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## G = C2×C23.37D4order 128 = 27

### Direct product of C2 and C23.37D4

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C4 — C2×C23.37D4
 Chief series C1 — C2 — C22 — C2×C4 — C22×C4 — C23×C4 — D4×C23 — C2×C23.37D4
 Lower central C1 — C2 — C4 — C2×C23.37D4
 Upper central C1 — C23 — C23×C4 — C2×C23.37D4
 Jennings C1 — C2 — C2 — C2×C4 — C2×C23.37D4

Generators and relations for C2×C23.37D4
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e4=d, f2=c, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, ebe-1=fbf-1=bd=db, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=cde3 >

Subgroups: 1100 in 504 conjugacy classes, 180 normal (16 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, D4, C23, C23, C23, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C24, C24, D4⋊C4, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C42⋊C2, C22×C8, C2×M4(2), C2×M4(2), C23×C4, C22×D4, C22×D4, C25, C2×D4⋊C4, C23.37D4, C2×C42⋊C2, C22×M4(2), D4×C23, C2×C23.37D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C24, C2×C22⋊C4, C8⋊C22, C23×C4, C22×D4, C23.37D4, C22×C22⋊C4, C2×C8⋊C22, C2×C23.37D4

Smallest permutation representation of C2×C23.37D4
On 32 points
Generators in S32
(1 19)(2 20)(3 21)(4 22)(5 23)(6 24)(7 17)(8 18)(9 27)(10 28)(11 29)(12 30)(13 31)(14 32)(15 25)(16 26)
(1 29)(2 26)(3 31)(4 28)(5 25)(6 30)(7 27)(8 32)(9 17)(10 22)(11 19)(12 24)(13 21)(14 18)(15 23)(16 20)
(1 29)(2 30)(3 31)(4 32)(5 25)(6 26)(7 27)(8 28)(9 17)(10 18)(11 19)(12 20)(13 21)(14 22)(15 23)(16 24)
(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 8 29 28)(2 27 30 7)(3 6 31 26)(4 25 32 5)(9 12 17 20)(10 19 18 11)(13 16 21 24)(14 23 22 15)

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

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

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

44 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 2L ··· 2S 4A ··· 4H 4I ··· 4P 8A ··· 8H order 1 2 ··· 2 2 2 2 2 2 ··· 2 4 ··· 4 4 ··· 4 8 ··· 8 size 1 1 ··· 1 2 2 2 2 4 ··· 4 2 ··· 2 4 ··· 4 4 ··· 4

44 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 4 type + + + + + + + + + image C1 C2 C2 C2 C2 C2 C4 D4 D4 C8⋊C22 kernel C2×C23.37D4 C2×D4⋊C4 C23.37D4 C2×C42⋊C2 C22×M4(2) D4×C23 C22×D4 C22×C4 C24 C22 # reps 1 4 8 1 1 1 16 7 1 4

Matrix representation of C2×C23.37D4 in GL8(𝔽17)

 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16
,
 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 16
,
 1 9 0 0 0 0 0 0 13 16 0 0 0 0 0 0 0 0 4 2 0 0 0 0 0 0 1 13 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 16 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0
,
 16 8 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 4 2 0 0 0 0 0 0 0 13 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0

G:=sub<GL(8,GF(17))| [16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16],[16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16],[1,13,0,0,0,0,0,0,9,16,0,0,0,0,0,0,0,0,4,1,0,0,0,0,0,0,2,13,0,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0],[16,0,0,0,0,0,0,0,8,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,2,13,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0] >;

C2×C23.37D4 in GAP, Magma, Sage, TeX

C_2\times C_2^3._{37}D_4
% in TeX

G:=Group("C2xC2^3.37D4");
// GroupNames label

G:=SmallGroup(128,1625);
// by ID

G=gap.SmallGroup(128,1625);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,1430,2804,1411,172]);
// Polycyclic

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

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