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G = C42.159D4  order 128 = 27

141st non-split extension by C42 of D4 acting via D4/C2=C22

p-group, metabelian, nilpotent (class 2), monomial

Aliases: C42.159D4, C24.196C23, C23.205C24, C22.442+ 1+4, C22.282- 1+4, C4.129(C4×D4), C22⋊Q8⋊16C4, C23.85(C22×C4), C22.96(C23×C4), C22.93(C22×D4), (C23×C4).291C22, (C2×C42).412C22, (C22×C4).470C23, C23.7Q8.26C2, C23.63C23⋊5C2, C24.C22.2C2, (C22×Q8).398C22, C23.67C23⋊13C2, C23.65C23⋊13C2, C2.C42.41C22, C2.4(C22.36C24), C2.3(C22.31C24), C2.3(C22.35C24), C2.8(C23.32C23), C2.5(C23.38C23), C2.12(C23.33C23), (C2×C4×Q8)⋊4C2, (C4×C4⋊C4)⋊25C2, C2.22(C2×C4×D4), C4⋊C4.103(C2×C4), (C2×C4).677(C2×D4), C22⋊C4.8(C2×C4), (C2×C4).26(C22×C4), (C2×Q8).148(C2×C4), C22.90(C2×C4○D4), (C2×C22⋊Q8).15C2, (C2×C4).646(C4○D4), (C2×C4⋊C4).178C22, (C22×C4).303(C2×C4), (C2×C42⋊C2).29C2, (C2×C22⋊C4).427C22, SmallGroup(128,1055)

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

Derived series
C1 — C22 — C42.159D4
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C23×C4 — C2×C42⋊C2 — C42.159D4
Lower central
C1 — C22 — C42.159D4
Upper central
C1 — C23 — C42.159D4
Jennings
C1 — C23 — C42.159D4

Generators and relations for C42.159D4
 G = < a,b,c,d | a4=b4=c4=1, d2=a2, ab=ba, ac=ca, dad-1=ab2, cbc-1=dbd-1=b-1, dcd-1=a2c-1 >

Subgroups: 444 in 278 conjugacy classes, 148 normal (42 characteristic)
C1, C2, C2, C4, C4, C22, C22, C2×C4, C2×C4, Q8, C23, C23, C23, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×Q8, C2×Q8, C24, C2.C42, C2×C42, C2×C42, C2×C22⋊C4, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C42⋊C2, C4×Q8, C22⋊Q8, C23×C4, C22×Q8, C4×C4⋊C4, C23.7Q8, C23.63C23, C24.C22, C23.65C23, C23.65C23, C23.67C23, C2×C42⋊C2, C2×C4×Q8, C2×C22⋊Q8, C42.159D4
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, 2+ 1+4, 2- 1+4, C2×C4×D4, C23.32C23, C23.33C23, C23.38C23, C22.31C24, C22.35C24, C22.36C24, C42.159D4

Smallest permutation representation of C42.159D4
►On 64 points
Generators in S64
(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)(33 34 35 36)(37 38 39 40)(41 42 43 44)(45 46 47 48)(49 50 51 52)(53 54 55 56)(57 58 59 60)(61 62 63 64)

(1 23 9 6)(2 24 10 7)(3 21 11 8)(4 22 12 5)(13 29 25 37)(14 30 26 38)(15 31 27 39)(16 32 28 40)(17 59 63 43)(18 60 64 44)(19 57 61 41)(20 58 62 42)(33 55 52 47)(34 56 49 48)(35 53 50 45)(36 54 51 46)

(1 41 25 53)(2 42 26 54)(3 43 27 55)(4 44 28 56)(5 18 32 49)(6 19 29 50)(7 20 30 51)(8 17 31 52)(9 57 13 45)(10 58 14 46)(11 59 15 47)(12 60 16 48)(21 63 39 33)(22 64 40 34)(23 61 37 35)(24 62 38 36)

(1 55 3 53)(2 48 4 46)(5 36 7 34)(6 52 8 50)(9 47 11 45)(10 56 12 54)(13 59 15 57)(14 44 16 42)(17 31 19 29)(18 40 20 38)(21 35 23 33)(22 51 24 49)(25 43 27 41)(26 60 28 58)(30 64 32 62)(37 63 39 61)
 
G:=sub<Sym(64)| (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)(33,34,35,36)(37,38,39,40)(41,42,43,44)(45,46,47,48)(49,50,51,52)(53,54,55,56)(57,58,59,60)(61,62,63,64), (1,23,9,6)(2,24,10,7)(3,21,11,8)(4,22,12,5)(13,29,25,37)(14,30,26,38)(15,31,27,39)(16,32,28,40)(17,59,63,43)(18,60,64,44)(19,57,61,41)(20,58,62,42)(33,55,52,47)(34,56,49,48)(35,53,50,45)(36,54,51,46), (1,41,25,53)(2,42,26,54)(3,43,27,55)(4,44,28,56)(5,18,32,49)(6,19,29,50)(7,20,30,51)(8,17,31,52)(9,57,13,45)(10,58,14,46)(11,59,15,47)(12,60,16,48)(21,63,39,33)(22,64,40,34)(23,61,37,35)(24,62,38,36), (1,55,3,53)(2,48,4,46)(5,36,7,34)(6,52,8,50)(9,47,11,45)(10,56,12,54)(13,59,15,57)(14,44,16,42)(17,31,19,29)(18,40,20,38)(21,35,23,33)(22,51,24,49)(25,43,27,41)(26,60,28,58)(30,64,32,62)(37,63,39,61)>;
 
