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G = C42.691C23order 128 = 27

106th non-split extension by C42 of C23 acting via C23/C22=C2

p-group, metabelian, nilpotent (class 2), monomial

Aliases: C42.691C23, C4.1662+ 1+4, D47(C2×C8), (C2×D4)⋊9C8, (C8×D4)⋊3C2, C233(C2×C8), (C4×C8)⋊4C22, C4⋊C895C22, D42(C22⋊C8), (C4×D4).32C4, C2.7(C23×C8), C4.19(C22×C8), C24.83(C2×C4), (C22×C8)⋊5C22, C22⋊C883C22, (C2×C4).669C24, (C2×C8).481C23, C42.220(C2×C4), (C22×D4).41C4, C22.2(C22×C8), (C4×D4).362C22, C2.3(Q8○M4(2)), C22.43(C23×C4), C42.12C423C2, (C23×C4).186C22, C23.149(C22×C4), (C2×C42).779C22, (C22×C4).1280C23, C2.4(C22.11C24), C4⋊C8(C4⋊C8), (C2×C4)⋊4(C2×C8), C22⋊C8(C4×D4), C4⋊C4(C22⋊C8), (C2×C4×D4).75C2, (C2×C4⋊C4).75C4, (C2×D4)(C22⋊C8), C4⋊C4.250(C2×C4), (C2×C22⋊C8)⋊17C2, C22⋊C8(C22⋊C8), C22⋊C4(C22⋊C8), (C2×D4).252(C2×C4), C22⋊C4.93(C2×C4), (C2×C22⋊C4).33C4, (C2×C4).499(C22×C4), (C22×C4).137(C2×C4), SmallGroup(128,1704)

Series: Derived Chief Lower central Upper central Jennings

C1C2 — C42.691C23
C1C2C4C2×C4C22×C4C23×C4C2×C4×D4 — C42.691C23
C1C2 — C42.691C23
C1C2×C4 — C42.691C23
C1C2C2C2×C4 — C42.691C23

Generators and relations for C42.691C23
 G = < a,b,c,d,e | a4=b4=d2=1, c2=b, e2=a2, ab=ba, ac=ca, dad=a-1, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece-1=a2c, ede-1=a2d >

Subgroups: 388 in 250 conjugacy classes, 174 normal (16 characteristic)
C1, C2 [×3], C2 [×10], C4 [×6], C4 [×7], C22, C22 [×10], C22 [×18], C8 [×8], C2×C4 [×2], C2×C4 [×12], C2×C4 [×17], D4 [×16], C23, C23 [×12], C23 [×4], C42 [×4], C22⋊C4 [×8], C4⋊C4 [×4], C2×C8 [×8], C2×C8 [×8], C22×C4 [×3], C22×C4 [×10], C22×C4 [×4], C2×D4 [×12], C24 [×2], C4×C8 [×4], C22⋊C8 [×12], C4⋊C8 [×4], C2×C42, C2×C22⋊C4 [×2], C2×C4⋊C4, C4×D4 [×8], C22×C8 [×8], C23×C4 [×2], C22×D4, C2×C22⋊C8 [×4], C42.12C4 [×2], C8×D4 [×8], C2×C4×D4, C42.691C23
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C8 [×8], C2×C4 [×28], C23 [×15], C2×C8 [×28], C22×C4 [×14], C24, C22×C8 [×14], C23×C4, 2+ 1+4 [×2], C22.11C24, C23×C8, Q8○M4(2), C42.691C23

Smallest permutation representation of C42.691C23
On 32 points
Generators in S32
(1 23 31 10)(2 24 32 11)(3 17 25 12)(4 18 26 13)(5 19 27 14)(6 20 28 15)(7 21 29 16)(8 22 30 9)
(1 3 5 7)(2 4 6 8)(9 11 13 15)(10 12 14 16)(17 19 21 23)(18 20 22 24)(25 27 29 31)(26 28 30 32)
(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)
(1 31)(2 32)(3 25)(4 26)(5 27)(6 28)(7 29)(8 30)
(1 14 31 19)(2 20 32 15)(3 16 25 21)(4 22 26 9)(5 10 27 23)(6 24 28 11)(7 12 29 17)(8 18 30 13)

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

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

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

68 conjugacy classes

class 1 2A2B2C2D···2M4A4B4C4D4E···4V8A···8AF
order12222···244444···48···8
size11112···211112···22···2

68 irreducible representations

dim111111111144
type++++++
imageC1C2C2C2C2C4C4C4C4C82+ 1+4Q8○M4(2)
kernelC42.691C23C2×C22⋊C8C42.12C4C8×D4C2×C4×D4C2×C22⋊C4C2×C4⋊C4C4×D4C22×D4C2×D4C4C2
# reps1428142823222

Matrix representation of C42.691C23 in GL5(𝔽17)

10000
00100
016000
00001
000160
,
130000
01000
00100
00010
00001
,
90000
00010
00001
01000
00100
,
160000
016000
00100
000160
00001
,
10000
001600
01000
00001
000160

G:=sub<GL(5,GF(17))| [1,0,0,0,0,0,0,16,0,0,0,1,0,0,0,0,0,0,0,16,0,0,0,1,0],[13,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[9,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,1,0,0,0,0,0,1,0,0],[16,0,0,0,0,0,16,0,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,16,0,0,0,0,0,0,0,16,0,0,0,1,0] >;

C42.691C23 in GAP, Magma, Sage, TeX

C_4^2._{691}C_2^3
% in TeX

G:=Group("C4^2.691C2^3");
// GroupNames label

G:=SmallGroup(128,1704);
// by ID

G=gap.SmallGroup(128,1704);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,219,675,124]);
// Polycyclic

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

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