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## G = C2×D4○D8order 128 = 27

### Direct product of C2 and D4○D8

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C4 — C2×D4○D8
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C2×C4○D4 — C2×2+ 1+4 — C2×D4○D8
 Lower central C1 — C2 — C4 — C2×D4○D8
 Upper central C1 — C22 — C2×C4○D4 — C2×D4○D8
 Jennings C1 — C2 — C2 — C4 — C2×D4○D8

Generators and relations for C2×D4○D8
G = < a,b,c,d,e | a2=b4=c2=e2=1, d4=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=b2d3 >

Subgroups: 1348 in 756 conjugacy classes, 428 normal (12 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, C2×C8, C2×C8, M4(2), D8, SD16, Q16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C4○D4, C4○D4, C24, C22×C8, C2×M4(2), C8○D4, C2×D8, C2×SD16, C2×Q16, C4○D8, C8⋊C22, C22×D4, C22×D4, C2×C4○D4, C2×C4○D4, C2×C4○D4, 2+ 1+4, 2+ 1+4, C2×C8○D4, C22×D8, C2×C4○D8, C2×C8⋊C22, D4○D8, C2×2+ 1+4, C2×D4○D8
Quotients: C1, C2, C22, D4, C23, C2×D4, C24, C22×D4, C25, D4○D8, D4×C23, C2×D4○D8

Smallest permutation representation of C2×D4○D8
On 32 points
Generators in S32
(1 28)(2 29)(3 30)(4 31)(5 32)(6 25)(7 26)(8 27)(9 22)(10 23)(11 24)(12 17)(13 18)(14 19)(15 20)(16 21)
(1 3 5 7)(2 4 6 8)(9 15 13 11)(10 16 14 12)(17 23 21 19)(18 24 22 20)(25 27 29 31)(26 28 30 32)
(1 14)(2 15)(3 16)(4 9)(5 10)(6 11)(7 12)(8 13)(17 26)(18 27)(19 28)(20 29)(21 30)(22 31)(23 32)(24 25)
(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 21)(2 20)(3 19)(4 18)(5 17)(6 24)(7 23)(8 22)(9 27)(10 26)(11 25)(12 32)(13 31)(14 30)(15 29)(16 28)

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

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

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

44 conjugacy classes

 class 1 2A 2B 2C 2D ··· 2I 2J ··· 2U 4A ··· 4H 4I 4J 4K 4L 8A 8B 8C 8D 8E ··· 8J order 1 2 2 2 2 ··· 2 2 ··· 2 4 ··· 4 4 4 4 4 8 8 8 8 8 ··· 8 size 1 1 1 1 2 ··· 2 4 ··· 4 2 ··· 2 4 4 4 4 2 2 2 2 4 ··· 4

44 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 4 type + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 D4 D4 D4 D4○D8 kernel C2×D4○D8 C2×C8○D4 C22×D8 C2×C4○D8 C2×C8⋊C22 D4○D8 C2×2+ 1+4 C2×D4 C2×Q8 C4○D4 C2 # reps 1 1 3 3 6 16 2 3 1 4 4

Matrix representation of C2×D4○D8 in GL6(𝔽17)

 16 0 0 0 0 0 0 16 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 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 16 0 0 0 0 1 0 0 0 0 0 16 16 16 15 0 0 0 1 1 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 1 2 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 16 16 16
,
 0 1 0 0 0 0 16 0 0 0 0 0 0 0 3 3 0 0 0 0 14 3 0 0 0 0 14 14 0 11 0 0 3 0 3 6
,
 16 0 0 0 0 0 0 1 0 0 0 0 0 0 16 16 16 15 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 1

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,16,0,0,0,16,0,16,1,0,0,0,0,16,1,0,0,0,0,15,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,1,16,0,0,1,1,0,16,0,0,2,0,0,16],[0,16,0,0,0,0,1,0,0,0,0,0,0,0,3,14,14,3,0,0,3,3,14,0,0,0,0,0,0,3,0,0,0,0,11,6],[16,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,16,0,1,0,0,0,16,1,0,0,0,0,15,0,0,1] >;

C2×D4○D8 in GAP, Magma, Sage, TeX

C_2\times D_4\circ D_8
% in TeX

G:=Group("C2xD4oD8");
// GroupNames label

G:=SmallGroup(128,2313);
// by ID

G=gap.SmallGroup(128,2313);
# by ID

G:=PCGroup([7,-2,2,2,2,2,-2,-2,477,521,4037,2028,124]);
// Polycyclic

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

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