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G = D48M4(2)  order 128 = 27

3rd semidirect product of D4 and M4(2) acting through Inn(D4)

p-group, metabelian, nilpotent (class 2), monomial

Aliases: D48M4(2), C42.306C23, C4.1252- 1+4, C4.1782+ 1+4, D43(C4⋊C8), (C8×D4)⋊50C2, C89D447C2, C84Q845C2, (C4×D4).38C4, (C4×Q8).35C4, C4.49(C8○D4), C4⋊C8.370C22, C42.235(C2×C4), (C4×C8).344C22, (C2×C8).446C23, (C2×C4).687C24, C4.37(C2×M4(2)), C4⋊M4(2)⋊37C2, C42⋊C2.36C4, C42.6C455C2, (C4×D4).365C22, (C4×Q8).286C22, C42.12C457C2, C8⋊C4.105C22, C22.8(C2×M4(2)), C22⋊C8.240C22, (C22×C8).453C22, (C2×C42).794C22, C23.153(C22×C4), C22.209(C23×C4), C2.24(C22×M4(2)), (C22×C4).1288C23, (C2×M4(2)).250C22, C2.45(C23.33C23), (C2×C4⋊C8)⋊51C2, C2.35(C2×C8○D4), C4⋊C4.234(C2×C4), (C2×C4○D4).30C4, (C4×C4○D4).20C2, (C2×D4).238(C2×C4), C22⋊C4.80(C2×C4), (C2×Q8).214(C2×C4), (C2×C4).281(C22×C4), (C22×C4).365(C2×C4), SmallGroup(128,1722)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — D48M4(2)
C1C2C4C2×C4C42C2×C42C4×C4○D4 — D48M4(2)
C1C22 — D48M4(2)
C1C2×C4 — D48M4(2)
C1C2C2C2×C4 — D48M4(2)

Generators and relations for D48M4(2)
 G = < a,b,c,d | a4=b2=c8=d2=1, bab=cac-1=a-1, ad=da, bc=cb, dbd=a2b, dcd=c5 >

Subgroups: 276 in 199 conjugacy classes, 136 normal (38 characteristic)
C1, C2 [×3], C2 [×5], C4 [×2], C4 [×4], C4 [×9], C22, C22 [×4], C22 [×7], C8 [×8], C2×C4 [×6], C2×C4 [×6], C2×C4 [×18], D4 [×4], D4 [×4], Q8 [×2], C23, C23 [×2], C42 [×4], C42 [×6], C22⋊C4 [×6], C4⋊C4 [×2], C4⋊C4 [×4], C2×C8 [×8], C2×C8 [×4], M4(2) [×2], C22×C4 [×3], C22×C4 [×6], C2×D4, C2×D4 [×2], C2×Q8, C4○D4 [×4], C4×C8 [×2], C8⋊C4 [×4], C22⋊C8 [×6], C4⋊C8 [×2], C4⋊C8 [×8], C2×C42, C2×C42 [×2], C42⋊C2, C42⋊C2 [×2], C4×D4 [×2], C4×D4 [×4], C4×Q8 [×2], C22×C8 [×4], C2×M4(2) [×2], C2×C4○D4, C2×C4⋊C8 [×2], C4⋊M4(2), C42.12C4, C42.6C4 [×2], C8×D4 [×2], C89D4 [×4], C84Q8 [×2], C4×C4○D4, D48M4(2)
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], M4(2) [×4], C22×C4 [×14], C24, C2×M4(2) [×6], C8○D4 [×2], C23×C4, 2+ 1+4, 2- 1+4, C23.33C23, C22×M4(2), C2×C8○D4, D48M4(2)

Smallest permutation representation of D48M4(2)
On 64 points
Generators in S64
(1 56 63 18)(2 19 64 49)(3 50 57 20)(4 21 58 51)(5 52 59 22)(6 23 60 53)(7 54 61 24)(8 17 62 55)(9 34 27 43)(10 44 28 35)(11 36 29 45)(12 46 30 37)(13 38 31 47)(14 48 32 39)(15 40 25 41)(16 42 26 33)
(1 27)(2 28)(3 29)(4 30)(5 31)(6 32)(7 25)(8 26)(9 63)(10 64)(11 57)(12 58)(13 59)(14 60)(15 61)(16 62)(17 42)(18 43)(19 44)(20 45)(21 46)(22 47)(23 48)(24 41)(33 55)(34 56)(35 49)(36 50)(37 51)(38 52)(39 53)(40 54)
(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 5)(3 7)(9 31)(10 28)(11 25)(12 30)(13 27)(14 32)(15 29)(16 26)(18 22)(20 24)(33 42)(34 47)(35 44)(36 41)(37 46)(38 43)(39 48)(40 45)(50 54)(52 56)(57 61)(59 63)

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

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

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

50 conjugacy classes

class 1 2A2B2C2D2E2F2G2H4A4B4C4D4E···4P4Q···4U8A···8H8I···8T
order12222222244444···44···48···88···8
size11112222411112···24···42···24···4

50 irreducible representations

dim11111111111112244
type++++++++++-
imageC1C2C2C2C2C2C2C2C2C4C4C4C4M4(2)C8○D42+ 1+42- 1+4
kernelD48M4(2)C2×C4⋊C8C4⋊M4(2)C42.12C4C42.6C4C8×D4C89D4C84Q8C4×C4○D4C42⋊C2C4×D4C4×Q8C2×C4○D4D4C4C4C4
# reps12112242166228811

Matrix representation of D48M4(2) in GL4(𝔽17) generated by

13000
13400
0010
0001
,
13800
13400
0010
0001
,
91600
9800
00102
00167
,
16000
16100
00160
00101
G:=sub<GL(4,GF(17))| [13,13,0,0,0,4,0,0,0,0,1,0,0,0,0,1],[13,13,0,0,8,4,0,0,0,0,1,0,0,0,0,1],[9,9,0,0,16,8,0,0,0,0,10,16,0,0,2,7],[16,16,0,0,0,1,0,0,0,0,16,10,0,0,0,1] >;

D48M4(2) in GAP, Magma, Sage, TeX

D_4\rtimes_8M_4(2)
% in TeX

G:=Group("D4:8M4(2)");
// GroupNames label

G:=SmallGroup(128,1722);
// by ID

G=gap.SmallGroup(128,1722);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,891,675,80,124]);
// Polycyclic

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

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