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## G = C22.75C25order 128 = 27

### 56th central stem extension by C22 of C25

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22 — C22.75C25
 Chief series C1 — C2 — C22 — C23 — C22×C4 — C23×C4 — Q8×C23 — C22.75C25
 Lower central C1 — C22 — C22.75C25
 Upper central C1 — C22 — C22.75C25
 Jennings C1 — C22 — C22.75C25

Generators and relations for C22.75C25
G = < a,b,c,d,e,f,g | a2=b2=c2=e2=g2=1, d2=f2=a, ab=ba, dcd-1=gcg=ac=ca, fdf-1=ad=da, ae=ea, af=fa, ag=ga, ece=bc=cb, bd=db, be=eb, bf=fb, bg=gb, cf=fc, de=ed, dg=gd, ef=fe, eg=ge, fg=gf >

Subgroups: 1068 in 746 conjugacy classes, 432 normal (10 characteristic)
C1, C2, C2 [×2], C2 [×10], C4 [×12], C4 [×18], C22, C22 [×6], C22 [×26], C2×C4 [×36], C2×C4 [×54], D4 [×36], Q8 [×16], Q8 [×36], C23, C23 [×6], C23 [×6], C42 [×12], C22⋊C4 [×36], C4⋊C4 [×36], C22×C4 [×30], C22×C4 [×12], C2×D4 [×18], C2×Q8 [×30], C2×Q8 [×60], C4○D4 [×40], C24, C42⋊C2 [×6], C4×D4 [×24], C4×Q8 [×8], C22≀C2 [×4], C4⋊D4 [×12], C22⋊Q8 [×36], C22.D4 [×12], C4.4D4 [×12], C4⋊Q8 [×12], C23×C4 [×3], C22×Q8, C22×Q8 [×15], C22×Q8 [×8], C2×C4○D4 [×10], 2- 1+4 [×8], C23.32C23, C22.19C24 [×6], C23.38C23 [×6], Q85D4 [×8], D4×Q8 [×8], Q8×C23, C2×2- 1+4, C22.75C25
Quotients: C1, C2 [×31], C22 [×155], D4 [×8], C23 [×155], C2×D4 [×28], C24 [×31], C22×D4 [×14], 2- 1+4 [×4], C25, D4×C23, C2×2- 1+4 [×2], C22.75C25

Smallest permutation representation of C22.75C25
On 32 points
Generators in S32
(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 15)(2 16)(3 13)(4 14)(5 17)(6 18)(7 19)(8 20)(9 25)(10 26)(11 27)(12 28)(21 29)(22 30)(23 31)(24 32)
(1 21)(2 24)(3 23)(4 22)(5 28)(6 27)(7 26)(8 25)(9 20)(10 19)(11 18)(12 17)(13 31)(14 30)(15 29)(16 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 15)(2 16)(3 13)(4 14)(5 17)(6 18)(7 19)(8 20)
(1 7 3 5)(2 6 4 8)(9 32 11 30)(10 31 12 29)(13 17 15 19)(14 20 16 18)(21 26 23 28)(22 25 24 27)
(1 15)(2 16)(3 13)(4 14)(5 17)(6 18)(7 19)(8 20)(9 27)(10 28)(11 25)(12 26)(21 31)(22 32)(23 29)(24 30)

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

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

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

44 conjugacy classes

 class 1 2A 2B 2C 2D ··· 2I 2J 2K 2L 2M 4A ··· 4L 4M ··· 4AD order 1 2 2 2 2 ··· 2 2 2 2 2 4 ··· 4 4 ··· 4 size 1 1 1 1 2 ··· 2 4 4 4 4 2 ··· 2 4 ··· 4

44 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 4 type + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 C2 D4 2- 1+4 kernel C22.75C25 C23.32C23 C22.19C24 C23.38C23 Q8⋊5D4 D4×Q8 Q8×C23 C2×2- 1+4 C2×Q8 C22 # reps 1 1 6 6 8 8 1 1 8 4

Matrix representation of C22.75C25 in GL6(𝔽5)

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

G:=sub<GL(6,GF(5))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[0,4,0,0,0,0,4,0,0,0,0,0,0,0,0,3,1,0,0,0,0,3,0,4,0,0,1,1,0,3,0,0,0,3,0,2],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,3,0,0,0,0,0,0,2,0,1,0,0,0,0,2,2,0,0,0,0,0,3],[4,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,2,0,0,0,4,0,2,0,0,0,0,1,0,0,0,0,0,0,1],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,4,3,0,0,0,1,0,3,3,0,0,0,0,1,1,0,0,0,0,3,4],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,2,0,0,0,4,0,2,0,0,0,0,1,0,0,0,0,0,0,1] >;

C22.75C25 in GAP, Magma, Sage, TeX

C_2^2._{75}C_2^5
% in TeX

G:=Group("C2^2.75C2^5");
// GroupNames label

G:=SmallGroup(128,2218);
// by ID

G=gap.SmallGroup(128,2218);
# by ID

G:=PCGroup([7,-2,2,2,2,2,-2,2,477,232,1430,570,136,1684]);
// Polycyclic

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

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