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## G = C2×D4⋊5D4order 128 = 27

### Direct product of C2 and D4⋊5D4

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22 — C2×D4⋊5D4
 Chief series C1 — C2 — C22 — C23 — C24 — C25 — D4×C23 — C2×D4⋊5D4
 Lower central C1 — C22 — C2×D4⋊5D4
 Upper central C1 — C23 — C2×D4⋊5D4
 Jennings C1 — C22 — C2×D4⋊5D4

Generators and relations for C2×D45D4
G = < a,b,c,d,e | a2=b4=c2=d4=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, dcd-1=ece=b2c, ede=d-1 >

Subgroups: 1692 in 950 conjugacy classes, 444 normal (32 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C2×C4, C2×C4, D4, D4, Q8, C23, C23, C23, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C4○D4, C24, C24, C24, C2×C42, C2×C22⋊C4, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C4×D4, C22≀C2, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C23×C4, C23×C4, C22×D4, C22×D4, C22×D4, C22×Q8, C2×C4○D4, C2×C4○D4, C25, C22×C22⋊C4, C2×C4×D4, C2×C22≀C2, C2×C4⋊D4, C2×C4⋊D4, C2×C22⋊Q8, C2×C22.D4, C2×C4.4D4, D45D4, D4×C23, C22×C4○D4, C2×D45D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C22×D4, C2×C4○D4, 2+ 1+4, C25, D45D4, D4×C23, C22×C4○D4, C2×2+ 1+4, C2×D45D4

Smallest permutation representation of C2×D45D4
On 32 points
Generators in S32
(1 7)(2 8)(3 5)(4 6)(9 24)(10 21)(11 22)(12 23)(13 17)(14 18)(15 19)(16 20)(25 30)(26 31)(27 32)(28 29)
(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 23)(2 22)(3 21)(4 24)(5 10)(6 9)(7 12)(8 11)(13 25)(14 28)(15 27)(16 26)(17 30)(18 29)(19 32)(20 31)
(1 30 24 20)(2 31 21 17)(3 32 22 18)(4 29 23 19)(5 27 11 14)(6 28 12 15)(7 25 9 16)(8 26 10 13)
(1 20)(2 17)(3 18)(4 19)(5 14)(6 15)(7 16)(8 13)(9 25)(10 26)(11 27)(12 28)(21 31)(22 32)(23 29)(24 30)

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

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

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

50 conjugacy classes

 class 1 2A ··· 2G 2H ··· 2S 2T ··· 2Y 4A ··· 4L 4M ··· 4X order 1 2 ··· 2 2 ··· 2 2 ··· 2 4 ··· 4 4 ··· 4 size 1 1 ··· 1 2 ··· 2 4 ··· 4 2 ··· 2 4 ··· 4

50 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 2 2 4 type + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 D4 C4○D4 2+ 1+4 kernel C2×D4⋊5D4 C22×C22⋊C4 C2×C4×D4 C2×C22≀C2 C2×C4⋊D4 C2×C22⋊Q8 C2×C22.D4 C2×C4.4D4 D4⋊5D4 D4×C23 C22×C4○D4 C2×D4 C23 C22 # reps 1 2 2 2 3 1 2 1 16 1 1 8 8 2

Matrix representation of C2×D45D4 in GL6(𝔽5)

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

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

C2×D45D4 in GAP, Magma, Sage, TeX

C_2\times D_4\rtimes_5D_4
% in TeX

G:=Group("C2xD4:5D4");
// GroupNames label

G:=SmallGroup(128,2195);
// by ID

G=gap.SmallGroup(128,2195);
# by ID

G:=PCGroup([7,-2,2,2,2,2,-2,2,224,477,1430,570]);
// Polycyclic

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

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