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## G = C2×C23.7D4order 128 = 27

### Direct product of C2 and C23.7D4

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C23 — C2×C23.7D4
 Chief series C1 — C2 — C22 — C23 — C24 — C22×D4 — C2×2+ 1+4 — C2×C23.7D4
 Lower central C1 — C2 — C23 — C2×C23.7D4
 Upper central C1 — C22 — C24 — C2×C23.7D4
 Jennings C1 — C2 — C23 — C2×C23.7D4

Generators and relations for C2×C23.7D4
G = < a,b,c,d,e,f | a2=b2=c2=d2=e4=1, f2=d, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, ebe-1=fbf-1=bcd, ece-1=cd=dc, cf=fc, de=ed, df=fd, fef-1=de-1 >

Subgroups: 836 in 397 conjugacy classes, 108 normal (8 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C2×C4, C2×C4, D4, Q8, C23, C23, C23, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C24, C24, C24, C23⋊C4, C2×C22⋊C4, C2×C22⋊C4, C2×C4⋊C4, C22.D4, C22.D4, C23×C4, C22×D4, C22×D4, C2×C4○D4, 2+ 1+4, 2+ 1+4, C2×C23⋊C4, C23.7D4, C2×C22.D4, C2×2+ 1+4, C2×C23.7D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C24, C22≀C2, C22×D4, C23.7D4, C2×C22≀C2, C2×C23.7D4

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

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

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

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

32 conjugacy classes

 class 1 2A 2B 2C 2D ··· 2I 2J ··· 2O 4A ··· 4J 4K ··· 4P order 1 2 2 2 2 ··· 2 2 ··· 2 4 ··· 4 4 ··· 4 size 1 1 1 1 2 ··· 2 4 ··· 4 4 ··· 4 8 ··· 8

32 irreducible representations

 dim 1 1 1 1 1 2 2 2 4 type + + + + + + + + image C1 C2 C2 C2 C2 D4 D4 D4 C23.7D4 kernel C2×C23.7D4 C2×C23⋊C4 C23.7D4 C2×C22.D4 C2×2+ 1+4 C22×C4 C2×D4 C24 C2 # reps 1 3 8 3 1 3 6 3 4

Matrix representation of C2×C23.7D4 in GL6(𝔽5)

 4 0 0 0 0 0 0 4 0 0 0 0 0 0 4 0 0 0 0 0 0 4 0 0 0 0 0 0 4 0 0 0 0 0 0 4
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 4 4 0 0 0 0 0 1 0 0 0 0 0 0 4 4 0 0 0 0 0 1
,
 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 4 0 0 0 0 0 0 4
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 4 0 0 0 0 0 0 4 0 0 0 0 0 0 4 0 0 0 0 0 0 4
,
 0 4 0 0 0 0 1 0 0 0 0 0 0 0 0 0 2 2 0 0 0 0 0 3 0 0 3 3 0 0 0 0 4 2 0 0
,
 4 0 0 0 0 0 0 1 0 0 0 0 0 0 3 0 0 0 0 0 4 2 0 0 0 0 0 0 3 0 0 0 0 0 0 3

G:=sub<GL(6,GF(5))| [4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,4,1,0,0,0,0,0,0,4,0,0,0,0,0,4,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[0,1,0,0,0,0,4,0,0,0,0,0,0,0,0,0,3,4,0,0,0,0,3,2,0,0,2,0,0,0,0,0,2,3,0,0],[4,0,0,0,0,0,0,1,0,0,0,0,0,0,3,4,0,0,0,0,0,2,0,0,0,0,0,0,3,0,0,0,0,0,0,3] >;

C2×C23.7D4 in GAP, Magma, Sage, TeX

C_2\times C_2^3._7D_4
% in TeX

G:=Group("C2xC2^3.7D4");
// GroupNames label

G:=SmallGroup(128,1756);
// by ID

G=gap.SmallGroup(128,1756);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,758,718,2028]);
// Polycyclic

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

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