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## G = C2×D4.10D4order 128 = 27

### Direct product of C2 and D4.10D4

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — C2×D4.10D4
 Chief series C1 — C2 — C22 — C2×C4 — C22×C4 — C22×Q8 — C2×2- 1+4 — C2×D4.10D4
 Lower central C1 — C2 — C2×C4 — C2×D4.10D4
 Upper central C1 — C22 — C22×C4 — C2×D4.10D4
 Jennings C1 — C2 — C2 — C2×C4 — C2×D4.10D4

Generators and relations for C2×D4.10D4
G = < a,b,c,d,e | a2=b4=c2=1, d4=e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=dbd-1=ebe-1=b-1, dcd-1=bc, ece-1=b-1c, ede-1=d3 >

Subgroups: 628 in 354 conjugacy classes, 108 normal (16 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, C42, C42, C4⋊C4, C2×C8, M4(2), M4(2), SD16, Q16, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C4○D4, C4○D4, C4.10D4, C4≀C2, C2×C42, C2×C4⋊C4, C4⋊Q8, C4⋊Q8, C2×M4(2), C2×SD16, C2×Q16, C8.C22, C8.C22, C22×Q8, C22×Q8, C2×C4○D4, C2×C4○D4, 2- 1+4, 2- 1+4, C2×C4.10D4, C2×C4≀C2, D4.10D4, C2×C4⋊Q8, C2×C8.C22, C2×2- 1+4, C2×D4.10D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C24, C22≀C2, C22×D4, D4.10D4, C2×C22≀C2, C2×D4.10D4

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

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

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

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

32 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 4A 4B 4C 4D 4E ··· 4P 4Q 4R 8A 8B 8C 8D order 1 2 2 2 2 2 2 2 2 2 4 4 4 4 4 ··· 4 4 4 8 8 8 8 size 1 1 1 1 2 2 4 4 4 4 2 2 2 2 4 ··· 4 8 8 8 8 8 8

32 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 4 type + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 D4 D4 D4 D4 D4.10D4 kernel C2×D4.10D4 C2×C4.10D4 C2×C4≀C2 D4.10D4 C2×C4⋊Q8 C2×C8.C22 C2×2- 1+4 C22×C4 C2×D4 C2×Q8 C4○D4 C2 # reps 1 1 2 8 1 2 1 2 2 4 4 4

Matrix representation of C2×D4.10D4 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
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 16 0 0 0 0 0 1 1 16 15 0 0 16 0 1 1
,
 16 0 0 0 0 0 0 16 0 0 0 0 0 0 10 10 8 14 0 0 16 16 11 2 0 0 16 7 0 0 0 0 10 9 1 8
,
 0 16 0 0 0 0 1 0 0 0 0 0 0 0 16 16 11 2 0 0 7 7 9 3 0 0 1 10 0 0 0 0 6 7 10 11
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 10 16 0 0 0 0 16 7 0 0 0 0 10 10 8 14 0 0 0 7 16 9

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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,16,1,16,0,0,1,0,1,0,0,0,0,0,16,1,0,0,0,0,15,1],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,10,16,16,10,0,0,10,16,7,9,0,0,8,11,0,1,0,0,14,2,0,8],[0,1,0,0,0,0,16,0,0,0,0,0,0,0,16,7,1,6,0,0,16,7,10,7,0,0,11,9,0,10,0,0,2,3,0,11],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,10,16,10,0,0,0,16,7,10,7,0,0,0,0,8,16,0,0,0,0,14,9] >;

C2×D4.10D4 in GAP, Magma, Sage, TeX

C_2\times D_4._{10}D_4
% in TeX

G:=Group("C2xD4.10D4");
// GroupNames label

G:=SmallGroup(128,1749);
// by ID

G=gap.SmallGroup(128,1749);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,758,352,2804,1411,718,172,2028]);
// Polycyclic

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

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