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G = C88M4(2)  order 128 = 27

2nd semidirect product of C8 and M4(2) acting via M4(2)/C2×C4=C2

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

Aliases: C88M4(2), C42.50Q8, C42.319D4, C42.623C23, (C4×C8).18C4, C82C824C2, (C22×C8).40C4, (C2×C4).69SD16, C4.98(C2×SD16), C4.10(C4.Q8), C4.5(C8.C4), (C22×C4).81Q8, C4⋊C8.214C22, C23.49(C4⋊C4), (C4×C8).424C22, C42.310(C2×C4), (C22×C4).574D4, C4.43(C2×M4(2)), C22.10(C4.Q8), C4⋊M4(2).20C2, C2.6(C4⋊M4(2)), (C2×C42).1041C22, (C2×C4×C8).52C2, C2.4(C2×C4.Q8), (C2×C4).74(C4⋊C4), (C2×C8).231(C2×C4), C2.6(C2×C8.C4), C22.80(C2×C4⋊C4), (C2×C4).150(C2×Q8), (C2×C4).1459(C2×D4), (C2×C4).505(C22×C4), (C22×C4).474(C2×C4), SmallGroup(128,298)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C88M4(2)
C1C2C22C2×C4C42C2×C42C2×C4×C8 — C88M4(2)
C1C2C2×C4 — C88M4(2)
C1C2×C4C2×C42 — C88M4(2)
C1C22C22C42 — C88M4(2)

Generators and relations for C88M4(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 [×3], C2 [×2], C4 [×8], C4 [×2], C22, C22 [×2], C22 [×2], C8 [×4], C8 [×6], C2×C4 [×6], C2×C4 [×4], C2×C4 [×4], C23, C42 [×4], C2×C8 [×4], C2×C8 [×8], M4(2) [×4], C22×C4 [×3], C4×C8 [×2], C4×C8 [×2], C4⋊C8 [×4], C4⋊C8 [×2], C2×C42, C22×C8 [×2], C2×M4(2) [×2], C82C8 [×4], C2×C4×C8, C4⋊M4(2) [×2], C88M4(2)
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], Q8 [×2], C23, C4⋊C4 [×4], M4(2) [×4], SD16 [×4], C22×C4, C2×D4, C2×Q8, C4.Q8 [×4], C8.C4 [×2], C2×C4⋊C4, C2×M4(2) [×2], C2×SD16 [×2], C4⋊M4(2), C2×C4.Q8, C2×C8.C4, C88M4(2)

Smallest permutation representation of C88M4(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 18 41 28 35 60 56 11)(2 21 42 31 36 63 49 14)(3 24 43 26 37 58 50 9)(4 19 44 29 38 61 51 12)(5 22 45 32 39 64 52 15)(6 17 46 27 40 59 53 10)(7 20 47 30 33 62 54 13)(8 23 48 25 34 57 55 16)
(9 26)(10 27)(11 28)(12 29)(13 30)(14 31)(15 32)(16 25)(17 59)(18 60)(19 61)(20 62)(21 63)(22 64)(23 57)(24 58)

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

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

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

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
kernelC88M4(2)C82C8C2×C4×C8C4⋊M4(2)C4×C8C22×C8C42C42C22×C4C22×C4C8C2×C4C4
# reps1412441111888

Matrix representation of C88M4(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] >;

C88M4(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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