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G = M4(2)⋊3Q8order 128 = 27

1st semidirect product of M4(2) and Q8 acting via Q8/C4=C2

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

Aliases: M4(2)⋊3Q8, C42.381C23, C8⋊Q84C2, (C2×C8)⋊4Q8, C8.3(C2×Q8), C83Q84C2, C4⋊C4.243D4, C4.28(C4⋊Q8), C22⋊C4.83D4, C8.5Q814C2, C22.9(C4⋊Q8), C4.14(C22×Q8), C4⋊C4.102C23, (C4×C8).177C22, (C2×C4).361C24, (C2×C8).272C23, C23.457(C2×D4), C4⋊Q8.112C22, C2.36(D4○SD16), C82M4(2).9C2, C2.D8.217C22, C4.Q8.160C22, C8⋊C4.122C22, (C22×C8).276C22, C22.621(C22×D4), C42.C2.17C22, M4(2)⋊C4.12C2, (C22×C4).1045C23, C23.25D4.19C2, C42⋊C2.146C22, (C2×M4(2)).275C22, C23.41C23.8C2, C2.31(C2×C4⋊Q8), (C2×C4).141(C2×D4), (C2×C4).143(C2×Q8), (C2×C4.Q8).12C2, (C2×C4⋊C4).632C22, SmallGroup(128,1895)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — M4(2)⋊3Q8
C1C2C4C2×C4C22×C4C22×C8C82M4(2) — M4(2)⋊3Q8
C1C2C2×C4 — M4(2)⋊3Q8
C1C22C42⋊C2 — M4(2)⋊3Q8
C1C2C2C2×C4 — M4(2)⋊3Q8

Generators and relations for M4(2)⋊3Q8
 G = < a,b,c,d | a8=b2=c4=1, d2=c2, bab=cac-1=a5, dad-1=a-1, cbc-1=a4b, bd=db, dcd-1=c-1 >

Subgroups: 284 in 170 conjugacy classes, 108 normal (20 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×2], C4 [×2], C4 [×12], C22, C22 [×2], C22 [×2], C8 [×8], C2×C4 [×2], C2×C4 [×8], C2×C4 [×10], Q8 [×4], C23, C42 [×2], C42 [×2], C22⋊C4 [×2], C22⋊C4 [×2], C4⋊C4 [×10], C4⋊C4 [×14], C2×C8 [×2], C2×C8 [×6], M4(2) [×4], C22×C4, C22×C4 [×2], C2×Q8 [×4], C4×C8 [×2], C8⋊C4 [×2], C4.Q8 [×2], C4.Q8 [×8], C2.D8 [×6], C2×C4⋊C4 [×2], C42⋊C2, C42⋊C2 [×2], C22⋊Q8 [×4], C42.C2 [×4], C42.C2 [×2], C4⋊Q8 [×4], C4⋊Q8 [×2], C22×C8, C2×M4(2), C82M4(2), C2×C4.Q8, C23.25D4, M4(2)⋊C4 [×2], C83Q8 [×2], C8.5Q8 [×2], C8⋊Q8 [×4], C23.41C23 [×2], M4(2)⋊3Q8
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, D4○SD16 [×2], M4(2)⋊3Q8

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

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

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

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

32 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G4H4I···4P8A8B8C8D8E···8J
order122222444444444···488888···8
size111122222244448···822224···4

32 irreducible representations

dim11111111122224
type+++++++++++--
imageC1C2C2C2C2C2C2C2C2D4D4Q8Q8D4○SD16
kernelM4(2)⋊3Q8C82M4(2)C2×C4.Q8C23.25D4M4(2)⋊C4C83Q8C8.5Q8C8⋊Q8C23.41C23C22⋊C4C4⋊C4C2×C8M4(2)C2
# reps11112224222444

Matrix representation of M4(2)⋊3Q8 in GL6(𝔽17)

1600000
0160000
00016710
00016010
00121200
0012501
,
100000
010000
0016000
0001600
000010
0001001
,
0160000
100000
000010
0007115
0016000
00881210
,
16100000
1010000
00101600
0016700
0007815
0051079

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,12,12,0,0,16,16,12,5,0,0,7,0,0,0,0,0,10,10,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,10,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,16,0,0,0,0,0,0,0,0,0,16,8,0,0,0,7,0,8,0,0,1,1,0,12,0,0,0,15,0,10],[16,10,0,0,0,0,10,1,0,0,0,0,0,0,10,16,0,5,0,0,16,7,7,10,0,0,0,0,8,7,0,0,0,0,15,9] >;

M4(2)⋊3Q8 in GAP, Magma, Sage, TeX

M_4(2)\rtimes_3Q_8
% in TeX

G:=Group("M4(2):3Q8");
// GroupNames label

G:=SmallGroup(128,1895);
// by ID

G=gap.SmallGroup(128,1895);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,112,253,120,758,184,1018,4037,124]);
// 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,c*b*c^-1=a^4*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

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