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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 [×2], C2 [×6], C4 [×4], C4 [×14], C22 [×3], C22 [×10], C8 [×4], C2×C4 [×2], C2×C4 [×8], C2×C4 [×40], D4 [×4], D4 [×18], Q8 [×4], Q8 [×24], C23, C23 [×2], C42 [×2], C42, C4⋊C4 [×8], C2×C8 [×2], M4(2) [×4], M4(2) [×2], SD16 [×8], Q16 [×8], C22×C4, C22×C4 [×2], C22×C4 [×8], C2×D4 [×2], C2×D4 [×4], C2×Q8 [×8], C2×Q8 [×27], C4○D4 [×8], C4○D4 [×36], C4.10D4 [×4], C4≀C2 [×8], C2×C42, C2×C4⋊C4 [×2], C4⋊Q8 [×4], C4⋊Q8 [×2], C2×M4(2) [×2], C2×SD16 [×2], C2×Q16 [×2], C8.C22 [×8], C8.C22 [×4], C22×Q8 [×2], C22×Q8 [×2], C2×C4○D4 [×2], C2×C4○D4 [×4], 2- 1+4 [×4], 2- 1+4 [×6], C2×C4.10D4, C2×C4≀C2 [×2], D4.10D4 [×8], C2×C4⋊Q8, C2×C8.C22 [×2], C2×2- 1+4, C2×D4.10D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×12], C23 [×15], C2×D4 [×18], C24, C22≀C2 [×4], C22×D4 [×3], D4.10D4 [×2], C2×C22≀C2, C2×D4.10D4

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

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

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

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

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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