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G = M4(2)⋊C8order 128 = 27

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

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

Aliases: M4(2)⋊2C8, C42.454D4, C4⋊C8.7C4, C4.6(C4×C8), C4.37C4≀C2, (C4×C8).11C4, C4.6(C8⋊C4), C22.8(C4⋊C8), (C2×C4).51C42, (C22×C4).68Q8, C23.35(C4⋊C4), C4.28(C22⋊C8), C42.302(C2×C4), (C22×C4).635D4, (C4×M4(2)).7C2, (C2×C4).38M4(2), C4.11(C8.C4), C2.2(C426C4), (C2×M4(2)).17C4, C42.12C4.4C2, C2.2(C4.C42), (C2×C42).1026C22, C2.7(C22.7C42), C22.19(C2.C42), (C2×C4×C8).3C2, (C2×C4).43(C2×C8), (C2×C4).65(C4⋊C4), (C22×C4).389(C2×C4), (C2×C4).370(C22⋊C4), SmallGroup(128,10)

Series: Derived Chief Lower central Upper central Jennings

C1C4 — M4(2)⋊C8
C1C2C22C2×C4C22×C4C2×C42C2×C4×C8 — M4(2)⋊C8
C1C2C4 — M4(2)⋊C8
C1C42C2×C42 — M4(2)⋊C8
C1C22C22C2×C42 — M4(2)⋊C8

Generators and relations for M4(2)⋊C8
 G = < a,b,c | a8=b2=c8=1, bab=a5, cac-1=a3b, bc=cb >

Subgroups: 128 in 86 conjugacy classes, 48 normal (26 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×6], C4 [×2], C22, C22 [×2], C22 [×2], C8 [×10], C2×C4 [×6], C2×C4 [×4], C2×C4 [×4], C23, C42 [×4], C2×C8 [×10], M4(2) [×4], M4(2) [×2], C22×C4 [×3], C4×C8 [×2], C4×C8 [×3], C8⋊C4, C22⋊C8, C4⋊C8 [×2], C2×C42, C22×C8, C2×M4(2) [×2], C2×C4×C8, C4×M4(2), C42.12C4, M4(2)⋊C8
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≀C2 [×2], C4⋊C8 [×2], C8.C4 [×2], C22.7C42, C426C4, C4.C42, M4(2)⋊C8

Smallest permutation representation of M4(2)⋊C8
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 63)(2 60)(3 57)(4 62)(5 59)(6 64)(7 61)(8 58)(9 22)(10 19)(11 24)(12 21)(13 18)(14 23)(15 20)(16 17)(25 40)(26 37)(27 34)(28 39)(29 36)(30 33)(31 38)(32 35)(41 50)(42 55)(43 52)(44 49)(45 54)(46 51)(47 56)(48 53)
(1 54 18 25 63 45 13 40)(2 48 23 39 64 49 10 32)(3 56 20 27 57 47 15 34)(4 42 17 33 58 51 12 26)(5 50 22 29 59 41 9 36)(6 44 19 35 60 53 14 28)(7 52 24 31 61 43 11 38)(8 46 21 37 62 55 16 30)

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

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

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

56 conjugacy classes

class 1 2A2B2C2D2E4A···4L4M···4R8A···8P8Q···8AF
order1222224···44···48···88···8
size1111221···12···22···24···4

56 irreducible representations

dim11111111222222
type++++++-
imageC1C2C2C2C4C4C4C8D4D4Q8M4(2)C4≀C2C8.C4
kernelM4(2)⋊C8C2×C4×C8C4×M4(2)C42.12C4C4×C8C4⋊C8C2×M4(2)M4(2)C42C22×C4C22×C4C2×C4C4C4
# reps111144416211488

Matrix representation of M4(2)⋊C8 in GL3(𝔽17) generated by

1600
004
010
,
100
010
0016
,
1500
0130
001
G:=sub<GL(3,GF(17))| [16,0,0,0,0,1,0,4,0],[1,0,0,0,1,0,0,0,16],[15,0,0,0,13,0,0,0,1] >;

M4(2)⋊C8 in GAP, Magma, Sage, TeX

M_4(2)\rtimes C_8
% in TeX

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

G:=SmallGroup(128,10);
// by ID

G=gap.SmallGroup(128,10);
# by ID

G:=PCGroup([7,-2,2,-2,2,2,-2,2,56,85,120,758,184,248,1684,172]);
// Polycyclic

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

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