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G = C22.33C25order 128 = 27

14th central stem extension by C22 of C25

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

Aliases: C22.33C25, C23.114C24, C42.539C23, C24.611C23, C42(C4⋊Q8), C42C22≀C2, (C4×D4)⋊92C22, C42(C4⋊D4), C42(C41D4), (C2×C4).36C24, C4⋊Q8107C22, (C4×Q8)⋊86C22, C42(C22⋊Q8), C41D458C22, C4⋊C4.460C23, (C2×C42)⋊43C22, (C22×C42)⋊25C2, C42(C4.4D4), C42(C42.C2), C42(C422C2), (C2×D4).446C23, C4.4D497C22, C22⋊C4.74C23, (C2×Q8).420C23, C42.C274C22, C42(C22.19C24), C22.19C2445C2, C42⋊C285C22, C422C252C22, C22≀C2.34C22, C4⋊D4.240C22, (C23×C4).708C22, C42(C22.D4), C22⋊Q8.240C22, C42(C22.26C24), (C22×C4).1297C23, C22.26C2458C2, C42(C22.19C24), C42(C23.37C23), C42(C23.36C23), C23.36C2366C2, C23.37C2359C2, C42(C22.26C24), C22.D4.41C22, C42(C23.36C23), C42(C23.37C23), (C4×C4○D4)⋊16C2, C4.71(C2×C4○D4), (C2×C4)⋊15(C4○D4), C22.7(C2×C4○D4), C2.15(C22×C4○D4), (C2×C4○D4).320C22, SmallGroup(128,2176)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C22.33C25
C1C2C22C2×C4C42C2×C42C22×C42 — C22.33C25
C1C22 — C22.33C25
C1C42 — C22.33C25
C1C22 — C22.33C25

Generators and relations for C22.33C25
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=e2=1, f2=b, g2=a, ab=ba, dcd=ac=ca, ad=da, ae=ea, af=fa, ag=ga, ece=bc=cb, bd=db, be=eb, bf=fb, bg=gb, cf=fc, cg=gc, de=ed, df=fd, dg=gd, ef=fe, eg=ge, fg=gf >

Subgroups: 812 in 606 conjugacy classes, 408 normal (8 characteristic)
C1, C2 [×3], C2 [×10], C4 [×12], C4 [×18], C22, C22 [×6], C22 [×26], C2×C4 [×36], C2×C4 [×54], D4 [×36], Q8 [×12], C23 [×7], C23 [×6], C42 [×2], C42 [×26], C22⋊C4 [×36], C4⋊C4 [×36], C22×C4 [×30], C22×C4 [×12], C2×D4 [×18], C2×Q8 [×6], C4○D4 [×24], C24, C2×C42 [×16], C42⋊C2 [×18], C4×D4 [×36], C4×Q8 [×12], C22≀C2 [×4], C4⋊D4 [×12], C22⋊Q8 [×12], C22.D4 [×12], C4.4D4 [×6], C42.C2 [×6], C422C2 [×8], C41D4, C4⋊Q8 [×3], C23×C4 [×3], C2×C4○D4 [×6], C22×C42, C4×C4○D4 [×6], C22.19C24 [×6], C23.36C23 [×12], C22.26C24 [×3], C23.37C23 [×3], C22.33C25
Quotients: C1, C2 [×31], C22 [×155], C23 [×155], C4○D4 [×12], C24 [×31], C2×C4○D4 [×18], C25, C22×C4○D4 [×3], C22.33C25

Smallest permutation representation of C22.33C25
On 32 points
Generators in S32
(1 15)(2 16)(3 13)(4 14)(5 18)(6 19)(7 20)(8 17)(9 22)(10 23)(11 24)(12 21)(25 31)(26 32)(27 29)(28 30)
(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 25)(2 26)(3 27)(4 28)(5 21)(6 22)(7 23)(8 24)(9 19)(10 20)(11 17)(12 18)(13 29)(14 30)(15 31)(16 32)
(1 3)(2 4)(5 7)(6 8)(9 24)(10 21)(11 22)(12 23)(13 15)(14 16)(17 19)(18 20)(25 29)(26 30)(27 31)(28 32)
(1 15)(2 16)(3 13)(4 14)(5 18)(6 19)(7 20)(8 17)(9 24)(10 21)(11 22)(12 23)(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 19 15 6)(2 20 16 7)(3 17 13 8)(4 18 14 5)(9 31 22 25)(10 32 23 26)(11 29 24 27)(12 30 21 28)

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

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

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

56 conjugacy classes

class 1 2A2B2C2D···2I2J2K2L2M4A···4L4M···4AD4AE···4AP
order12222···222224···44···44···4
size11112···244441···12···24···4

56 irreducible representations

dim11111112
type+++++++
imageC1C2C2C2C2C2C2C4○D4
kernelC22.33C25C22×C42C4×C4○D4C22.19C24C23.36C23C22.26C24C23.37C23C2×C4
# reps1166123324

Matrix representation of C22.33C25 in GL4(𝔽5) generated by

1000
0100
0040
0004
,
4000
0400
0010
0001
,
0100
1000
0001
0010
,
4000
0400
0010
0004
,
1000
0400
0040
0004
,
2000
0200
0010
0001
,
4000
0400
0020
0002
G:=sub<GL(4,GF(5))| [1,0,0,0,0,1,0,0,0,0,4,0,0,0,0,4],[4,0,0,0,0,4,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,0,1,0,0,1,0],[4,0,0,0,0,4,0,0,0,0,1,0,0,0,0,4],[1,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[2,0,0,0,0,2,0,0,0,0,1,0,0,0,0,1],[4,0,0,0,0,4,0,0,0,0,2,0,0,0,0,2] >;

C22.33C25 in GAP, Magma, Sage, TeX

C_2^2._{33}C_2^5
% in TeX

G:=Group("C2^2.33C2^5");
// GroupNames label

G:=SmallGroup(128,2176);
// by ID

G=gap.SmallGroup(128,2176);
# by ID

G:=PCGroup([7,-2,2,2,2,2,-2,2,477,1430,248,102]);
// Polycyclic

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

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