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G = Q8⋊M4(2)  order 128 = 27

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

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

Aliases: Q83M4(2), C42.49D4, C42.608C23, Q8⋊C830C2, (C4×C8).6C22, (C4×Q8).15C4, (C2×C4).54Q16, C4.52(C2×Q16), C42.61(C2×C4), C4.94(C2×SD16), C4⋊C8.193C22, (C2×C4).114SD16, (C22×C4).731D4, C4.19(C2×M4(2)), (C22×Q8).20C4, C4.24(Q8⋊C4), (C4×Q8).251C22, C4⋊M4(2).12C2, (C2×C42).164C22, C23.170(C22⋊C4), C42.12C4.18C2, C2.13(C24.4C4), C22.20(Q8⋊C4), C2.10(C42⋊C22), (C2×C4×Q8).5C2, (C2×C4⋊C4).41C4, C4⋊C4.180(C2×C4), C2.5(C2×Q8⋊C4), (C2×C4).1450(C2×D4), (C2×Q8).175(C2×C4), (C2×C4).79(C22⋊C4), (C2×C4).313(C22×C4), (C22×C4).186(C2×C4), C22.163(C2×C22⋊C4), SmallGroup(128,219)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — Q8⋊M4(2)
C1C2C22C2×C4C42C2×C42C2×C4×Q8 — Q8⋊M4(2)
C1C2C2×C4 — Q8⋊M4(2)
C1C2×C4C2×C42 — Q8⋊M4(2)
C1C22C22C42 — Q8⋊M4(2)

Generators and relations for Q8⋊M4(2)
 G = < a,b,c,d | a4=c8=d2=1, b2=a2, bab-1=cac-1=a-1, ad=da, cbc-1=a-1b, bd=db, dcd=c5 >

Subgroups: 212 in 124 conjugacy classes, 58 normal (28 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×4], C4 [×9], C22, C22 [×2], C22 [×2], C8 [×4], C2×C4 [×6], C2×C4 [×4], C2×C4 [×13], Q8 [×4], Q8 [×6], C23, C42 [×4], C42 [×4], C4⋊C4 [×2], C4⋊C4 [×5], C2×C8 [×4], M4(2) [×2], C22×C4 [×3], C22×C4 [×2], C2×Q8 [×2], C2×Q8 [×5], C4×C8 [×2], C22⋊C8, C4⋊C8 [×2], C4⋊C8 [×2], C2×C42, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C4×Q8 [×4], C4×Q8 [×2], C2×M4(2), C22×Q8, Q8⋊C8 [×4], C4⋊M4(2), C42.12C4, C2×C4×Q8, Q8⋊M4(2)
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], M4(2) [×4], SD16 [×2], Q16 [×2], C22×C4, C2×D4 [×2], Q8⋊C4 [×4], C2×C22⋊C4, C2×M4(2) [×2], C2×SD16, C2×Q16, C24.4C4, C2×Q8⋊C4, C42⋊C22, Q8⋊M4(2)

Smallest permutation representation of Q8⋊M4(2)
On 64 points
Generators in S64
(1 31 51 21)(2 22 52 32)(3 25 53 23)(4 24 54 26)(5 27 55 17)(6 18 56 28)(7 29 49 19)(8 20 50 30)(9 33 64 44)(10 45 57 34)(11 35 58 46)(12 47 59 36)(13 37 60 48)(14 41 61 38)(15 39 62 42)(16 43 63 40)
(1 63 51 16)(2 33 52 44)(3 57 53 10)(4 35 54 46)(5 59 55 12)(6 37 56 48)(7 61 49 14)(8 39 50 42)(9 32 64 22)(11 26 58 24)(13 28 60 18)(15 30 62 20)(17 36 27 47)(19 38 29 41)(21 40 31 43)(23 34 25 45)
(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)
(2 6)(4 8)(9 13)(11 15)(18 22)(20 24)(26 30)(28 32)(33 37)(35 39)(42 46)(44 48)(50 54)(52 56)(58 62)(60 64)

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

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

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

38 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E···4J4K···4T8A···8H8I8J8K8L
order12222244444···44···48···88888
size11112211112···24···44···48888

38 irreducible representations

dim11111111222224
type+++++++-
imageC1C2C2C2C2C4C4C4D4D4SD16Q16M4(2)C42⋊C22
kernelQ8⋊M4(2)Q8⋊C8C4⋊M4(2)C42.12C4C2×C4×Q8C2×C4⋊C4C4×Q8C22×Q8C42C22×C4C2×C4C2×C4Q8C2
# reps14111242224482

Matrix representation of Q8⋊M4(2) in GL4(𝔽17) generated by

4000
41300
0010
0001
,
8100
3900
00160
00016
,
13800
13400
00515
00212
,
1000
0100
0010
00516
G:=sub<GL(4,GF(17))| [4,4,0,0,0,13,0,0,0,0,1,0,0,0,0,1],[8,3,0,0,1,9,0,0,0,0,16,0,0,0,0,16],[13,13,0,0,8,4,0,0,0,0,5,2,0,0,15,12],[1,0,0,0,0,1,0,0,0,0,1,5,0,0,0,16] >;

Q8⋊M4(2) in GAP, Magma, Sage, TeX

Q_8\rtimes M_4(2)
% in TeX

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

G:=SmallGroup(128,219);
// by ID

G=gap.SmallGroup(128,219);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,232,1430,1123,570,136,172]);
// Polycyclic

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

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