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G = C42.61D4order 128 = 27

43rd non-split extension by C42 of D4 acting via D4/C2=C22

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

Aliases: C42.61D4, (C4×D4)⋊7C4, C4⋊Q812C4, C41D49C4, (C2×C4).16D8, C428C42C2, C42.78(C2×C4), (C2×C4).24SD16, C22.11(C2×D8), C22.SD162C2, C23.507(C2×D4), (C22×C4).218D4, C4.35(D4⋊C4), C22.29(C2×SD16), C42.12C417C2, C4⋊D4.141C22, C22⋊C8.166C22, (C22×C4).639C23, (C2×C42).181C22, C22.26C24.8C2, C2.C42.7C22, C2.22(C42⋊C22), C2.21(C23.C23), C4⋊C4.17(C2×C4), (C2×D4).14(C2×C4), C2.9(C2×D4⋊C4), (C2×C4).1163(C2×D4), (C2×C4).129(C22×C4), (C2×C4).239(C22⋊C4), C22.193(C2×C22⋊C4), SmallGroup(128,249)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.61D4
C1C2C22C23C22×C4C2×C42C22.26C24 — C42.61D4
C1C22C2×C4 — C42.61D4
C1C22C2×C42 — C42.61D4
C1C2C22C22×C4 — C42.61D4

Generators and relations for C42.61D4
 G = < a,b,c,d | a4=b4=c4=1, d2=b-1, ab=ba, cac-1=ab2, dad-1=a-1b2, cbc-1=a2b, bd=db, dcd-1=b-1c-1 >

Subgroups: 316 in 132 conjugacy classes, 52 normal (28 characteristic)
C1, C2 [×3], C2 [×4], C4 [×4], C4 [×8], C22 [×3], C22 [×8], C8 [×2], C2×C4 [×6], C2×C4 [×4], C2×C4 [×14], D4 [×10], Q8 [×2], C23, C23 [×2], C42 [×4], C22⋊C4 [×4], C4⋊C4 [×2], C4⋊C4 [×3], 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, C4×C8, C22⋊C8 [×2], C4⋊C8, C2×C42, C2×C4⋊C4, C4×D4 [×2], C4×D4, C4⋊D4 [×2], C4⋊D4, C4.4D4, C41D4, C4⋊Q8, C2×C4○D4, C22.SD16 [×4], C428C4, C42.12C4, C22.26C24, C42.61D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], D8 [×2], SD16 [×2], C22×C4, C2×D4 [×2], D4⋊C4 [×4], C2×C22⋊C4, C2×D8, C2×SD16, C23.C23, C2×D4⋊C4, C42⋊C22, C42.61D4

Smallest permutation representation of C42.61D4
On 32 points
Generators in S32
(1 9 30 17)(2 22 31 14)(3 11 32 19)(4 24 25 16)(5 13 26 21)(6 18 27 10)(7 15 28 23)(8 20 29 12)
(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 26 13)(2 24)(3 11 28 23)(4 10)(5 21 30 9)(6 20)(7 15 32 19)(8 14)(12 27)(16 31)(18 25)(22 29)
(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)| (1,9,30,17)(2,22,31,14)(3,11,32,19)(4,24,25,16)(5,13,26,21)(6,18,27,10)(7,15,28,23)(8,20,29,12), (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), (1,17,26,13)(2,24)(3,11,28,23)(4,10)(5,21,30,9)(6,20)(7,15,32,19)(8,14)(12,27)(16,31)(18,25)(22,29), (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( (1,9,30,17)(2,22,31,14)(3,11,32,19)(4,24,25,16)(5,13,26,21)(6,18,27,10)(7,15,28,23)(8,20,29,12), (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), (1,17,26,13)(2,24)(3,11,28,23)(4,10)(5,21,30,9)(6,20)(7,15,32,19)(8,14)(12,27)(16,31)(18,25)(22,29), (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([(1,9,30,17),(2,22,31,14),(3,11,32,19),(4,24,25,16),(5,13,26,21),(6,18,27,10),(7,15,28,23),(8,20,29,12)], [(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)], [(1,17,26,13),(2,24),(3,11,28,23),(4,10),(5,21,30,9),(6,20),(7,15,32,19),(8,14),(12,27),(16,31),(18,25),(22,29)], [(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···4H4I4J4K···4P8A···8H
order122222224···4444···48···8
size111122882···2448···84···4

32 irreducible representations

dim11111111222244
type++++++++
imageC1C2C2C2C2C4C4C4D4D4D8SD16C23.C23C42⋊C22
kernelC42.61D4C22.SD16C428C4C42.12C4C22.26C24C4×D4C41D4C4⋊Q8C42C22×C4C2×C4C2×C4C2C2
# reps14111422224422

Matrix representation of C42.61D4 in GL6(𝔽17)

010000
1600000
000100
001000
000201
0015010
,
0160000
100000
004000
000400
000040
000004
,
040000
400000
0001600
001000
00012013
0012040
,
5120000
550000
000909
009080
000009
000090

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

C42.61D4 in GAP, Magma, Sage, TeX

C_4^2._{61}D_4
% in TeX

G:=Group("C4^2.61D4");
// GroupNames label

G:=SmallGroup(128,249);
// by ID

G=gap.SmallGroup(128,249);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,1430,387,184,1123,1018,248,1971]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^4=c^4=1,d^2=b^-1,a*b=b*a,c*a*c^-1=a*b^2,d*a*d^-1=a^-1*b^2,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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