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G = C8.6C42order 128 = 27

6th non-split extension by C8 of C42 acting via C42/C22=C22

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

Aliases: C8.6C42, C42.382D4, C8:C4.5C4, C8.C4:6C4, C4.174(C4xD4), C22.4(C4xQ8), C4.27(C2xC42), (C22xC4).13Q8, C23.78(C2xQ8), C42.135(C2xC4), (C4xM4(2)).22C2, M4(2).22(C2xC4), C4.C42.8C2, C4.51(C42:C2), (C22xC8).211C22, (C2xC42).242C22, C2.3(M4(2).C4), (C22xC4).1319C23, (C2xM4(2)).313C22, C2.17(C4xC4:C4), (C2xC8:C4).3C2, (C2xC4).81(C4:C4), (C2xC8).141(C2xC4), C22.62(C2xC4:C4), (C2xC8.C4).9C2, (C2xC4).1510(C2xD4), (C2xC4).550(C4oD4), (C2xC4).526(C22xC4), SmallGroup(128,510)

Series: Derived Chief Lower central Upper central Jennings

C1C4 — C8.6C42
C1C2C4C2xC4C22xC4C2xC42C4xM4(2) — C8.6C42
C1C2C4 — C8.6C42
C1C2xC4C2xC42 — C8.6C42
C1C2C2C22xC4 — C8.6C42

Generators and relations for C8.6C42
 G = < a,b,c | a8=1, b4=c4=a4, bab-1=a3, cac-1=a5, cbc-1=a2b >

Subgroups: 148 in 110 conjugacy classes, 76 normal (14 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C8, C8, C2xC4, C2xC4, C2xC4, C23, C42, C2xC8, C2xC8, M4(2), M4(2), C22xC4, C22xC4, C4xC8, C8:C4, C8:C4, C8.C4, C2xC42, C22xC8, C2xM4(2), C4.C42, C2xC8:C4, C4xM4(2), C2xC8.C4, C8.6C42
Quotients: C1, C2, C4, C22, C2xC4, D4, Q8, C23, C42, C4:C4, C22xC4, C2xD4, C2xQ8, C4oD4, C2xC42, C2xC4:C4, C42:C2, C4xD4, C4xQ8, C4xC4:C4, M4(2).C4, C8.6C42

Smallest permutation representation of C8.6C42
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 41 7 43 5 45 3 47)(2 44 8 46 6 48 4 42)(9 35 11 33 13 39 15 37)(10 38 12 36 14 34 16 40)(17 26 19 32 21 30 23 28)(18 29 20 27 22 25 24 31)(49 63 55 57 53 59 51 61)(50 58 56 60 54 62 52 64)
(1 52 27 38 5 56 31 34)(2 49 28 35 6 53 32 39)(3 54 29 40 7 50 25 36)(4 51 30 37 8 55 26 33)(9 48 61 17 13 44 57 21)(10 45 62 22 14 41 58 18)(11 42 63 19 15 46 59 23)(12 47 64 24 16 43 60 20)

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

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

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

44 conjugacy classes

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

44 irreducible representations

dim11111112224
type++++++-
imageC1C2C2C2C2C4C4D4Q8C4oD4M4(2).C4
kernelC8.6C42C4.C42C2xC8:C4C4xM4(2)C2xC8.C4C8:C4C8.C4C42C22xC4C2xC4C2
# reps121228162244

Matrix representation of C8.6C42 in GL6(F17)

7160000
16100000
005800
0081200
0000315
00001514
,
0130000
400000
000010
000001
0013000
0001300
,
640000
4110000
00151400
0014200
0000812
0000129

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

C8.6C42 in GAP, Magma, Sage, TeX

C_8._6C_4^2
% in TeX

G:=Group("C8.6C4^2");
// GroupNames label

G:=SmallGroup(128,510);
// by ID

G=gap.SmallGroup(128,510);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,112,141,232,1430,142,1018,248,1411,172,124]);
// Polycyclic

G:=Group<a,b,c|a^8=1,b^4=c^4=a^4,b*a*b^-1=a^3,c*a*c^-1=a^5,c*b*c^-1=a^2*b>;
// generators/relations

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