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G = C23.21C42order 128 = 27

3rd non-split extension by C23 of C42 acting via C42/C2×C4=C2

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

Aliases: C23.21C42, C23.5M4(2), C22⋊C85C4, C22⋊C42C8, (C22×C8)⋊1C4, C22.6(C4×C8), C23.5(C2×C8), (C22×C4).4Q8, C22.9(C4⋊C8), C24.36(C2×C4), C4.32(C23⋊C4), C23.37(C4⋊C4), (C22×C4).637D4, (C23×C4).2C22, C22.5(C8⋊C4), C22.10(C22⋊C8), C2.2(C23.9D4), C2.2(M4(2)⋊4C4), C23.135(C22⋊C4), C2.11(C22.7C42), C22.23(C2.C42), (C2×C4).67(C4⋊C4), (C2×C22⋊C8).2C2, (C4×C22⋊C4).1C2, (C2×C22⋊C4).14C4, (C22×C4).424(C2×C4), (C2×C4).371(C22⋊C4), SmallGroup(128,14)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C23.21C42
C1C2C22C2×C4C22×C4C23×C4C2×C22⋊C8 — C23.21C42
C1C2C22 — C23.21C42
C1C2×C4C23×C4 — C23.21C42
C1C2C22C23×C4 — C23.21C42

Generators and relations for C23.21C42
 G = < a,b,c,d,e | a2=b2=c2=1, d4=e4=c, dad-1=ab=ba, ac=ca, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede-1=abd >

Subgroups: 240 in 114 conjugacy classes, 46 normal (26 characteristic)
C1, C2 [×3], C2 [×6], C4 [×2], C4 [×7], C22 [×3], C22 [×4], C22 [×10], C8 [×4], C2×C4 [×4], C2×C4 [×17], C23 [×3], C23 [×4], C23 [×2], C42 [×2], C22⋊C4 [×4], C22⋊C4 [×2], C2×C8 [×8], C22×C4 [×6], C22×C4 [×4], C24, C2.C42, C22⋊C8 [×2], C22⋊C8 [×3], C2×C42, C2×C22⋊C4 [×2], C22×C8 [×2], C22×C8, C23×C4, C4×C22⋊C4, C2×C22⋊C8 [×2], C23.21C42
Quotients: C1, C2 [×3], C4 [×6], C22, C8 [×4], C2×C4 [×3], D4 [×3], Q8, C42, C22⋊C4 [×3], C4⋊C4 [×3], C2×C8 [×2], M4(2) [×2], C2.C42, C4×C8, C8⋊C4, C22⋊C8 [×2], C23⋊C4 [×2], C4⋊C8 [×2], C22.7C42, C23.9D4, M4(2)⋊4C4, C23.21C42

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

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

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

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

44 conjugacy classes

class 1 2A2B2C2D···2I4A4B4C4D4E···4J4K···4R8A···8P
order12222···244444···44···48···8
size11112···211112···24···44···4

44 irreducible representations

dim111111122244
type++++-+
imageC1C2C2C4C4C4C8D4Q8M4(2)C23⋊C4M4(2)⋊4C4
kernelC23.21C42C4×C22⋊C4C2×C22⋊C8C22⋊C8C2×C22⋊C4C22×C8C22⋊C4C22×C4C22×C4C23C4C2
# reps1124441631422

Matrix representation of C23.21C42 in GL6(𝔽17)

1600000
0160000
0016200
000100
0000162
000001
,
100000
010000
0016000
0001600
0000160
0000016
,
1600000
0160000
001000
000100
000010
000001
,
1590000
120000
0000115
0000116
0016000
0016100
,
1590000
020000
000010
000001
0016000
0001600

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

C23.21C42 in GAP, Magma, Sage, TeX

C_2^3._{21}C_4^2
% in TeX

G:=Group("C2^3.21C4^2");
// GroupNames label

G:=SmallGroup(128,14);
// by ID

G=gap.SmallGroup(128,14);
# by ID

G:=PCGroup([7,-2,2,-2,2,2,-2,2,56,85,120,758,570,248,172]);
// Polycyclic

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

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