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G = C2×M4(2)⋊4C4  order 128 = 27

Direct product of C2 and M4(2)⋊4C4

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

Aliases: C2×M4(2)⋊4C4, C23.32C42, (C22×C8)⋊7C4, C24.48(C2×C4), (C2×C4).70C42, (C22×C4).41Q8, C23.65(C4⋊C4), M4(2)⋊21(C2×C4), (C2×M4(2))⋊17C4, (C22×C4).758D4, C22.8(C2×C42), C4○(M4(2)⋊4C4), C42⋊C2.19C4, C23.63(C22×C4), (C23×C4).223C22, (C22×C4).648C23, C23.191(C22⋊C4), C4.23(C2.C42), (C22×M4(2)).17C2, C42⋊C2.258C22, (C2×M4(2)).301C22, C22.33(C2.C42), (C2×C8)⋊3(C2×C4), C4.31(C2×C4⋊C4), C22.16(C2×C4⋊C4), (C2×C4).114(C2×Q8), (C2×C4).128(C4⋊C4), (C2×C4).1298(C2×D4), (C2×C22⋊C4).21C4, C22⋊C4.48(C2×C4), C4.105(C2×C22⋊C4), (C22×C4).256(C2×C4), (C2×C4).519(C22×C4), C22.27(C2×C22⋊C4), (C2×C4).116(C22⋊C4), (C2×C42⋊C2).11C2, C2.20(C2×C2.C42), SmallGroup(128,475)

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

Derived series
C1 — C22 — C2×M4(2)⋊4C4
Chief series
C1 — C2 — C4 — C2×C4 — C22×C4 — C23×C4 — C22×M4(2) — C2×M4(2)⋊4C4
Lower central
C1 — C2 — C22 — C2×M4(2)⋊4C4
Upper central
C1 — C2×C4 — C23×C4 — C2×M4(2)⋊4C4
Jennings
C1 — C2 — C2 — C22×C4 — C2×M4(2)⋊4C4

Generators and relations for C2×M4(2)⋊4C4
 G = < a,b,c,d | a2=b8=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=b5, dbd-1=bc, dcd-1=b4c >

Subgroups: 308 in 194 conjugacy classes, 108 normal (72 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, C23, C23, C42, C22⋊C4, C22⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), M4(2), C22×C4, C22×C4, C24, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C42⋊C2, C22×C8, C22×C8, C2×M4(2), C2×M4(2), C23×C4, M4(2)⋊4C4, C2×C42⋊C2, C22×M4(2), C2×M4(2)⋊4C4
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, C2.C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, M4(2)⋊4C4, C2×C2.C42, C2×M4(2)⋊4C4

Smallest permutation representation of C2×M4(2)⋊4C4
►On 32 points
Generators in S32
(1 15)(2 16)(3 9)(4 10)(5 11)(6 12)(7 13)(8 14)(17 27)(18 28)(19 29)(20 30)(21 31)(22 32)(23 25)(24 26)

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

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

44 conjugacy classes

class 1 2A2B2C2D···2I4A4B4C4D4E···4J4K···4R8A···8P
order12222···244444···44···48···8
size11112···211112···24···44···4

44 irreducible representations

dim11111111224
type+++++-
imageC1C2C2C2C4C4C4C4D4Q8M4(2)⋊4C4
kernelC2×M4(2)⋊4C4M4(2)⋊4C4C2×C42⋊C2C22×M4(2)C2×C22⋊C4C42⋊C2C22×C8C2×M4(2)C22×C4C22×C4C2
# reps141244412624

Matrix representation of C2×M4(2)⋊4C4 ►in GL6(𝔽17)

1600000
0160000
001000
000100
000010
000001
,
0160000
100000
0000130
000004
0001600
001000
,
1600000
0160000
000100
001000
000001
000010
,
1300000
040000
001000
0001600
0000016
000010

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

C2×M4(2)⋊4C4 in GAP, Magma, Sage, TeX 

C_2\times M_4(2)\rtimes_4C_4
 
% in TeX
 
G:=Group("C2xM4(2):4C4");
 
// GroupNames label
 
G:=SmallGroup(128,475);
 
// by ID
 
G=gap.SmallGroup(128,475);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,2,-2,112,141,232,2019,1411,124]);
 
// 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^5,d*b*d^-1=b*c,d*c*d^-1=b^4*c>;
 
// generators/relations
 

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