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G = M4(2):1C8order 128 = 27

1st semidirect product of M4(2) and C8 acting via C8/C4=C2

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

Aliases: M4(2):1C8, C42.15Q8, C42.419D4, C42.622C23, C8:1(C2xC8), C8:2C8:7C2, C8:1C8:11C2, C4.18(C4:C8), C8:C4.4C4, C4.27(C22xC8), (C22xC4).11Q8, C4:C8.266C22, C23.48(C4:C4), C22.12(C4:C8), (C4xC8).137C22, C42.121(C2xC4), (C22xC4).667D4, (C4xM4(2)).1C2, (C2xC4).18M4(2), C4.42(C2xM4(2)), C4.137(C8:C22), (C2xM4(2)).10C4, C4.131(C8.C22), (C2xC42).224C22, C2.1(M4(2).C4), C2.1(M4(2):C4), C42.12C4.29C2, C2.7(C2xC4:C8), (C2xC4:C8).18C2, (C2xC4).19(C2xC8), (C2xC8).56(C2xC4), C22.48(C2xC4:C4), (C2xC4).149(C2xQ8), (C2xC4).116(C4:C4), (C2xC4).1458(C2xD4), (C22xC4).246(C2xC4), (C2xC4).504(C22xC4), SmallGroup(128,297)

Series: Derived Chief Lower central Upper central Jennings

C1C4 — M4(2):1C8
C1C2C22C2xC4C42C2xC42C4xM4(2) — M4(2):1C8
C1C2C4 — M4(2):1C8
C1C2xC4C2xC42 — M4(2):1C8
C1C22C22C42 — M4(2):1C8

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

Subgroups: 132 in 90 conjugacy classes, 62 normal (32 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C8, C8, C2xC4, C2xC4, C2xC4, C23, C42, C2xC8, C2xC8, M4(2), M4(2), C22xC4, C4xC8, C4xC8, C8:C4, C22:C8, C4:C8, C4:C8, C4:C8, C2xC42, C22xC8, C2xM4(2), C8:2C8, C8:1C8, C4xM4(2), C2xC4:C8, C42.12C4, M4(2):1C8
Quotients: C1, C2, C4, C22, C8, C2xC4, D4, Q8, C23, C4:C4, C2xC8, M4(2), C22xC4, C2xD4, C2xQ8, C4:C8, C2xC4:C4, C22xC8, C2xM4(2), C8:C22, C8.C22, C2xC4:C8, M4(2):C4, M4(2).C4, M4(2):1C8

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

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

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

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

44 conjugacy classes

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

44 irreducible representations

dim11111111122222444
type+++++++-+-+-
imageC1C2C2C2C2C2C4C4C8D4Q8D4Q8M4(2)C8:C22C8.C22M4(2).C4
kernelM4(2):1C8C8:2C8C8:1C8C4xM4(2)C2xC4:C8C42.12C4C8:C4C2xM4(2)M4(2)C42C42C22xC4C22xC4C2xC4C4C4C2
# reps122111441611114112

Matrix representation of M4(2):1C8 in GL6(F17)

0160000
100000
00160150
0000161
0011610
000010
,
100000
010000
001000
000100
00160160
00160016
,
020000
200000
0091400
0016800
00710716
000101610

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

M4(2):1C8 in GAP, Magma, Sage, TeX

M_4(2)\rtimes_1C_8
% in TeX

G:=Group("M4(2):1C8");
// GroupNames label

G:=SmallGroup(128,297);
// by ID

G=gap.SmallGroup(128,297);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,64,387,1123,136,172]);
// Polycyclic

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

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