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G = C42.445D4order 128 = 27

78th non-split extension by C42 of D4 acting via D4/C22=C2

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

Aliases: C42.445D4, C42.324C23, D4.5(C2×D4), Q8.5(C2×D4), C4○D4.29D4, C42Q1619C2, D4.D42C2, C4⋊C8.39C22, C45(C8.C22), (C2×C8).14C23, C4.70(C22×D4), C4⋊C4.380C23, C4⋊M4(2)⋊5C2, (C2×C4).243C24, C23.655(C2×D4), (C22×C4).423D4, C4⋊Q8.257C22, (C2×Q8).37C23, C4.105(C4⋊D4), (C2×D4).386C23, (C4×D4).311C22, C23.38D46C2, (C4×Q8).292C22, (C2×Q16).52C22, (C2×SD16).4C22, C22.78(C4⋊D4), (C22×C4).973C23, (C2×C42).812C22, Q8⋊C4.23C22, C22.503(C22×D4), (C2×M4(2)).50C22, (C22×Q8).271C22, C42⋊C2.312C22, (C2×C4⋊Q8)⋊30C2, (C4×C4○D4).22C2, C4.153(C2×C4○D4), C2.61(C2×C4⋊D4), (C2×C4).1422(C2×D4), C2.16(C2×C8.C22), (C2×C4).274(C4○D4), (C2×C8.C22).10C2, (C2×C4○D4).299C22, SmallGroup(128,1771)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.445D4
C1C2C4C2×C4C22×C4C2×C4○D4C4×C4○D4 — C42.445D4
C1C2C2×C4 — C42.445D4
C1C22C2×C42 — C42.445D4
C1C2C2C2×C4 — C42.445D4

Generators and relations for C42.445D4
 G = < a,b,c,d | a4=b4=1, c4=d2=b2, ab=ba, cac-1=dad-1=a-1, cbc-1=dbd-1=b-1, dcd-1=c3 >

Subgroups: 420 in 240 conjugacy classes, 104 normal (20 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×2], C4 [×6], C4 [×10], C22, C22 [×2], C22 [×6], C8 [×4], C2×C4 [×2], C2×C4 [×8], C2×C4 [×23], D4 [×2], D4 [×5], Q8 [×2], Q8 [×13], C23, C23, C42 [×2], C42 [×2], C42 [×3], C22⋊C4 [×3], C4⋊C4 [×2], C4⋊C4 [×10], C2×C8 [×4], M4(2) [×4], SD16 [×8], Q16 [×8], C22×C4, C22×C4 [×2], C22×C4 [×5], C2×D4, C2×D4, C2×Q8, C2×Q8 [×4], C2×Q8 [×10], C4○D4 [×4], C4○D4 [×2], Q8⋊C4 [×8], C4⋊C8 [×4], C2×C42, C2×C42, C2×C4⋊C4 [×2], C42⋊C2, C42⋊C2, C4×D4 [×2], C4×D4 [×2], C4×Q8 [×2], C4⋊Q8 [×4], C4⋊Q8 [×2], C2×M4(2) [×2], C2×SD16 [×4], C2×Q16 [×4], C8.C22 [×8], C22×Q8 [×2], C2×C4○D4, C23.38D4 [×2], C4⋊M4(2), D4.D4 [×4], C42Q16 [×4], C4×C4○D4, C2×C4⋊Q8, C2×C8.C22 [×2], C42.445D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×8], C23 [×15], C2×D4 [×12], C4○D4 [×2], C24, C4⋊D4 [×4], C8.C22 [×4], C22×D4 [×2], C2×C4○D4, C2×C4⋊D4, C2×C8.C22 [×2], C42.445D4

Smallest permutation representation of C42.445D4
On 64 points
Generators in S64
(1 59 51 18)(2 19 52 60)(3 61 53 20)(4 21 54 62)(5 63 55 22)(6 23 56 64)(7 57 49 24)(8 17 50 58)(9 25 37 42)(10 43 38 26)(11 27 39 44)(12 45 40 28)(13 29 33 46)(14 47 34 30)(15 31 35 48)(16 41 36 32)
(1 47 5 43)(2 44 6 48)(3 41 7 45)(4 46 8 42)(9 21 13 17)(10 18 14 22)(11 23 15 19)(12 20 16 24)(25 54 29 50)(26 51 30 55)(27 56 31 52)(28 53 32 49)(33 58 37 62)(34 63 38 59)(35 60 39 64)(36 57 40 61)
(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 4 5 8)(2 7 6 3)(9 10 13 14)(11 16 15 12)(17 18 21 22)(19 24 23 20)(25 26 29 30)(27 32 31 28)(33 34 37 38)(35 40 39 36)(41 48 45 44)(42 43 46 47)(49 56 53 52)(50 51 54 55)(57 64 61 60)(58 59 62 63)

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

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

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

32 conjugacy classes

class 1 2A2B2C2D2E2F2G4A···4H4I···4P4Q4R4S4T8A8B8C8D
order122222224···44···444448888
size111122442···24···488888888

32 irreducible representations

dim1111111122224
type+++++++++++-
imageC1C2C2C2C2C2C2C2D4D4D4C4○D4C8.C22
kernelC42.445D4C23.38D4C4⋊M4(2)D4.D4C42Q16C4×C4○D4C2×C4⋊Q8C2×C8.C22C42C22×C4C4○D4C2×C4C4
# reps1214411222444

Matrix representation of C42.445D4 in GL6(𝔽17)

440000
0130000
0000160
00161162
001000
00116016
,
1600000
0160000
000010
00116115
0016000
0016101
,
1210000
1050000
0010706
001610611
0016108
001081014
,
5160000
7120000
0010706
0017116
0016108
0016070

G:=sub<GL(6,GF(17))| [4,0,0,0,0,0,4,13,0,0,0,0,0,0,0,16,1,1,0,0,0,1,0,16,0,0,16,16,0,0,0,0,0,2,0,16],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,1,16,16,0,0,0,16,0,1,0,0,1,1,0,0,0,0,0,15,0,1],[12,10,0,0,0,0,1,5,0,0,0,0,0,0,10,16,16,10,0,0,7,10,1,8,0,0,0,6,0,10,0,0,6,11,8,14],[5,7,0,0,0,0,16,12,0,0,0,0,0,0,10,1,16,16,0,0,7,7,1,0,0,0,0,11,0,7,0,0,6,6,8,0] >;

C42.445D4 in GAP, Magma, Sage, TeX

C_4^2._{445}D_4
% in TeX

G:=Group("C4^2.445D4");
// GroupNames label

G:=SmallGroup(128,1771);
// by ID

G=gap.SmallGroup(128,1771);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,120,758,2019,248,4037,1027,124]);
// Polycyclic

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

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