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G = C23.32C23order 64 = 26

5th non-split extension by C23 of C23 acting via C23/C22=C2

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

Aliases: C23.32C23, C42.34C22, C22.12C24, C2.12- 1+4, (C2×Q8)⋊9C4, (C4×Q8)⋊5C2, Q8(C22⋊C4), Q8.8(C2×C4), C2.8(C23×C4), C4⋊C4.80C22, C4.20(C22×C4), (C22×Q8).7C2, (C2×C4).128C23, (C2×Q8).72C22, C42⋊C2.10C2, C22⋊C4.28C22, (C22×C4).57C22, C22.12(C22×C4), (C2×C4).31(C2×C4), SmallGroup(64,200)

Series: Derived Chief Lower central Upper central Jennings

C1C2 — C23.32C23
C1C2C22C23C22×C4C22×Q8 — C23.32C23
C1C2 — C23.32C23
C1C22 — C23.32C23
C1C22 — C23.32C23

Generators and relations for C23.32C23
 G = < a,b,c,d,e,f | a2=b2=c2=1, d2=c, e2=f2=b, dad-1=ab=ba, ac=ca, ae=ea, af=fa, bc=cb, ede-1=bd=db, fef-1=be=eb, bf=fb, cd=dc, ce=ec, cf=fc, df=fd >

Subgroups: 145 in 133 conjugacy classes, 121 normal (6 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×12], C4 [×8], C22, C22 [×2], C22 [×2], C2×C4 [×26], Q8 [×16], C23, C42 [×12], C22⋊C4 [×4], C4⋊C4 [×12], C22×C4 [×3], C2×Q8 [×12], C42⋊C2 [×6], C4×Q8 [×8], C22×Q8, C23.32C23
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], C22×C4 [×14], C24, C23×C4, 2- 1+4 [×2], C23.32C23

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

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

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

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

C23.32C23 is a maximal subgroup of
(C2×Q8).211D4  C4.10D42C4  C4⋊Q815C4  C4.4D413C4  C23.4C24  C42.276C23  C42.278C23  C42.279C23  C42.16C23  C42.17C23  C42.21C23  C42.355C23  C42.360C23  C42.361C23  C22.14C25  C4×2- 1+4  C22.75C25  C22.76C25  C22.84C25  C22.99C25  C22.105C25  C22.113C25  (C2×Q8)⋊C12  D5.2- 1+4
 C42.D2p: C42.6D4  C42.7D4  C42.87D6  C42.125D6  C42.87D10  C42.125D10  C42.87D14  C42.125D14 ...
 C2p.2- 1+4: C22.91C25  C22.98C25  C22.107C25  C23.146C24  C6.422- 1+4  C10.422- 1+4  C14.422- 1+4 ...
C23.32C23 is a maximal quotient of
C24.524C23  Q84C42  C23.192C24  C23.195C24  C24.545C23  C23.202C24  C424Q8  C23.214C24  C23.218C24  Q8×C22⋊C4  C23.226C24  C23.233C24  C23.238C24  C23.244C24  C23.247C24  C23.253C24  C24.227C23  C23.263C24  C23.264C24  D5.2- 1+4
 C42.D2p: C42.159D4  C42.161D4  C42.87D6  C42.125D6  C42.87D10  C42.125D10  C42.87D14  C42.125D14 ...
 C2p.2- 1+4: C42.695C23  C42.302C23  Q8.4M4(2)  C42.696C23  C42.304C23  C42.305C23  C6.422- 1+4  C10.422- 1+4 ...

34 conjugacy classes

class 1 2A2B2C2D2E4A···4AB
order1222224···4
size1111222···2

34 irreducible representations

dim111114
type++++-
imageC1C2C2C2C42- 1+4
kernelC23.32C23C42⋊C2C4×Q8C22×Q8C2×Q8C2
# reps1681162

Matrix representation of C23.32C23 in GL5(𝔽5)

40000
01000
00100
00040
02204
,
10000
04000
00400
00040
00004
,
40000
04000
00400
00040
00004
,
30000
00010
02243
04000
00003
,
10000
02000
00300
00030
00122
,
40000
00100
04000
02243
03011

G:=sub<GL(5,GF(5))| [4,0,0,0,0,0,1,0,0,2,0,0,1,0,2,0,0,0,4,0,0,0,0,0,4],[1,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,4],[4,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,4],[3,0,0,0,0,0,0,2,4,0,0,0,2,0,0,0,1,4,0,0,0,0,3,0,3],[1,0,0,0,0,0,2,0,0,0,0,0,3,0,1,0,0,0,3,2,0,0,0,0,2],[4,0,0,0,0,0,0,4,2,3,0,1,0,2,0,0,0,0,4,1,0,0,0,3,1] >;

C23.32C23 in GAP, Magma, Sage, TeX

C_2^3._{32}C_2^3
% in TeX

G:=Group("C2^3.32C2^3");
// GroupNames label

G:=SmallGroup(64,200);
// by ID

G=gap.SmallGroup(64,200);
# by ID

G:=PCGroup([6,-2,2,2,2,-2,2,192,217,103,188,86,579]);
// Polycyclic

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

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