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G = C42.20D4  order 128 = 27

2nd non-split extension by C42 of D4 acting via D4/C2=C22

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

Aliases: C42.20D4, (C2×C8)⋊1C8, (C4×C8).1C4, C4.17(C4×C8), (C22×C4).2Q8, C22.1(C4⋊C8), (C22×C8).12C4, (C2×C4).79C42, C4.12(C8⋊C4), C23.34(C4⋊C4), C4.16(C22⋊C8), C42.289(C2×C4), (C22×C4).117D4, (C2×C4).64M4(2), C2.2(C4.9C42), C42.12C4.2C2, (C2×C42).122C22, C2.1(C4.10C42), C2.4(C22.7C42), C22.16(C2.C42), (C2×C4).67(C2×C8), (C2×C4).63(C4⋊C4), (C2×C8⋊C4).13C2, (C22×C4).461(C2×C4), (C2×C4).288(C22⋊C4), SmallGroup(128,7)

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

Derived series
C1 — C4 — C42.20D4
Chief series
C1 — C2 — C22 — C2×C4 — C42 — C2×C42 — C2×C8⋊C4 — C42.20D4
Lower central
C1 — C4 — C42.20D4
Upper central
C1 — C2×C4 — C42.20D4
Jennings
C1 — C22 — C22 — C2×C42 — C42.20D4

Generators and relations for C42.20D4
 G = < a,b,c,d | a4=b4=1, c4=a2, d2=a, ab=ba, cac-1=ab2, ad=da, cbc-1=dbd-1=b-1, dcd-1=a-1bc3 >

Subgroups: 120 in 76 conjugacy classes, 44 normal (16 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, C23, C42, C2×C8, C2×C8, C22×C4, C4×C8, C8⋊C4, C22⋊C8, C4⋊C8, C2×C42, C22×C8, C2×C8⋊C4, C42.12C4, C42.20D4
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, Q8, C42, C22⋊C4, C4⋊C4, C2×C8, M4(2), C2.C42, C4×C8, C8⋊C4, C22⋊C8, C4⋊C8, C22.7C42, C4.9C42, C4.10C42, C42.20D4

Smallest permutation representation of C42.20D4
►On 64 points
Generators in S64
(1 3 5 7)(2 34 6 38)(4 36 8 40)(9 11 13 15)(10 48 14 44)(12 42 16 46)(17 31 21 27)(18 20 22 24)(19 25 23 29)(26 28 30 32)(33 35 37 39)(41 43 45 47)(49 51 53 55)(50 58 54 62)(52 60 56 64)(57 59 61 63)

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

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

44 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E···4N8A···8X
order12222244444···48···8
size11112211112···24···4

44 irreducible representations

dim111111222244
type+++++-
imageC1C2C2C4C4C8D4D4Q8M4(2)C4.9C42C4.10C42
kernelC42.20D4C2×C8⋊C4C42.12C4C4×C8C22×C8C2×C8C42C22×C4C22×C4C2×C4C2C2
# reps1128416211422

Matrix representation of C42.20D4 ►in GL6(𝔽17)

1300000
0130000
0001140
001030
000010
00001616
,
100000
010000
0040316
0004141
0000130
0000013
,
0150000
1500000
0010151311
0072814
00134141
00013108
,
800000
090000
00161220
001570
002200
001501213

G:=sub<GL(6,GF(17))| [13,0,0,0,0,0,0,13,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,14,3,1,16,0,0,0,0,0,16],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,3,14,13,0,0,0,16,1,0,13],[0,15,0,0,0,0,15,0,0,0,0,0,0,0,10,7,13,0,0,0,15,2,4,13,0,0,13,8,14,10,0,0,11,14,1,8],[8,0,0,0,0,0,0,9,0,0,0,0,0,0,16,1,2,15,0,0,12,5,2,0,0,0,2,7,0,12,0,0,0,0,0,13] >;
 

C42.20D4 in GAP, Magma, Sage, TeX 

C_4^2._{20}D_4
 
% in TeX
 
G:=Group("C4^2.20D4");
 
// GroupNames label
 
G:=SmallGroup(128,7);
 
// by ID
 
G=gap.SmallGroup(128,7);
 
# by ID
 
G:=PCGroup([7,-2,2,-2,2,2,-2,2,56,85,120,184,570,248,1684,102]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^4=b^4=1,c^4=a^2,d^2=a,a*b=b*a,c*a*c^-1=a*b^2,a*d=d*a,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=a^-1*b*c^3>;
 
// generators/relations
 

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