0 0 0 0 0 0 16 0 0 0 0 0 0 4

G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16],[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],[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],[0,12,0,0,0,0,10,10,0,0,0,0,0,0,12,12,0,0,0,0,5,12,0,0,0,0,0,0,0,16,0,0,0,0,4,0],[16,16,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,4] >;

C2×C23.31D4 in GAP, Magma, Sage, TeX

C_2\times C_2^3._{31}D_4
% in TeX

G:=Group("C2xC2^3.31D4");
// GroupNames label

G:=SmallGroup(128,231);
// by ID

G=gap.SmallGroup(128,231);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,232,1123,1018,248,1971]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^4=d,f^2=b,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,b*d=d*b,e*b*e^-1=b*c*d,b*f=f*b,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=b*c*d*e^3>;
// generators/relations

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