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G = C42.693C23order 128 = 27

108th non-split extension by C42 of C23 acting via C23/C22=C2

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

Aliases: C42.693C23, C4.1692+ 1+4, C86D440C2, C4⋊C891C22, (C4×C8)⋊60C22, (C4×D4).35C4, (C2×C4)⋊4M4(2), C24.86(C2×C4), C22⋊C846C22, (C2×C8).433C23, (C2×C4).672C24, C42.223(C2×C4), (C22×D4).44C4, C4.15(C2×M4(2)), C24.4C436C2, (C4×D4).299C22, C42.12C453C2, C2.28(Q8○M4(2)), (C2×M4(2))⋊46C22, (C22×C4).940C23, (C23×C4).531C22, C23.106(C22×C4), C22.196(C23×C4), (C2×C42).782C22, C22.29(C2×M4(2)), C2.20(C22×M4(2)), C2.46(C22.11C24), (C2×C4×D4).78C2, (C2×C4⋊C4).78C4, C4⋊C4.230(C2×C4), (C2×D4).236(C2×C4), (C2×C22⋊C4).52C4, C22⋊C4.78(C2×C4), (C22×C4).353(C2×C4), (C2×C4).298(C22×C4), SmallGroup(128,1707)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C42.693C23
C1C2C4C2×C4C22×C4C23×C4C2×C4×D4 — C42.693C23
C1C22 — C42.693C23
C1C2×C4 — C42.693C23
C1C2C2C2×C4 — C42.693C23

Generators and relations for C42.693C23
 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, dcd=b2c, ece-1=a2c, ede-1=a2d >

Subgroups: 388 in 226 conjugacy classes, 134 normal (16 characteristic)
C1, C2 [×3], C2 [×6], C4 [×6], C4 [×7], C22, C22 [×2], C22 [×22], C8 [×8], C2×C4 [×2], C2×C4 [×12], C2×C4 [×17], D4 [×8], C23, C23 [×4], C23 [×8], C42 [×4], C22⋊C4 [×8], C4⋊C4 [×4], C2×C8 [×8], M4(2) [×8], C22×C4 [×3], C22×C4 [×10], C22×C4 [×4], C2×D4 [×4], C2×D4 [×4], C24 [×2], C4×C8 [×4], C22⋊C8 [×12], C4⋊C8 [×4], C2×C42, C2×C22⋊C4 [×2], C2×C4⋊C4, C4×D4 [×8], C2×M4(2) [×8], C23×C4 [×2], C22×D4, C24.4C4 [×4], C42.12C4 [×2], C86D4 [×8], C2×C4×D4, C42.693C23
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], M4(2) [×4], C22×C4 [×14], C24, C2×M4(2) [×6], C23×C4, 2+ 1+4 [×2], C22.11C24, C22×M4(2), Q8○M4(2), C42.693C23

Smallest permutation representation of C42.693C23
On 32 points
Generators in S32
(1 19 27 10)(2 20 28 11)(3 21 29 12)(4 22 30 13)(5 23 31 14)(6 24 32 15)(7 17 25 16)(8 18 26 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 27)(2 32)(3 29)(4 26)(5 31)(6 28)(7 25)(8 30)(9 13)(11 15)(18 22)(20 24)
(1 14 27 23)(2 24 28 15)(3 16 29 17)(4 18 30 9)(5 10 31 19)(6 20 32 11)(7 12 25 21)(8 22 26 13)

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

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

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

44 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I4A4B4C4D4E···4N4O4P4Q4R8A···8P
order122222222244444···444448···8
size111122444411112···244444···4

44 irreducible representations

dim111111111244
type++++++
imageC1C2C2C2C2C4C4C4C4M4(2)2+ 1+4Q8○M4(2)
kernelC42.693C23C24.4C4C42.12C4C86D4C2×C4×D4C2×C22⋊C4C2×C4⋊C4C4×D4C22×D4C2×C4C4C2
# reps142814282822

Matrix representation of C42.693C23 in GL6(𝔽17)

100000
010000
001200
00161600
000012
00001616
,
1300000
0130000
004000
000400
000040
000004
,
0160000
400000
000010
000001
004000
000400
,
100000
0160000
00161500
000100
000012
0000016
,
100000
010000
001200
00161600
00001615
000011

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

C42.693C23 in GAP, Magma, Sage, TeX

C_4^2._{693}C_2^3
% in TeX

G:=Group("C4^2.693C2^3");
// GroupNames label

G:=SmallGroup(128,1707);
// by ID

G=gap.SmallGroup(128,1707);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,1430,891,2467,124]);
// 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,d*c*d=b^2*c,e*c*e^-1=a^2*c,e*d*e^-1=a^2*d>;
// generators/relations

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