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G = C42.20D4order 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

C1C4 — C42.20D4
C1C2C22C2×C4C42C2×C42C2×C8⋊C4 — C42.20D4
C1C4 — C42.20D4
C1C2×C4 — C42.20D4
C1C22C22C2×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 [×3], C2 [×2], C4 [×2], C4 [×2], C4 [×4], C22, C22 [×2], C22 [×2], C8 [×8], C2×C4 [×2], C2×C4 [×8], C2×C4 [×2], C23, C42 [×4], C2×C8 [×4], C2×C8 [×8], C22×C4 [×3], C4×C8 [×4], C8⋊C4 [×2], C22⋊C8 [×2], C4⋊C8 [×2], C2×C42, C22×C8 [×2], C2×C8⋊C4, C42.12C4 [×2], C42.20D4
Quotients: C1, C2 [×3], C4 [×6], C22, C8 [×4], C2×C4 [×3], D4 [×3], Q8, C42, C22⋊C4 [×3], C4⋊C4 [×3], C2×C8 [×2], M4(2) [×2], C2.C42, C4×C8, C8⋊C4, C22⋊C8 [×2], C4⋊C8 [×2], 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 18 6 22)(4 20 8 24)(9 11 13 15)(10 62 14 58)(12 64 16 60)(17 19 21 23)(25 35 29 39)(26 28 30 32)(27 37 31 33)(34 36 38 40)(41 43 45 47)(42 56 46 52)(44 50 48 54)(49 51 53 55)(57 59 61 63)
(1 57 23 15)(2 16 24 58)(3 59 17 9)(4 10 18 60)(5 61 19 11)(6 12 20 62)(7 63 21 13)(8 14 22 64)(25 44 33 56)(26 49 34 45)(27 46 35 50)(28 51 36 47)(29 48 37 52)(30 53 38 41)(31 42 39 54)(32 55 40 43)
(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 37 5 31 7 33)(2 47 18 41 6 43 22 45)(4 53 20 55 8 49 24 51)(9 52 11 42 13 56 15 46)(10 30 62 32 14 26 58 28)(12 40 64 34 16 36 60 38)(17 29 19 39 21 25 23 35)(44 57 50 59 48 61 54 63)

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

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

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

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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