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G = C42.91D4order 128 = 27

73rd non-split extension by C42 of D4 acting via D4/C2=C22

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

Aliases: C42.91D4, C42.18Q8, C42.628C23, C4⋊C8.12C4, C81C827C2, C4.84(C2×D8), (C2×C4).132D8, C4.56(C2×Q16), (C2×C4).59Q16, C4.5(C2.D8), C4.44(C8○D4), C22⋊C8.11C4, (C4×C8).37C22, (C22×C4).30Q8, C4⋊C8.268C22, C23.53(C4⋊C4), C42.124(C2×C4), (C22×C4).748D4, C22.5(C2.D8), C4⋊M4(2).24C2, (C2×C42).227C22, C2.6(M4(2).C4), C42.12C4.33C2, C2.6(C42.6C22), (C2×C4⋊C8).20C2, C2.5(C2×C2.D8), (C2×C8).27(C2×C4), (C2×C4).35(C4⋊C4), C22.85(C2×C4⋊C4), (C2×C4).155(C2×Q8), (C2×C4).1464(C2×D4), (C22×C4).249(C2×C4), (C2×C4).510(C22×C4), SmallGroup(128,303)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.91D4
C1C2C22C2×C4C42C2×C42C42.12C4 — C42.91D4
C1C2C2×C4 — C42.91D4
C1C2×C4C2×C42 — C42.91D4
C1C22C22C42 — C42.91D4

Generators and relations for C42.91D4
 G = < a,b,c,d | a4=b4=1, c4=a2, d2=b, ab=ba, cac-1=ab2, dad-1=a-1b2, bc=cb, bd=db, dcd-1=a2c3 >

Subgroups: 132 in 84 conjugacy classes, 54 normal (26 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×4], C4 [×3], C22, C22 [×2], C22 [×2], C8 [×8], C2×C4 [×6], C2×C4 [×4], C2×C4 [×3], C23, C42 [×4], C2×C8 [×4], C2×C8 [×6], M4(2) [×2], C22×C4 [×3], C4×C8 [×2], C22⋊C8 [×2], C4⋊C8 [×6], C4⋊C8 [×2], C2×C42, C22×C8, C2×M4(2), C81C8 [×4], C2×C4⋊C8, C4⋊M4(2), C42.12C4, C42.91D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], Q8 [×2], C23, C4⋊C4 [×4], D8 [×2], Q16 [×2], C22×C4, C2×D4, C2×Q8, C2.D8 [×4], C2×C4⋊C4, C8○D4 [×2], C2×D8, C2×Q16, C42.6C22, C2×C2.D8, M4(2).C4, C42.91D4

Smallest permutation representation of C42.91D4
On 64 points
Generators in S64
(1 7 5 3)(2 56 6 52)(4 50 8 54)(9 17 13 21)(10 16 14 12)(11 19 15 23)(18 24 22 20)(25 60 29 64)(26 32 30 28)(27 62 31 58)(33 39 37 35)(34 48 38 44)(36 42 40 46)(41 47 45 43)(49 55 53 51)(57 63 61 59)
(1 35 49 43)(2 36 50 44)(3 37 51 45)(4 38 52 46)(5 39 53 47)(6 40 54 48)(7 33 55 41)(8 34 56 42)(9 60 19 31)(10 61 20 32)(11 62 21 25)(12 63 22 26)(13 64 23 27)(14 57 24 28)(15 58 17 29)(16 59 18 30)
(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 29 35 15 49 58 43 17)(2 28 36 14 50 57 44 24)(3 27 37 13 51 64 45 23)(4 26 38 12 52 63 46 22)(5 25 39 11 53 62 47 21)(6 32 40 10 54 61 48 20)(7 31 33 9 55 60 41 19)(8 30 34 16 56 59 42 18)

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

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

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

38 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E···4J4K4L8A···8P8Q8R8S8T
order12222244444···4448···88888
size11112211112···2444···48888

38 irreducible representations

dim111111122222224
type++++++-+-+-
imageC1C2C2C2C2C4C4D4Q8D4Q8D8Q16C8○D4M4(2).C4
kernelC42.91D4C81C8C2×C4⋊C8C4⋊M4(2)C42.12C4C22⋊C8C4⋊C8C42C42C22×C4C22×C4C2×C4C2×C4C4C2
# reps141114411114482

Matrix representation of C42.91D4 in GL4(𝔽17) generated by

0100
16000
00160
0011
,
16000
01600
0040
0004
,
14300
141400
001615
0011
,
7100
11000
001513
0002
G:=sub<GL(4,GF(17))| [0,16,0,0,1,0,0,0,0,0,16,1,0,0,0,1],[16,0,0,0,0,16,0,0,0,0,4,0,0,0,0,4],[14,14,0,0,3,14,0,0,0,0,16,1,0,0,15,1],[7,1,0,0,1,10,0,0,0,0,15,0,0,0,13,2] >;

C42.91D4 in GAP, Magma, Sage, TeX

C_4^2._{91}D_4
% in TeX

G:=Group("C4^2.91D4");
// GroupNames label

G:=SmallGroup(128,303);
// by ID

G=gap.SmallGroup(128,303);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,288,723,1123,136,172]);
// Polycyclic

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

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