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G = C42.383D4order 128 = 27

16th non-split extension by C42 of D4 acting via D4/C22=C2

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

Aliases: C42.383D4, C42.272C23, C4○D85C4, D811(C2×C4), C4(D8⋊C4), Q1611(C2×C4), SD167(C2×C4), C4.186(C4×D4), D8⋊C432C2, C22.5(C4×D4), C4(Q16⋊C4), C4.20(C23×C4), C8.19(C22×C4), D4.4(C22×C4), Q16⋊C432C2, C4(SD16⋊C4), Q8.4(C22×C4), C4⋊C4.360C23, (C2×C8).411C23, (C2×C4).200C24, C23.381(C2×D4), (C22×C4).168D4, SD16⋊C453C2, (C2×D8).156C22, (C4×D4).291C22, (C2×D4).369C23, (C4×Q8).274C22, (C2×Q8).342C23, C4.Q8.125C22, C2.D8.211C22, C8⋊C4.111C22, C23.24D437C2, C23.25D423C2, C2.3(D8⋊C22), (C22×C8).439C22, (C2×C42).765C22, (C2×Q16).151C22, C22.144(C22×D4), D4⋊C4.195C22, (C22×C4).1516C23, Q8⋊C4.195C22, (C2×SD16).106C22, C42⋊C2.296C22, C2.60(C2×C4×D4), (C2×C8)⋊13(C2×C4), C4○D48(C2×C4), (C4×C4○D4)⋊4C2, (C2×C8⋊C4)⋊6C2, C4.8(C2×C4○D4), (C2×C4○D8).14C2, (C2×C4)(Q16⋊C4), (C2×C4).1212(C2×D4), (C2×C4).692(C4○D4), (C2×C4).471(C22×C4), (C2×C4○D4).290C22, SmallGroup(128,1675)

Series: Derived Chief Lower central Upper central Jennings

C1C4 — C42.383D4
C1C2C22C2×C4C22×C4C2×C42C4×C4○D4 — C42.383D4
C1C2C4 — C42.383D4
C1C2×C4C2×C42 — C42.383D4
C1C2C2C2×C4 — C42.383D4

Generators and relations for C42.383D4
 G = < a,b,c,d | a4=b4=1, c4=b2, d2=a2, ab=ba, cac-1=dad-1=ab2, bc=cb, bd=db, dcd-1=a2b2c3 >

Subgroups: 396 in 246 conjugacy classes, 140 normal (20 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×2], C4 [×2], C4 [×12], C22, C22 [×2], C22 [×10], C8 [×4], C8 [×2], C2×C4 [×2], C2×C4 [×8], C2×C4 [×24], D4 [×4], D4 [×10], Q8 [×4], Q8 [×2], C23, C23 [×2], C42 [×4], C42 [×6], C22⋊C4 [×6], C4⋊C4 [×4], C4⋊C4 [×4], C2×C8 [×2], C2×C8 [×6], C2×C8 [×2], D8 [×4], SD16 [×8], Q16 [×4], C22×C4, C22×C4 [×2], C22×C4 [×6], C2×D4 [×2], C2×D4 [×2], C2×Q8 [×2], C4○D4 [×8], C4○D4 [×4], C8⋊C4 [×4], D4⋊C4 [×4], Q8⋊C4 [×4], C4.Q8 [×2], C2.D8 [×2], C2×C42, C2×C42 [×2], C42⋊C2 [×2], C42⋊C2 [×2], C4×D4 [×4], C4×D4 [×4], C4×Q8 [×4], C22×C8 [×2], C2×D8, C2×SD16 [×2], C2×Q16, C4○D8 [×8], C2×C4○D4 [×2], C2×C8⋊C4, C23.24D4 [×2], C23.25D4, SD16⋊C4 [×4], Q16⋊C4 [×2], D8⋊C4 [×2], C4×C4○D4 [×2], C2×C4○D8, C42.383D4
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], D4 [×4], C23 [×15], C22×C4 [×14], C2×D4 [×6], C4○D4 [×2], C24, C4×D4 [×4], C23×C4, C22×D4, C2×C4○D4, C2×C4×D4, D8⋊C22 [×2], C42.383D4

Smallest permutation representation of C42.383D4
On 64 points
Generators in S64
(1 10 58 34)(2 15 59 39)(3 12 60 36)(4 9 61 33)(5 14 62 38)(6 11 63 35)(7 16 64 40)(8 13 57 37)(17 53 41 26)(18 50 42 31)(19 55 43 28)(20 52 44 25)(21 49 45 30)(22 54 46 27)(23 51 47 32)(24 56 48 29)
(1 53 5 49)(2 54 6 50)(3 55 7 51)(4 56 8 52)(9 48 13 44)(10 41 14 45)(11 42 15 46)(12 43 16 47)(17 38 21 34)(18 39 22 35)(19 40 23 36)(20 33 24 37)(25 61 29 57)(26 62 30 58)(27 63 31 59)(28 64 32 60)
(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 21 58 45)(2 44 59 20)(3 19 60 43)(4 42 61 18)(5 17 62 41)(6 48 63 24)(7 23 64 47)(8 46 57 22)(9 27 33 54)(10 53 34 26)(11 25 35 52)(12 51 36 32)(13 31 37 50)(14 49 38 30)(15 29 39 56)(16 55 40 28)

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

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

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

44 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I4A4B4C4D4E···4N4O···4Z8A···8H
order122222222244444···44···48···8
size111122444411112···24···44···4

44 irreducible representations

dim11111111112224
type+++++++++++
imageC1C2C2C2C2C2C2C2C2C4D4D4C4○D4D8⋊C22
kernelC42.383D4C2×C8⋊C4C23.24D4C23.25D4SD16⋊C4Q16⋊C4D8⋊C4C4×C4○D4C2×C4○D8C4○D8C42C22×C4C2×C4C2
# reps112142221162244

Matrix representation of C42.383D4 in GL6(𝔽17)

1300000
0130000
00013611
004060
0000134
000094
,
1600000
0160000
0013000
0001300
0000130
0000013
,
490000
4130000
0011941
006366
0021563
0040114
,
1300000
1340000
00016116
00160110
0000116
0000016

G:=sub<GL(6,GF(17))| [13,0,0,0,0,0,0,13,0,0,0,0,0,0,0,4,0,0,0,0,13,0,0,0,0,0,6,6,13,9,0,0,11,0,4,4],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,13],[4,4,0,0,0,0,9,13,0,0,0,0,0,0,11,6,2,4,0,0,9,3,15,0,0,0,4,6,6,1,0,0,1,6,3,14],[13,13,0,0,0,0,0,4,0,0,0,0,0,0,0,16,0,0,0,0,16,0,0,0,0,0,11,11,1,0,0,0,6,0,16,16] >;

C42.383D4 in GAP, Magma, Sage, TeX

C_4^2._{383}D_4
% in TeX

G:=Group("C4^2.383D4");
// GroupNames label

G:=SmallGroup(128,1675);
// by ID

G=gap.SmallGroup(128,1675);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,1430,184,248,2804,1411,172]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^4=1,c^4=b^2,d^2=a^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a*b^2,b*c=c*b,b*d=d*b,d*c*d^-1=a^2*b^2*c^3>;
// generators/relations

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