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G = C2×C8.26D4  order 128 = 27

Direct product of C2 and C8.26D4

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

Aliases: C2×C8.26D4, C42.282C23, M4(2).31C23, C4○D8⋊8C4, (C2×D8)⋊19C4, D8⋊13(C2×C4), C4.49(C4×D4), C4○(C8.26D4), (C2×Q16)⋊19C4, Q16⋊13(C2×C4), SD16⋊9(C2×C4), C8.129(C2×D4), (C2×C8).394D4, C4≀C2⋊17C22, C22.7(C4×D4), (C2×SD16)⋊11C4, C8○D4⋊18C22, C4.31(C23×C4), C8.25(C22×C4), C8⋊C4⋊40C22, (C2×C8).246C23, (C2×C4).211C24, C4○D4.23C23, C4○D8.23C22, D4.13(C22×C4), C4.202(C22×D4), Q8.13(C22×C4), C8.C4⋊11C22, C23.257(C4○D4), (C22×C8).253C22, (C2×C42).768C22, (C22×C4).1518C23, (C2×M4(2)).358C22, C2.71(C2×C4×D4), (C2×C4≀C2)⋊31C2, (C2×C8⋊C4)⋊7C2, (C2×C8○D4)⋊26C2, (C2×C8).98(C2×C4), C4○D4.22(C2×C4), (C2×C4○D8).17C2, (C2×C8.C4)⋊23C2, C22.2(C2×C4○D4), (C2×D4).178(C2×C4), (C2×C4).1578(C2×D4), (C2×Q8).161(C2×C4), (C2×C4).694(C4○D4), (C2×C4).473(C22×C4), (C2×C4○D4).294C22, SmallGroup(128,1686)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series
C1 — C4 — C2×C8.26D4
Chief series
C1 — C2 — C4 — C2×C4 — C22×C4 — C22×C8 — C2×C8○D4 — C2×C8.26D4
Lower central
C1 — C2 — C4 — C2×C8.26D4
Upper central
C1 — C2×C4 — C22×C8 — C2×C8.26D4
Jennings
C1 — C2 — C2 — C2×C4 — C2×C8.26D4

Generators and relations for C2×C8.26D4
 G = < a,b,c,d | a2=b8=c4=1, d2=b2, ab=ba, ac=ca, ad=da, cbc-1=dbd-1=b5, dcd-1=b2c-1 >

Subgroups: 348 in 232 conjugacy classes, 140 normal (28 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, C42, C42, C2×C8, C2×C8, C2×C8, M4(2), M4(2), D8, SD16, Q16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C4○D4, C8⋊C4, C4≀C2, C8.C4, C2×C42, C22×C8, C22×C8, C2×M4(2), C2×M4(2), C8○D4, C8○D4, C2×D8, C2×SD16, C2×Q16, C4○D8, C2×C4○D4, C2×C8⋊C4, C2×C4≀C2, C2×C8.C4, C8.26D4, C2×C8○D4, C2×C4○D8, C2×C8.26D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22×C4, C2×D4, C4○D4, C24, C4×D4, C23×C4, C22×D4, C2×C4○D4, C8.26D4, C2×C4×D4, C2×C8.26D4

Smallest permutation representation of C2×C8.26D4
►On 32 points
Generators in S32
(1 19)(2 20)(3 21)(4 22)(5 23)(6 24)(7 17)(8 18)(9 28)(10 29)(11 30)(12 31)(13 32)(14 25)(15 26)(16 27)

(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)

(2 6)(4 8)(9 11 13 15)(10 16 14 12)(18 22)(20 24)(25 31 29 27)(26 28 30 32)

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

44 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I4A4B4C4D4E4F4G···4N8A···8H8I···8T
order12222222224444444···48···88···8
size11112244441111224···42···24···4

44 irreducible representations

dim111111111112224
type++++++++
imageC1C2C2C2C2C2C2C4C4C4C4D4C4○D4C4○D4C8.26D4
kernelC2×C8.26D4C2×C8⋊C4C2×C4≀C2C2×C8.C4C8.26D4C2×C8○D4C2×C4○D8C2×D8C2×SD16C2×Q16C4○D8C2×C8C2×C4C23C2
# reps112182124284224

Matrix representation of C2×C8.26D4 ►in GL6(𝔽17)

1600000
0160000
001000
000100
000010
000001
,
100000
010000
000200
002000
0013042
00841113
,
400000
0130000
001000
0001600
000040
00100113
,
0130000
400000
0000130
009084
0016000
002190

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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,2,13,8,0,0,2,0,0,4,0,0,0,0,4,11,0,0,0,0,2,13],[4,0,0,0,0,0,0,13,0,0,0,0,0,0,1,0,0,10,0,0,0,16,0,0,0,0,0,0,4,1,0,0,0,0,0,13],[0,4,0,0,0,0,13,0,0,0,0,0,0,0,0,9,16,2,0,0,0,0,0,1,0,0,13,8,0,9,0,0,0,4,0,0] >;
 

C2×C8.26D4 in GAP, Magma, Sage, TeX 

C_2\times C_8._{26}D_4
 
% in TeX
 
G:=Group("C2xC8.26D4");
 
// GroupNames label
 
G:=SmallGroup(128,1686);
 
// by ID
 
G=gap.SmallGroup(128,1686);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,1430,184,2804,1411,172,124]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^2=b^8=c^4=1,d^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d^-1=b^5,d*c*d^-1=b^2*c^-1>;
 
// generators/relations
 

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