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G = C22.14C25order 128 = 27

10th central extension by C22 of C25

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

Aliases: C22.14C25, C23.108C24, C42.592C23, C24.473C23, (C4×D4)⋊87C22, D4(C42⋊C2), C4.40(C23×C4), C2.10(C24×C4), (C4×Q8)⋊83C22, Q8(C42⋊C2), C4⋊C4.515C23, (C2×C4).160C24, (C2×C42)⋊40C22, D4.24(C22×C4), C22.4(C23×C4), Q8.25(C22×C4), C4(C22.11C24), (C2×D4).498C23, (C2×Q8).481C23, C22.11C2425C2, C22⋊C4.127C23, C2.1(C2.C25), C23.111(C22×C4), (C22×C4).617C23, (C23×C4).577C22, C42⋊C2104C22, C4(C23.32C23), C4(C23.33C23), (C22×D4).579C22, (C22×Q8).482C22, C23.33C2333C2, C23.32C2320C2, C4⋊C4(C4○D4), (C4×C4○D4)⋊13C2, C4○D418(C2×C4), (C2×C4○D4)⋊24C4, (C2×D4)⋊53(C2×C4), C22⋊C4(C4○D4), (C2×Q8)⋊44(C2×C4), C4⋊C4(C42⋊C2), (C22×C4)⋊42(C2×C4), (C2×C4⋊C4)⋊120C22, C4○D4(C42⋊C2), (C2×C42⋊C2)⋊50C2, (C2×C4).283(C22×C4), (C22×C4○D4).26C2, (C2×C4○D4).317C22, C42⋊C2(C42⋊C2), (C2×C22⋊C4).476C22, (C2×C4)(C23.32C23), C42⋊C2(C2×C4○D4), SmallGroup(128,2160)

Series: Derived Chief Lower central Upper central Jennings

C1C2 — C22.14C25
C1C2C22C2×C4C22×C4C23×C4C22×C4○D4 — C22.14C25
C1C2 — C22.14C25
C1C2×C4 — C22.14C25
C1C22 — C22.14C25

Generators and relations for C22.14C25
 G = < a,b,c,d,e,f,g | a2=b2=d2=e2=f2=1, c2=b, g2=a, ab=ba, dcd=fcf=ac=ca, ede=ad=da, ae=ea, af=fa, ag=ga, bc=cb, bd=db, be=eb, bf=fb, bg=gb, ce=ec, cg=gc, df=fd, dg=gd, ef=fe, eg=ge, fg=gf >

Subgroups: 940 in 770 conjugacy classes, 684 normal (10 characteristic)
C1, C2, C2 [×2], C2 [×14], C4 [×16], C4 [×16], C22, C22 [×14], C22 [×26], C2×C4 [×2], C2×C4 [×86], C2×C4 [×24], D4 [×48], Q8 [×16], C23, C23 [×18], C23 [×6], C42 [×32], C22⋊C4 [×32], C4⋊C4 [×32], C22×C4, C22×C4 [×63], C2×D4 [×36], C2×Q8 [×12], C4○D4 [×64], C24 [×3], C2×C42 [×12], C2×C22⋊C4 [×12], C2×C4⋊C4 [×12], C42⋊C2 [×40], C4×D4 [×48], C4×Q8 [×16], C23×C4 [×3], C22×D4 [×3], C22×Q8, C2×C4○D4 [×24], C2×C42⋊C2 [×6], C4×C4○D4 [×8], C22.11C24 [×6], C23.32C23 [×2], C23.33C23 [×8], C22×C4○D4, C22.14C25
Quotients: C1, C2 [×31], C4 [×16], C22 [×155], C2×C4 [×120], C23 [×155], C22×C4 [×140], C24 [×31], C23×C4 [×30], C25, C24×C4, C2.C25 [×2], C22.14C25

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

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

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

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

68 conjugacy classes

class 1 2A2B2C2D···2Q4A4B4C4D4E···4AX
order12222···244444···4
size11112···211112···2

68 irreducible representations

dim111111114
type+++++++
imageC1C2C2C2C2C2C2C4C2.C25
kernelC22.14C25C2×C42⋊C2C4×C4○D4C22.11C24C23.32C23C23.33C23C22×C4○D4C2×C4○D4C2
# reps1686281324

Matrix representation of C22.14C25 in GL5(𝔽5)

10000
04000
00400
00040
00004
,
40000
01000
00100
00010
00001
,
30000
00113
00001
01204
00100
,
10000
01042
01404
00023
00043
,
40000
03433
03202
00040
00031
,
10000
01003
00100
00040
00004
,
40000
03000
00300
00030
00003

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

C22.14C25 in GAP, Magma, Sage, TeX

C_2^2._{14}C_2^5
% in TeX

G:=Group("C2^2.14C2^5");
// GroupNames label

G:=SmallGroup(128,2160);
// by ID

G=gap.SmallGroup(128,2160);
# by ID

G:=PCGroup([7,-2,2,2,2,2,-2,2,448,477,387,1123,102]);
// Polycyclic

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