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G = C2×D8⋊2C4  order 128 = 27

Direct product of C2 and D8⋊2C4

direct product, p-group, metabelian, nilpotent (class 4), monomial

Aliases: C2×D8⋊2C4, C23.42D8, M5(2)⋊13C22, C4○D8⋊3C4, D8⋊9(C2×C4), (C2×D8)⋊15C4, Q16⋊9(C2×C4), (C2×C4).29D8, (C2×Q16)⋊15C4, (C2×C8).121D4, C8.116(C2×D4), C8.6(C22×C4), C8.8(C22⋊C4), C4.63(C2×SD16), (C2×C4).51SD16, C22.17(C2×D8), C4.Q8⋊40C22, (C2×M5(2))⋊15C2, (C2×C8).224C23, C4○D8.16C22, (C22×C4).334D4, C4.39(D4⋊C4), (C22×C8).233C22, C22.54(D4⋊C4), (C2×C4.Q8)⋊4C2, (C2×C8).83(C2×C4), (C2×C4○D8).11C2, (C2×C4).269(C2×D4), C4.56(C2×C22⋊C4), C2.34(C2×D4⋊C4), (C2×C4).273(C22⋊C4), SmallGroup(128,876)

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

Derived series
C1 — C8 — C2×D8⋊2C4
Chief series
C1 — C2 — C4 — C2×C4 — C2×C8 — C22×C8 — C2×C4○D8 — C2×D8⋊2C4
Lower central
C1 — C2 — C4 — C8 — C2×D8⋊2C4
Upper central
C1 — C22 — C22×C4 — C22×C8 — C2×D8⋊2C4
Jennings
C1 — C2 — C2 — C2 — C2 — C4 — C4 — C2×C8 — C2×D8⋊2C4

Generators and relations for C2×D8⋊2C4
 G = < a,b,c,d | a2=b8=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=b-1, dbd-1=b3, dcd-1=b5c >

Subgroups: 276 in 116 conjugacy classes, 52 normal (34 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, D4, Q8, C23, C23, C16, C4⋊C4, C2×C8, D8, D8, SD16, Q16, Q16, C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, C4.Q8, C4.Q8, C2×C16, M5(2), M5(2), C2×C4⋊C4, C22×C8, C2×D8, C2×SD16, C2×Q16, C4○D8, C4○D8, C2×C4○D4, D8⋊2C4, C2×C4.Q8, C2×M5(2), C2×C4○D8, C2×D8⋊2C4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, D8, SD16, C22×C4, C2×D4, D4⋊C4, C2×C22⋊C4, C2×D8, C2×SD16, D8⋊2C4, C2×D4⋊C4, C2×D8⋊2C4

Smallest permutation representation of C2×D8⋊2C4
►On 32 points
Generators in S32
(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 9)(8 10)(17 29)(18 30)(19 31)(20 32)(21 25)(22 26)(23 27)(24 28)

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

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

32 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E···4J8A8B8C8D8E8F16A···16H
order1222222244444···488888816···16
size1111228822228···82222444···4

32 irreducible representations

dim11111111222224
type+++++++++
imageC1C2C2C2C2C4C4C4D4D4D8SD16D8D8⋊2C4
kernelC2×D8⋊2C4D8⋊2C4C2×C4.Q8C2×M5(2)C2×C4○D8C2×D8C2×Q16C4○D8C2×C8C22×C4C2×C4C2×C4C23C2
# reps14111224312424

Matrix representation of C2×D8⋊2C4 ►in GL6(𝔽17)

1600000
0160000
0016000
0001600
0000160
0000016
,
1600000
0160000
0012500
00121200
000055
0000125
,
010000
100000
000055
0000125
0012500
00121200
,
400000
0130000
000100
001000
000055
0000512

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],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,12,12,0,0,0,0,5,12,0,0,0,0,0,0,5,12,0,0,0,0,5,5],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,12,12,0,0,0,0,5,12,0,0,5,12,0,0,0,0,5,5,0,0],[4,0,0,0,0,0,0,13,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,5,5,0,0,0,0,5,12] >;
 

C2×D8⋊2C4 in GAP, Magma, Sage, TeX 

C_2\times D_8\rtimes_2C_4
 
% in TeX
 
G:=Group("C2xD8:2C4");
 
// GroupNames label
 
G:=SmallGroup(128,876);
 
// by ID
 
G=gap.SmallGroup(128,876);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,-2,-2,112,141,1123,1466,136,1411,4037,2028,124]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^2=b^8=c^2=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,d*b*d^-1=b^3,d*c*d^-1=b^5*c>;
 
// generators/relations
 

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