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G = C42.389D4order 128 = 27

22nd non-split extension by C42 of D4 acting via D4/C22=C2

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

Aliases: C42.389D4, C4⋊C8.9C4, (C2×C4).117D8, C22⋊C8.9C4, (C2×C4).51Q16, C42.39(C2×C4), (C2×C4).13C42, (C2×C4).91SD16, C4.9(C8.C4), (C22×C4).24Q8, C23.44(C4⋊C4), (C22×C4).723D4, C4.43(D4⋊C4), C22.2(C4.Q8), C22.2(C2.D8), C4.23(Q8⋊C4), C4⋊M4(2).7C2, C2.8(C4.C42), C2.8(C22.4Q16), (C2×C42).136C22, C2.9(M4(2)⋊4C4), C42.12C4.10C2, C22.51(C2.C42), (C2×C4⋊C8).7C2, (C2×C4).97(C4⋊C4), (C22×C4).156(C2×C4), (C2×C4).374(C22⋊C4), SmallGroup(128,33)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.389D4
C1C2C22C2×C4C42C2×C42C42.12C4 — C42.389D4
C1C22C2×C4 — C42.389D4
C1C2×C4C2×C42 — C42.389D4
C1C22C22C2×C42 — C42.389D4

Generators and relations for C42.389D4
 G = < a,b,c,d | a4=b4=1, c4=a2, d2=a, ab=ba, ac=ca, ad=da, cbc-1=dbd-1=a2b-1, dcd-1=a-1bc3 >

Subgroups: 128 in 74 conjugacy classes, 42 normal (34 characteristic)
C1, C2 [×3], C2 [×2], C4 [×4], C4 [×2], C4 [×3], C22 [×3], C22 [×2], C8 [×6], C2×C4 [×10], C2×C4 [×3], C23, C42 [×4], C2×C8 [×8], M4(2) [×2], C22×C4 [×3], C4×C8, C22⋊C8 [×2], C4⋊C8 [×4], C4⋊C8 [×3], C2×C42, C22×C8, C2×M4(2), C2×C4⋊C8, C4⋊M4(2), C42.12C4, C42.389D4
Quotients: C1, C2 [×3], C4 [×6], C22, C2×C4 [×3], D4 [×3], Q8, C42, C22⋊C4 [×3], C4⋊C4 [×3], D8, SD16 [×2], Q16, C2.C42, D4⋊C4 [×2], Q8⋊C4 [×2], C4.Q8, C2.D8, C8.C4 [×2], C22.4Q16, C4.C42, M4(2)⋊4C4, C42.389D4

Smallest permutation representation of C42.389D4
On 64 points
Generators in S64
(1 7 5 3)(2 8 6 4)(9 15 13 11)(10 16 14 12)(17 23 21 19)(18 24 22 20)(25 27 29 31)(26 28 30 32)(33 35 37 39)(34 36 38 40)(41 43 45 47)(42 44 46 48)(49 51 53 55)(50 52 54 56)(57 63 61 59)(58 64 62 60)
(1 60 16 21)(2 18 9 57)(3 62 10 23)(4 20 11 59)(5 64 12 17)(6 22 13 61)(7 58 14 19)(8 24 15 63)(25 44 39 52)(26 49 40 41)(27 46 33 54)(28 51 34 43)(29 48 35 56)(30 53 36 45)(31 42 37 50)(32 55 38 47)
(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 27 7 29 5 31 3 25)(2 47 8 41 6 43 4 45)(9 55 15 49 13 51 11 53)(10 39 16 33 14 35 12 37)(17 46 23 48 21 42 19 44)(18 34 24 36 22 38 20 40)(26 57 28 63 30 61 32 59)(50 58 52 64 54 62 56 60)

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

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

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

38 conjugacy classes

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

38 irreducible representations

dim11111122222224
type++++++-+-
imageC1C2C2C2C4C4D4D4Q8D8SD16Q16C8.C4M4(2)⋊4C4
kernelC42.389D4C2×C4⋊C8C4⋊M4(2)C42.12C4C22⋊C8C4⋊C8C42C22×C4C22×C4C2×C4C2×C4C2×C4C4C2
# reps11114821124282

Matrix representation of C42.389D4 in GL4(𝔽17) generated by

4000
0400
00160
00016
,
4000
0400
0001
00160
,
8000
14200
001016
00167
,
81600
9900
001610
00101
G:=sub<GL(4,GF(17))| [4,0,0,0,0,4,0,0,0,0,16,0,0,0,0,16],[4,0,0,0,0,4,0,0,0,0,0,16,0,0,1,0],[8,14,0,0,0,2,0,0,0,0,10,16,0,0,16,7],[8,9,0,0,16,9,0,0,0,0,16,10,0,0,10,1] >;

C42.389D4 in GAP, Magma, Sage, TeX

C_4^2._{389}D_4
% in TeX

G:=Group("C4^2.389D4");
// GroupNames label

G:=SmallGroup(128,33);
// by ID

G=gap.SmallGroup(128,33);
# by ID

G:=PCGroup([7,-2,2,-2,2,2,-2,2,56,85,120,758,723,184,248,3924,102]);
// Polycyclic

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

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