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## G = C23.33C23order 64 = 26

### 6th non-split extension by C23 of C23 acting via C23/C22=C2

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2 — C23.33C23
 Chief series C1 — C2 — C22 — C2×C4 — C22×C4 — C2×C4○D4 — C23.33C23
 Lower central C1 — C2 — C23.33C23
 Upper central C1 — C22 — C23.33C23
 Jennings C1 — C22 — C23.33C23

Generators and relations for C23.33C23
G = < a,b,c,d,e,f | a2=b2=c2=e2=1, d2=c, f2=b, eae=ab=ba, ac=ca, ad=da, af=fa, bc=cb, fdf-1=bd=db, fef-1=be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed >

Subgroups: 185 in 147 conjugacy classes, 121 normal (10 characteristic)
C1, C2 [×3], C2 [×6], C4 [×8], C4 [×8], C22, C22 [×6], C22 [×6], C2×C4, C2×C4 [×23], C2×C4 [×6], D4 [×12], Q8 [×4], C23 [×3], C42 [×6], C22⋊C4 [×6], C4⋊C4, C4⋊C4 [×9], C22×C4 [×9], C2×D4 [×3], C2×Q8, C4○D4 [×8], C2×C4⋊C4 [×3], C42⋊C2 [×3], C4×D4 [×6], C4×Q8 [×2], C2×C4○D4, C23.33C23
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], C22×C4 [×14], C24, C23×C4, 2+ 1+4, 2- 1+4, C23.33C23

Smallest permutation representation of C23.33C23
On 32 points
Generators in S32
(5 20)(6 17)(7 18)(8 19)(21 29)(22 30)(23 31)(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)(2 32)(3 29)(4 30)(5 14)(6 15)(7 16)(8 13)(9 19)(10 20)(11 17)(12 18)(21 25)(22 26)(23 27)(24 28)
(1 15 27 11)(2 12 28 16)(3 13 25 9)(4 10 26 14)(5 22 20 30)(6 31 17 23)(7 24 18 32)(8 29 19 21)

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

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

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

34 conjugacy classes

 class 1 2A 2B 2C 2D ··· 2I 4A ··· 4X order 1 2 2 2 2 ··· 2 4 ··· 4 size 1 1 1 1 2 ··· 2 2 ··· 2

34 irreducible representations

 dim 1 1 1 1 1 1 1 4 4 type + + + + + + + - image C1 C2 C2 C2 C2 C2 C4 2+ 1+4 2- 1+4 kernel C23.33C23 C2×C4⋊C4 C42⋊C2 C4×D4 C4×Q8 C2×C4○D4 C4○D4 C2 C2 # reps 1 3 3 6 2 1 16 1 1

Matrix representation of C23.33C23 in GL5(𝔽5)

 4 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 3 4 0 0 1 4 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
,
 2 0 0 0 0 0 0 3 0 0 0 3 0 0 0 0 0 0 3 4 0 0 0 0 2
,
 4 0 0 0 0 0 0 4 4 0 0 4 0 4 2 0 0 0 1 3 0 0 0 0 4
,
 1 0 0 0 0 0 0 1 0 0 0 4 0 0 0 0 0 0 4 2 0 0 0 4 1

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

C23.33C23 in GAP, Magma, Sage, TeX

C_2^3._{33}C_2^3
% in TeX

G:=Group("C2^3.33C2^3");
// GroupNames label

G:=SmallGroup(64,201);
// by ID

G=gap.SmallGroup(64,201);
# by ID

G:=PCGroup([6,-2,2,2,2,-2,2,192,217,188,579,69]);
// Polycyclic

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

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