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G = M4(2)⋊6Q8order 128 = 27

4th semidirect product of M4(2) and Q8 acting via Q8/C4=C2

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

Aliases: M4(2)⋊6Q8, C42.254D4, C42.720C23, C8⋊Q87C2, C8.5(C2×Q8), C4.43(C4⋊Q8), C8.5Q816C2, C4.17(C22×Q8), C4⋊C4.105C23, (C2×C4).364C24, (C4×C8).180C22, (C2×C8).275C23, (C22×C4).474D4, C23.687(C2×D4), (C4×M4(2)).5C2, C4⋊Q8.288C22, C22.19(C4⋊Q8), C2.D8.95C22, C4.Q8.25C22, C8⋊C4.125C22, (C2×C42).861C22, C22.624(C22×D4), C2.41(D8⋊C22), M4(2)⋊C4.15C2, (C22×C4).1048C23, C42.C2.119C22, C42⋊C2.149C22, (C2×M4(2)).278C22, C23.37C23.34C2, C2.34(C2×C4⋊Q8), (C2×C4).697(C2×D4), (C2×C4).108(C2×Q8), (C2×C4⋊C4).635C22, (C2×C42.C2).35C2, SmallGroup(128,1898)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — M4(2)⋊6Q8
C1C2C4C2×C4C22×C4C2×M4(2)C4×M4(2) — M4(2)⋊6Q8
C1C2C2×C4 — M4(2)⋊6Q8
C1C22C2×C42 — M4(2)⋊6Q8
C1C2C2C2×C4 — M4(2)⋊6Q8

Generators and relations for M4(2)⋊6Q8
 G = < a,b,c,d | a8=b2=c4=1, d2=c2, bab=cac-1=a5, dad-1=a-1, bc=cb, dbd-1=a4b, dcd-1=c-1 >

Subgroups: 276 in 174 conjugacy classes, 108 normal (14 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×4], C4 [×12], C22, C22 [×2], C22 [×2], C8 [×8], C2×C4 [×2], C2×C4 [×8], C2×C4 [×14], Q8 [×4], C23, C42 [×2], C42 [×2], C42 [×2], C22⋊C4 [×2], C4⋊C4 [×8], C4⋊C4 [×16], C2×C8 [×4], M4(2) [×8], C22×C4, C22×C4 [×2], C22×C4 [×2], C2×Q8 [×2], C4×C8 [×2], C8⋊C4 [×2], C4.Q8 [×8], C2.D8 [×8], C2×C42, C2×C4⋊C4 [×2], C2×C4⋊C4 [×2], C42⋊C2 [×2], C4×Q8 [×2], C22⋊Q8 [×2], C42.C2 [×6], C42.C2 [×2], C4⋊Q8 [×2], C2×M4(2) [×2], C4×M4(2), M4(2)⋊C4 [×4], C8.5Q8 [×4], C8⋊Q8 [×4], C2×C42.C2, C23.37C23, M4(2)⋊6Q8
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], Q8 [×8], C23 [×15], C2×D4 [×6], C2×Q8 [×12], C24, C4⋊Q8 [×4], C22×D4, C22×Q8 [×2], C2×C4⋊Q8, D8⋊C22 [×2], M4(2)⋊6Q8

Smallest permutation representation of M4(2)⋊6Q8
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 19)(10 24)(11 21)(12 18)(13 23)(14 20)(15 17)(16 22)(25 55)(26 52)(27 49)(28 54)(29 51)(30 56)(31 53)(32 50)(33 57)(34 62)(35 59)(36 64)(37 61)(38 58)(39 63)(40 60)
(1 20 47 10)(2 17 48 15)(3 22 41 12)(4 19 42 9)(5 24 43 14)(6 21 44 11)(7 18 45 16)(8 23 46 13)(25 33 55 57)(26 38 56 62)(27 35 49 59)(28 40 50 64)(29 37 51 61)(30 34 52 58)(31 39 53 63)(32 36 54 60)
(1 29 47 51)(2 28 48 50)(3 27 41 49)(4 26 42 56)(5 25 43 55)(6 32 44 54)(7 31 45 53)(8 30 46 52)(9 38 19 62)(10 37 20 61)(11 36 21 60)(12 35 22 59)(13 34 23 58)(14 33 24 57)(15 40 17 64)(16 39 18 63)

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

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

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

32 conjugacy classes

class 1 2A2B2C2D2E4A···4H4I4J4K···4R8A···8H
order1222224···4444···48···8
size1111222···2448···84···4

32 irreducible representations

dim11111112224
type++++++++-+
imageC1C2C2C2C2C2C2D4Q8D4D8⋊C22
kernelM4(2)⋊6Q8C4×M4(2)M4(2)⋊C4C8.5Q8C8⋊Q8C2×C42.C2C23.37C23C42M4(2)C22×C4C2
# reps11444112824

Matrix representation of M4(2)⋊6Q8 in GL6(𝔽17)

1150000
1160000
000040
0044139
0001300
0004013
,
100000
010000
001000
000100
0000160
0011016
,
1620000
1610000
004000
000400
0000130
0044013
,
830000
190000
001313158
006675
006400
001321115

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

M4(2)⋊6Q8 in GAP, Magma, Sage, TeX

M_4(2)\rtimes_6Q_8
% in TeX

G:=Group("M4(2):6Q8");
// GroupNames label

G:=SmallGroup(128,1898);
// by ID

G=gap.SmallGroup(128,1898);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,112,253,568,758,184,521,4037,124]);
// Polycyclic

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

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