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G = C42.55D4order 128 = 27

37th non-split extension by C42 of D4 acting via D4/C2=C22

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

Aliases: C42.55D4, C4⋊Q88C4, (C4×D4)⋊3C4, C4.13C4≀C2, C41D47C4, C424C45C2, C42.72(C2×C4), (C22×C4).734D4, C23.495(C2×D4), C42.6C429C2, C22.SD1618C2, C4⋊D4.133C22, C22⋊C8.128C22, C22.34(C8⋊C22), (C22×C4).627C23, (C2×C42).175C22, C2.7(C23.37D4), C22.26C24.6C2, C2.C42.504C22, C2.15(C23.C23), C4⋊C4.5(C2×C4), C2.22(C2×C4≀C2), (C2×D4).6(C2×C4), (C2×C4).1151(C2×D4), (C2×C4).117(C22×C4), (C2×C4).317(C22⋊C4), C22.181(C2×C22⋊C4), SmallGroup(128,237)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.55D4
C1C2C22C23C22×C4C2×C42C22.26C24 — C42.55D4
C1C22C2×C4 — C42.55D4
C1C22C2×C42 — C42.55D4
C1C2C22C22×C4 — C42.55D4

Generators and relations for C42.55D4
 G = < a,b,c,d | a4=b4=c4=1, d2=b-1, ab=ba, ac=ca, dad-1=a-1, cbc-1=a2b, bd=db, dcd-1=b-1c-1 >

Subgroups: 316 in 136 conjugacy classes, 46 normal (26 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×11], C22, C22 [×2], C22 [×8], C8 [×2], C2×C4 [×6], C2×C4 [×19], D4 [×10], Q8 [×2], C23, C23 [×2], C42 [×4], C42 [×4], C22⋊C4 [×4], C4⋊C4 [×2], C4⋊C4, C2×C8 [×2], C22×C4 [×3], C22×C4 [×4], C2×D4 [×2], C2×D4 [×4], C2×Q8, C4○D4 [×4], C2.C42 [×2], C2.C42, C8⋊C4, C22⋊C8 [×2], C4⋊C8, C2×C42, C2×C42, C4×D4 [×2], C4×D4, C4⋊D4 [×2], C4⋊D4, C4.4D4, C41D4, C4⋊Q8, C2×C4○D4, C22.SD16 [×4], C424C4, C42.6C4, C22.26C24, C42.55D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×2], C4≀C2 [×2], C2×C22⋊C4, C8⋊C22 [×2], C23.C23, C23.37D4, C2×C4≀C2, C42.55D4

Smallest permutation representation of C42.55D4
On 32 points
Generators in S32
(1 14 31 21)(2 22 32 15)(3 16 25 23)(4 24 26 9)(5 10 27 17)(6 18 28 11)(7 12 29 19)(8 20 30 13)
(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 31 29 27)(26 32 30 28)
(1 17 31 10)(2 20 6 24)(3 12 25 19)(4 15 8 11)(5 21 27 14)(7 16 29 23)(9 32 13 28)(18 26 22 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)

G:=sub<Sym(32)| (1,14,31,21)(2,22,32,15)(3,16,25,23)(4,24,26,9)(5,10,27,17)(6,18,28,11)(7,12,29,19)(8,20,30,13), (1,7,5,3)(2,8,6,4)(9,15,13,11)(10,16,14,12)(17,23,21,19)(18,24,22,20)(25,31,29,27)(26,32,30,28), (1,17,31,10)(2,20,6,24)(3,12,25,19)(4,15,8,11)(5,21,27,14)(7,16,29,23)(9,32,13,28)(18,26,22,30), (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)>;

G:=Group( (1,14,31,21)(2,22,32,15)(3,16,25,23)(4,24,26,9)(5,10,27,17)(6,18,28,11)(7,12,29,19)(8,20,30,13), (1,7,5,3)(2,8,6,4)(9,15,13,11)(10,16,14,12)(17,23,21,19)(18,24,22,20)(25,31,29,27)(26,32,30,28), (1,17,31,10)(2,20,6,24)(3,12,25,19)(4,15,8,11)(5,21,27,14)(7,16,29,23)(9,32,13,28)(18,26,22,30), (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) );

G=PermutationGroup([(1,14,31,21),(2,22,32,15),(3,16,25,23),(4,24,26,9),(5,10,27,17),(6,18,28,11),(7,12,29,19),(8,20,30,13)], [(1,7,5,3),(2,8,6,4),(9,15,13,11),(10,16,14,12),(17,23,21,19),(18,24,22,20),(25,31,29,27),(26,32,30,28)], [(1,17,31,10),(2,20,6,24),(3,12,25,19),(4,15,8,11),(5,21,27,14),(7,16,29,23),(9,32,13,28),(18,26,22,30)], [(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)])

32 conjugacy classes

class 1 2A2B2C2D2E2F2G4A···4H4I···4R4S4T8A8B8C8D
order122222224···44···4448888
size111122882···24···4888888

32 irreducible representations

dim1111111122244
type++++++++
imageC1C2C2C2C2C4C4C4D4D4C4≀C2C8⋊C22C23.C23
kernelC42.55D4C22.SD16C424C4C42.6C4C22.26C24C4×D4C41D4C4⋊Q8C42C22×C4C4C22C2
# reps1411142222822

Matrix representation of C42.55D4 in GL6(𝔽17)

1600000
0160000
004025
000400
0000130
0000013
,
400000
040000
0016083
0021914
0000131
000024
,
100000
0130000
001313016
000400
0000160
000091
,
0130000
100000
001616816
000091
00411111
000400

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,2,0,13,0,0,0,5,0,0,13],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,16,2,0,0,0,0,0,1,0,0,0,0,8,9,13,2,0,0,3,14,1,4],[1,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,13,4,0,0,0,0,0,0,16,9,0,0,16,0,0,1],[0,1,0,0,0,0,13,0,0,0,0,0,0,0,16,0,4,0,0,0,16,0,11,4,0,0,8,9,1,0,0,0,16,1,11,0] >;

C42.55D4 in GAP, Magma, Sage, TeX

C_4^2._{55}D_4
% in TeX

G:=Group("C4^2.55D4");
// GroupNames label

G:=SmallGroup(128,237);
// by ID

G=gap.SmallGroup(128,237);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,387,184,1123,1018,248,1971]);
// Polycyclic

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

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