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G = C2×D4○D8  order 128 = 27

Direct product of C2 and D4○D8

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

Aliases: C2×D4○D8, D8⋊8C23, C4.8C25, C8.3C24, Q16⋊8C23, D4.5C24, Q8.5C24, SD16⋊4C23, M4(2)⋊7C23, 2+ (1+4)⋊8C22, D8○(C2×D4), D4○(C2×D8), Q8○(C2×Q16), Q16○(C2×Q8), (C2×C8)⋊6C23, C4○D4.38D4, D4.61(C2×D4), C4○D8⋊9C22, C4○D4⋊1C23, Q8.63(C2×D4), (C2×D4).357D4, C8○D4⋊13C22, (C2×D8)⋊56C22, (C2×D4)⋊11C23, (C22×D8)⋊23C2, (C2×Q8).276D4, C2.43(D4×C23), C8⋊C22⋊12C22, (C2×C4).614C24, (C22×C8)⋊28C22, (C2×Q16)⋊64C22, C4.125(C22×D4), C23.485(C2×D4), (C2×SD16)⋊62C22, (C22×D4)⋊50C22, (C2×Q8).475C23, C22.17(C22×D4), (C2×M4(2))⋊59C22, (C2×2+ (1+4))⋊13C2, (C22×C4).1225C23, (C2×Q8)○(C2×Q16), (C2×C4○D8)⋊30C2, (C2×C8○D4)⋊10C2, (C2×C8⋊C22)⋊34C2, (C2×C4).1113(C2×D4), (C2×C4○D4)⋊56C22, SmallGroup(128,2313)

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

Subgroups: 1348 in 756 conjugacy classes, 428 normal (12 characteristic)
C1, C2, C2 [×2], C2 [×18], C4 [×2], C4 [×6], C4 [×4], C22, C22 [×6], C22 [×54], C8 [×8], C2×C4, C2×C4 [×15], C2×C4 [×26], D4 [×24], D4 [×54], Q8 [×8], Q8 [×2], C23 [×3], C23 [×42], C2×C8, C2×C8 [×15], M4(2) [×12], D8 [×36], SD16 [×24], Q16 [×4], C22×C4 [×3], C22×C4 [×6], C2×D4 [×33], C2×D4 [×72], C2×Q8, C2×Q8 [×2], C4○D4 [×32], C4○D4 [×28], C24 [×6], C22×C8 [×3], C2×M4(2) [×3], C8○D4 [×8], C2×D8 [×33], C2×SD16 [×6], C2×Q16, C4○D8 [×24], C8⋊C22 [×48], C22×D4 [×6], C22×D4 [×6], C2×C4○D4, C2×C4○D4 [×6], C2×C4○D4 [×2], 2+ (1+4) [×16], 2+ (1+4) [×8], C2×C8○D4, C22×D8 [×3], C2×C4○D8 [×3], C2×C8⋊C22 [×6], D4○D8 [×16], C2×2+ (1+4) [×2], C2×D4○D8

Quotients:
C1, C2 [×31], C22 [×155], D4 [×8], C23 [×155], C2×D4 [×28], C24 [×31], C22×D4 [×14], C25, D4○D8 [×2], D4×C23, C2×D4○D8

Generators and relations
 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 >

Smallest permutation representation
►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)])
 

Matrix representation ►G ⊆ GL6(𝔽17)

1600000
0160000
001000
000100
000010
000001
,
100000
010000
0001600
001000
0016161615
000111
,
100000
010000
001112
000010
000100
000161616
,
010000
1600000
003300
0014300
001414011
003036
,
1600000
010000
0016161615
000010
000100
000001

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] >;
 

44 conjugacy classes

class 1 2A2B2C2D···2I2J···2U4A···4H4I4J4K4L8A8B8C8D8E···8J
order12222···22···24···4444488888···8
size11112···24···42···2444422224···4

44 irreducible representations

dim11111112224
type+++++++++++
imageC1C2C2C2C2C2C2D4D4D4D4○D8
kernelC2×D4○D8C2×C8○D4C22×D8C2×C4○D8C2×C8⋊C22D4○D8C2×2+ (1+4)C2×D4C2×Q8C4○D4C2
# reps113361623144

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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