G:=Group( (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)(33,34,35,36)(37,38,39,40)(41,42,43,44)(45,46,47,48)(49,50,51,52)(53,54,55,56)(57,58,59,60)(61,62,63,64), (1,23,9,6)(2,24,10,7)(3,21,11,8)(4,22,12,5)(13,29,25,37)(14,30,26,38)(15,31,27,39)(16,32,28,40)(17,59,63,43)(18,60,64,44)(19,57,61,41)(20,58,62,42)(33,55,52,47)(34,56,49,48)(35,53,50,45)(36,54,51,46), (1,41,25,53)(2,42,26,54)(3,43,27,55)(4,44,28,56)(5,18,32,49)(6,19,29,50)(7,20,30,51)(8,17,31,52)(9,57,13,45)(10,58,14,46)(11,59,15,47)(12,60,16,48)(21,63,39,33)(22,64,40,34)(23,61,37,35)(24,62,38,36), (1,55,3,53)(2,48,4,46)(5,36,7,34)(6,52,8,50)(9,47,11,45)(10,56,12,54)(13,59,15,57)(14,44,16,42)(17,31,19,29)(18,40,20,38)(21,35,23,33)(22,51,24,49)(25,43,27,41)(26,60,28,58)(30,64,32,62)(37,63,39,61) );
 
G=PermutationGroup([[(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),(33,34,35,36),(37,38,39,40),(41,42,43,44),(45,46,47,48),(49,50,51,52),(53,54,55,56),(57,58,59,60),(61,62,63,64)], [(1,23,9,6),(2,24,10,7),(3,21,11,8),(4,22,12,5),(13,29,25,37),(14,30,26,38),(15,31,27,39),(16,32,28,40),(17,59,63,43),(18,60,64,44),(19,57,61,41),(20,58,62,42),(33,55,52,47),(34,56,49,48),(35,53,50,45),(36,54,51,46)], [(1,41,25,53),(2,42,26,54),(3,43,27,55),(4,44,28,56),(5,18,32,49),(6,19,29,50),(7,20,30,51),(8,17,31,52),(9,57,13,45),(10,58,14,46),(11,59,15,47),(12,60,16,48),(21,63,39,33),(22,64,40,34),(23,61,37,35),(24,62,38,36)], [(1,55,3,53),(2,48,4,46),(5,36,7,34),(6,52,8,50),(9,47,11,45),(10,56,12,54),(13,59,15,57),(14,44,16,42),(17,31,19,29),(18,40,20,38),(21,35,23,33),(22,51,24,49),(25,43,27,41),(26,60,28,58),(30,64,32,62),(37,63,39,61)]])
 

44 conjugacy classes

class 1 2A···2G2H2I4A···4L4M···4AH
order12···2224···44···4
size11···1442···24···4

44 irreducible representations

dim111111111112244
type++++++++++++-
imageC1C2C2C2C2C2C2C2C2C2C4D4C4○D42+ 1+42- 1+4
kernelC42.159D4C4×C4⋊C4C23.7Q8C23.63C23C24.C22C23.65C23C23.67C23C2×C42⋊C2C2×C4×Q8C2×C22⋊Q8C22⋊Q8C42C2×C4C22C22
# reps1112431111164413

Matrix representation of C42.159D4 ►in GL8(𝔽5)

20000000
02000000
00400000
00040000
00000010
00003212
00004000
00000003
,
40000000
04000000
00400000
00040000
00000100
00004000
00003212
00002044
,
01000000
10000000
00010000
00400000
00000200
00002000
00001424
00000003
,
01000000
40000000
00010000
00100000
00000200
00002000
00004131
00001402

G:=sub<GL(8,GF(5))| [2,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,3,4,0,0,0,0,0,0,2,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,2,0,3],[4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,3,2,0,0,0,0,1,0,2,0,0,0,0,0,0,0,1,4,0,0,0,0,0,0,2,4],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,2,1,0,0,0,0,0,2,0,4,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,4,3],[0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,2,4,1,0,0,0,0,2,0,1,4,0,0,0,0,0,0,3,0,0,0,0,0,0,0,1,2] >;
 

C42.159D4 in GAP, Magma, Sage, TeX 

C_4^2._{159}D_4
 
% in TeX
 
G:=Group("C4^2.159D4");
 
// GroupNames label
 
G:=SmallGroup(128,1055);
 
// by ID
 
G=gap.SmallGroup(128,1055);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,456,758,219,675,80]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^4=b^4=c^4=1,d^2=a^2,a*b=b*a,a*c=c*a,d*a*d^-1=a*b^2,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=a^2*c^-1>;
 
// generators/relations
 

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