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G = C22.126C25order 128 = 27

107th central stem extension by C22 of C25

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

Aliases: C23.67C24, C24.146C23, C42.109C23, C22.126C25, C4.852+ 1+4, C22.222+ 1+4, D4220C2, (C4×D4)⋊61C22, C4⋊D491C22, C4⋊C4.314C23, C233D412C2, C41D423C22, (C2×C4).116C24, (C23×C4)⋊50C22, C22≀C213C22, (C2×D4).318C23, C4.4D437C22, (C22×D4)⋊43C22, C22⋊C4.44C23, C22⋊Q8100C22, (C2×Q8).302C23, C42.C220C22, C22.19C2440C2, C22.32C2415C2, C422C213C22, C42⋊C254C22, C22.29C2428C2, C22.54C245C2, (C22×C4).386C23, C2.55(C2×2+ 1+4), C22.D416C22, C22.34C2418C2, C22.47C2428C2, (C2×C4⋊D4)⋊74C2, (C2×C4⋊C4)⋊86C22, (C2×C4○D4)⋊45C22, (C2×C22⋊C4)⋊59C22, SmallGroup(128,2269)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C22.126C25
C1C2C22C2×C4C22×C4C23×C4C22.19C24 — C22.126C25
C1C22 — C22.126C25
C1C22 — C22.126C25
C1C22 — C22.126C25

Generators and relations for C22.126C25
 G = < a,b,c,d,e,f,g | a2=b2=c2=e2=f2=1, d2=g2=a, ab=ba, dcd-1=gcg-1=ac=ca, fdf=ad=da, ae=ea, af=fa, ag=ga, ece=fcf=bc=cb, ede=bd=db, be=eb, bf=fb, bg=gb, dg=gd, ef=fe, eg=ge, fg=gf >

Subgroups: 1228 in 635 conjugacy classes, 384 normal (16 characteristic)
C1, C2, C2 [×2], C2 [×15], C4 [×2], C4 [×17], C22, C22 [×4], C22 [×57], C2×C4 [×2], C2×C4 [×16], C2×C4 [×25], D4 [×67], Q8, C23, C23 [×12], C23 [×33], C42 [×6], C22⋊C4 [×46], C4⋊C4 [×22], C22×C4 [×2], C22×C4 [×18], C22×C4 [×2], C2×D4, C2×D4 [×50], C2×D4 [×32], C2×Q8, C4○D4 [×2], C24, C24 [×8], C2×C22⋊C4 [×4], C2×C4⋊C4 [×2], C42⋊C2, C42⋊C2 [×2], C4×D4 [×16], C22≀C2 [×26], C4⋊D4 [×46], C22⋊Q8 [×2], C22.D4 [×18], C4.4D4 [×4], C42.C2 [×2], C422C2 [×8], C41D4 [×6], C23×C4, C22×D4 [×16], C2×C4○D4, C2×C4⋊D4 [×2], C22.19C24, C233D4 [×4], C22.29C24 [×2], C22.32C24 [×4], C22.34C24 [×2], D42 [×8], C22.47C24 [×4], C22.54C24 [×4], C22.126C25
Quotients: C1, C2 [×31], C22 [×155], C23 [×155], C24 [×31], 2+ 1+4 [×6], C25, C2×2+ 1+4 [×3], C22.126C25

Smallest permutation representation of C22.126C25
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 27)(2 28)(3 25)(4 26)(5 20)(6 17)(7 18)(8 19)(9 29)(10 30)(11 31)(12 32)(13 21)(14 22)(15 23)(16 24)
(1 23)(2 22)(3 21)(4 24)(5 29)(6 32)(7 31)(8 30)(9 20)(10 19)(11 18)(12 17)(13 25)(14 28)(15 27)(16 26)
(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 11)(2 32)(3 9)(4 30)(5 21)(6 14)(7 23)(8 16)(10 26)(12 28)(13 20)(15 18)(17 22)(19 24)(25 29)(27 31)
(1 11)(2 10)(3 9)(4 12)(5 21)(6 24)(7 23)(8 22)(13 20)(14 19)(15 18)(16 17)(25 29)(26 32)(27 31)(28 30)
(1 21 3 23)(2 22 4 24)(5 9 7 11)(6 10 8 12)(13 25 15 27)(14 26 16 28)(17 30 19 32)(18 31 20 29)

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

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

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

38 conjugacy classes

class 1 2A2B2C2D2E2F2G2H···2R4A4B4C···4S
order122222222···2444···4
size111122224···4224···4

38 irreducible representations

dim111111111144
type++++++++++++
imageC1C2C2C2C2C2C2C2C2C22+ 1+42+ 1+4
kernelC22.126C25C2×C4⋊D4C22.19C24C233D4C22.29C24C22.32C24C22.34C24D42C22.47C24C22.54C24C4C22
# reps121424284424

Matrix representation of C22.126C25 in GL8(𝔽5)

40000000
04000000
00400000
00040000
00004000
00000400
00000040
00000004
,
40000000
04000000
00400000
00040000
00001000
00000100
00000010
00000001
,
01000000
10000000
01040000
10400000
00000100
00001000
00000204
00002040
,
10300000
01030000
10400000
01040000
00001040
00000104
00002040
00000204
,
03000000
20000000
03020000
20300000
00004000
00000400
00000040
00000004
,
03000000
20000000
03020000
20300000
00004000
00000400
00003010
00000301
,
01000000
40000000
00010000
00400000
00000100
00004000
00000001
00000040

G:=sub<GL(8,GF(5))| [4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,1,0,1,0,0,0,0,1,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,1,0,2,0,0,0,0,1,0,2,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0],[1,0,1,0,0,0,0,0,0,1,0,1,0,0,0,0,3,0,4,0,0,0,0,0,0,3,0,4,0,0,0,0,0,0,0,0,1,0,2,0,0,0,0,0,0,1,0,2,0,0,0,0,4,0,4,0,0,0,0,0,0,4,0,4],[0,2,0,2,0,0,0,0,3,0,3,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[0,2,0,2,0,0,0,0,3,0,3,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,4,0,3,0,0,0,0,0,0,4,0,3,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0] >;

C22.126C25 in GAP, Magma, Sage, TeX

C_2^2._{126}C_2^5
% in TeX

G:=Group("C2^2.126C2^5");
// GroupNames label

G:=SmallGroup(128,2269);
// by ID

G=gap.SmallGroup(128,2269);
# by ID

G:=PCGroup([7,-2,2,2,2,2,-2,2,477,232,1430,723,2019,570,1684,102]);
// Polycyclic

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

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