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G = C42.44D4order 128 = 27

26th non-split extension by C42 of D4 acting via D4/C2=C22

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

Aliases: C42.44D4, (C2×C42).18C4, (C2×C4).4M4(2), (C22×Q8).17C4, C4.15(C4.10D4), C22⋊C8.123C22, C42.6C4.13C2, C23.168(C22×C4), (C22×C4).431C23, (C2×C42).150C22, C22.20(C2×M4(2)), C2.11(C24.4C4), C2.8(C23.C23), C22.M4(2).10C2, (C2×C4×Q8).2C2, (C2×C4⋊C4).35C4, (C2×C4).1128(C2×D4), (C22×C4).70(C2×C4), C2.6(C2×C4.10D4), (C2×C4⋊C4).738C22, (C2×C4).314(C22⋊C4), C22.149(C2×C22⋊C4), SmallGroup(128,199)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C42.44D4
C1C2C22C2×C4C22×C4C2×C42C2×C4×Q8 — C42.44D4
C1C2C23 — C42.44D4
C1C22C2×C42 — C42.44D4
C1C2C22C22×C4 — C42.44D4

Generators and relations for C42.44D4
 G = < a,b,c,d | a4=b4=1, c4=b2, d2=b, ab=ba, cac-1=dad-1=ab2, cbc-1=a2b-1, bd=db, dcd-1=a2bc3 >

Subgroups: 204 in 118 conjugacy classes, 50 normal (16 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, Q8, C23, C42, C42, C4⋊C4, C2×C8, C22×C4, C22×C4, C2×Q8, C8⋊C4, C22⋊C8, C4⋊C8, C2×C42, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C4×Q8, C22×Q8, C22.M4(2), C42.6C4, C2×C4×Q8, C42.44D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, M4(2), C22×C4, C2×D4, C4.10D4, C2×C22⋊C4, C2×M4(2), C24.4C4, C23.C23, C2×C4.10D4, C42.44D4

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

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

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

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

32 conjugacy classes

class 1 2A2B2C2D2E4A···4H4I···4R8A···8H
order1222224···44···48···8
size1111222···24···48···8

32 irreducible representations

dim11111112244
type+++++-
imageC1C2C2C2C4C4C4D4M4(2)C4.10D4C23.C23
kernelC42.44D4C22.M4(2)C42.6C4C2×C4×Q8C2×C42C2×C4⋊C4C22×Q8C42C2×C4C4C2
# reps14214224822

Matrix representation of C42.44D4 in GL6(𝔽17)

400000
8130000
0016000
0001600
0011910
0081101
,
1300000
0130000
000100
0016000
000001
0000160
,
1160000
660000
0010280
002709
004272
00213210
,
1160000
1660000
0068150
0096015
0035119
00123811

G:=sub<GL(6,GF(17))| [4,8,0,0,0,0,0,13,0,0,0,0,0,0,16,0,11,8,0,0,0,16,9,11,0,0,0,0,1,0,0,0,0,0,0,1],[13,0,0,0,0,0,0,13,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0],[11,6,0,0,0,0,6,6,0,0,0,0,0,0,10,2,4,2,0,0,2,7,2,13,0,0,8,0,7,2,0,0,0,9,2,10],[11,16,0,0,0,0,6,6,0,0,0,0,0,0,6,9,3,12,0,0,8,6,5,3,0,0,15,0,11,8,0,0,0,15,9,11] >;

C42.44D4 in GAP, Magma, Sage, TeX

C_4^2._{44}D_4
% in TeX

G:=Group("C4^2.44D4");
// GroupNames label

G:=SmallGroup(128,199);
// by ID

G=gap.SmallGroup(128,199);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,232,1430,520,1123,851,172]);
// Polycyclic

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

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