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G = C8:8M4(2)  order 128 = 27

2nd semidirect product of C8 and M4(2) acting via M4(2)/C2xC4=C2

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

Aliases: C8:8M4(2), C42.50Q8, C42.319D4, C42.623C23, (C4xC8).18C4, C8:2C8:24C2, (C22xC8).40C4, (C2xC4).69SD16, C4.98(C2xSD16), C4.10(C4.Q8), C4.5(C8.C4), (C22xC4).81Q8, C4:C8.214C22, C23.49(C4:C4), (C4xC8).424C22, C42.310(C2xC4), (C22xC4).574D4, C4.43(C2xM4(2)), C22.10(C4.Q8), C4:M4(2).20C2, C2.6(C4:M4(2)), (C2xC42).1041C22, (C2xC4xC8).52C2, C2.4(C2xC4.Q8), (C2xC4).74(C4:C4), (C2xC8).231(C2xC4), C2.6(C2xC8.C4), C22.80(C2xC4:C4), (C2xC4).150(C2xQ8), (C2xC4).1459(C2xD4), (C2xC4).505(C22xC4), (C22xC4).474(C2xC4), SmallGroup(128,298)

Series: Derived Chief Lower central Upper central Jennings

C1C2xC4 — C8:8M4(2)
C1C2C22C2xC4C42C2xC42C2xC4xC8 — C8:8M4(2)
C1C2C2xC4 — C8:8M4(2)
C1C2xC4C2xC42 — C8:8M4(2)
C1C22C22C42 — C8:8M4(2)

Generators and relations for C8:8M4(2)
 G = < a,b,c | a8=b8=c2=1, bab-1=a3, ac=ca, cbc=b5 >

Subgroups: 140 in 92 conjugacy classes, 60 normal (24 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C8, C8, C2xC4, C2xC4, C2xC4, C23, C42, C2xC8, C2xC8, M4(2), C22xC4, C4xC8, C4xC8, C4:C8, C4:C8, C2xC42, C22xC8, C2xM4(2), C8:2C8, C2xC4xC8, C4:M4(2), C8:8M4(2)
Quotients: C1, C2, C4, C22, C2xC4, D4, Q8, C23, C4:C4, M4(2), SD16, C22xC4, C2xD4, C2xQ8, C4.Q8, C8.C4, C2xC4:C4, C2xM4(2), C2xSD16, C4:M4(2), C2xC4.Q8, C2xC8.C4, C8:8M4(2)

Smallest permutation representation of C8:8M4(2)
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 24 47 28 35 60 54 9)(2 19 48 31 36 63 55 12)(3 22 41 26 37 58 56 15)(4 17 42 29 38 61 49 10)(5 20 43 32 39 64 50 13)(6 23 44 27 40 59 51 16)(7 18 45 30 33 62 52 11)(8 21 46 25 34 57 53 14)
(9 28)(10 29)(11 30)(12 31)(13 32)(14 25)(15 26)(16 27)(17 61)(18 62)(19 63)(20 64)(21 57)(22 58)(23 59)(24 60)

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

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

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

44 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E···4N8A···8P8Q···8X
order12222244444···48···88···8
size11112211112···22···28···8

44 irreducible representations

dim1111112222222
type+++++-+-
imageC1C2C2C2C4C4D4Q8D4Q8M4(2)SD16C8.C4
kernelC8:8M4(2)C8:2C8C2xC4xC8C4:M4(2)C4xC8C22xC8C42C42C22xC4C22xC4C8C2xC4C4
# reps1412441111888

Matrix representation of C8:8M4(2) in GL4(F17) generated by

9000
01500
0010
0001
,
0100
4000
0001
00130
,
1000
01600
0010
00016
G:=sub<GL(4,GF(17))| [9,0,0,0,0,15,0,0,0,0,1,0,0,0,0,1],[0,4,0,0,1,0,0,0,0,0,0,13,0,0,1,0],[1,0,0,0,0,16,0,0,0,0,1,0,0,0,0,16] >;

C8:8M4(2) in GAP, Magma, Sage, TeX

C_8\rtimes_8M_4(2)
% in TeX

G:=Group("C8:8M4(2)");
// GroupNames label

G:=SmallGroup(128,298);
// by ID

G=gap.SmallGroup(128,298);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,64,1430,1123,136,172]);
// Polycyclic

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

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