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G = C22.38C25order 128 = 27

19th central stem extension by C22 of C25

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

Aliases: C23.18C24, C22.38C25, C42.541C23, C24.480C23, (C22×C4)⋊44D4, C4⋊Q874C22, (C4×D4)⋊30C22, C4(C233D4), (C2×C4).41C24, C2.17(D4×C23), C233D418C2, C4⋊D463C22, C41D443C22, C4⋊C4.282C23, (C2×C42)⋊45C22, (C23×C4)⋊31C22, C23.355(C2×D4), C4.173(C22×D4), C22⋊C4.6C23, C22⋊Q874C22, C22≀C227C22, C22.3(C22×D4), C4(C22.29C24), (C2×D4).291C23, C4.4D464C22, (C22×D4)⋊60C22, (C2×Q8).423C23, (C22×Q8)⋊60C22, C22.19C2413C2, C22.29C2437C2, C42⋊C288C22, C2.4(C2.C25), C4(C23.38C23), C4(C22.31C24), C22.26C2426C2, (C22×C4).1179C23, C22.D432C22, C23.38C2339C2, C22.31C2430C2, (C2×C4).662(C2×D4), (C2×C4⋊C4)⋊129C22, (C22×C4○D4)⋊15C2, (C2×C4○D4)⋊69C22, (C2×C4)(C233D4), (C2×C42⋊C2)⋊55C2, (C2×C4)(C22.29C24), (C2×C22⋊C4).529C22, (C2×C4)(C22.31C24), SmallGroup(128,2181)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C22.38C25
C1C2C22C2×C4C22×C4C23×C4C22×C4○D4 — C22.38C25
C1C22 — C22.38C25
C1C2×C4 — C22.38C25
C1C22 — C22.38C25

Generators and relations for C22.38C25
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=e2=f2=1, g2=a, ab=ba, dcd=fcf=ac=ca, ede=ad=da, ae=ea, af=fa, ag=ga, ece=bc=cb, bd=db, be=eb, bf=fb, bg=gb, cg=gc, df=fd, dg=gd, ef=fe, eg=ge, fg=gf >

Subgroups: 1244 in 778 conjugacy classes, 428 normal (12 characteristic)
C1, C2, C2 [×2], C2 [×14], C4 [×8], C4 [×16], C22, C22 [×6], C22 [×50], C2×C4 [×2], C2×C4 [×42], C2×C4 [×44], D4 [×64], Q8 [×16], C23, C23 [×14], C23 [×18], C42 [×8], C22⋊C4 [×40], C4⋊C4 [×24], C22×C4 [×2], C22×C4 [×36], C22×C4 [×16], C2×D4 [×40], C2×D4 [×24], C2×Q8 [×8], C2×Q8 [×8], C4○D4 [×64], C24, C24 [×4], C2×C42 [×2], C2×C22⋊C4 [×2], C2×C4⋊C4 [×2], C42⋊C2 [×8], C4×D4 [×16], C22≀C2 [×16], C4⋊D4 [×32], C22⋊Q8 [×16], C22.D4 [×16], C4.4D4 [×8], C41D4 [×4], C4⋊Q8 [×4], C23×C4, C23×C4 [×4], C22×D4 [×6], C22×Q8 [×2], C2×C4○D4 [×16], C2×C4○D4 [×16], C2×C42⋊C2, C22.19C24 [×8], C22.26C24 [×4], C233D4 [×4], C22.29C24 [×4], C23.38C23 [×4], C22.31C24 [×4], C22×C4○D4 [×2], C22.38C25
Quotients: C1, C2 [×31], C22 [×155], D4 [×8], C23 [×155], C2×D4 [×28], C24 [×31], C22×D4 [×14], C25, D4×C23, C2.C25 [×2], C22.38C25

Smallest permutation representation of C22.38C25
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 11)(2 12)(3 9)(4 10)(5 29)(6 30)(7 31)(8 32)(13 18)(14 19)(15 20)(16 17)(21 26)(22 27)(23 28)(24 25)
(1 13)(2 14)(3 15)(4 16)(5 22)(6 23)(7 24)(8 21)(9 20)(10 17)(11 18)(12 19)(25 31)(26 32)(27 29)(28 30)
(1 19)(2 20)(3 17)(4 18)(5 23)(6 24)(7 21)(8 22)(9 16)(10 13)(11 14)(12 15)(25 30)(26 31)(27 32)(28 29)
(1 22)(2 23)(3 24)(4 21)(5 18)(6 19)(7 20)(8 17)(9 25)(10 26)(11 27)(12 28)(13 29)(14 30)(15 31)(16 32)
(1 14)(2 15)(3 16)(4 13)(5 26)(6 27)(7 28)(8 25)(9 17)(10 18)(11 19)(12 20)(21 29)(22 30)(23 31)(24 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)

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

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

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

44 conjugacy classes

class 1 2A2B2C2D···2I2J···2Q4A4B4C4D4E···4J4K···4Z
order12222···22···244444···44···4
size11112···24···411112···24···4

44 irreducible representations

dim11111111124
type++++++++++
imageC1C2C2C2C2C2C2C2C2D4C2.C25
kernelC22.38C25C2×C42⋊C2C22.19C24C22.26C24C233D4C22.29C24C23.38C23C22.31C24C22×C4○D4C22×C4C2
# reps11844444284

Matrix representation of C22.38C25 in GL6(𝔽5)

100000
010000
004000
000400
000040
000004
,
400000
040000
001000
000100
000010
000001
,
100000
040000
000300
002000
000003
000020
,
100000
010000
000100
001000
000004
000040
,
040000
400000
000010
000001
001000
000100
,
100000
010000
000100
001000
000001
000010
,
400000
040000
003000
000300
000030
000003

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],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,4,0,0,0,0,0,0,0,2,0,0,0,0,3,0,0,0,0,0,0,0,0,2,0,0,0,0,3,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,4,0],[0,4,0,0,0,0,4,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,0,3] >;

C22.38C25 in GAP, Magma, Sage, TeX

C_2^2._{38}C_2^5
% in TeX

G:=Group("C2^2.38C2^5");
// GroupNames label

G:=SmallGroup(128,2181);
// by ID

G=gap.SmallGroup(128,2181);
# by ID

G:=PCGroup([7,-2,2,2,2,2,-2,2,477,1430,387,1123,102]);
// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=e^2=f^2=1,g^2=a,a*b=b*a,d*c*d=f*c*f=a*c=c*a,e*d*e=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*g=g*c,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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