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G = C249D4order 128 = 27

4th semidirect product of C24 and D4 acting via D4/C2=C22

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

Aliases: C249D4, C25.50C22, C23.513C24, C24.359C23, C22.2922+ 1+4, (D4×C23)⋊6C2, (C22×C4)⋊31D4, C243C419C2, C232D422C2, C23.187(C2×D4), (C22×D4)⋊8C22, C22.54C22≀C2, C23.10D452C2, C23.34D440C2, C2.21(C233D4), (C22×C4).851C23, (C23×C4).416C22, C22.338(C22×D4), C2.C4228C22, C2.31(C22.29C24), (C2×C22≀C2)⋊9C2, (C2×C4⋊C4)⋊24C22, (C2×C4).373(C2×D4), C2.25(C2×C22≀C2), (C2×C22⋊C4)⋊22C22, (C2×C22.D4)⋊24C2, SmallGroup(128,1345)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C249D4
C1C2C22C23C24C22×D4C2×C22≀C2 — C249D4
C1C23 — C249D4
C1C23 — C249D4
C1C23 — C249D4

Generators and relations for C249D4
 G = < a,b,c,d,e,f | a2=b2=c2=d2=e4=f2=1, ab=ba, faf=ac=ca, ad=da, eae-1=acd, ebe-1=bc=cb, bd=db, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef=e-1 >

Subgroups: 1348 in 567 conjugacy classes, 116 normal (14 characteristic)
C1, C2, C2 [×6], C2 [×13], C4 [×11], C22, C22 [×10], C22 [×91], C2×C4 [×4], C2×C4 [×33], D4 [×52], C23, C23 [×14], C23 [×99], C22⋊C4 [×26], C4⋊C4 [×4], C22×C4, C22×C4 [×12], C22×C4 [×4], C2×D4 [×78], C24 [×2], C24 [×12], C24 [×16], C2.C42 [×4], C2×C22⋊C4, C2×C22⋊C4 [×16], C2×C4⋊C4 [×2], C22≀C2 [×8], C22.D4 [×4], C23×C4, C22×D4, C22×D4 [×8], C22×D4 [×12], C25 [×2], C243C4 [×2], C23.34D4, C232D4 [×4], C23.10D4 [×4], C2×C22≀C2 [×2], C2×C22.D4, D4×C23, C249D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×12], C23 [×15], C2×D4 [×18], C24, C22≀C2 [×4], C22×D4 [×3], 2+ 1+4 [×4], C2×C22≀C2, C233D4 [×4], C22.29C24 [×2], C249D4

Smallest permutation representation of C249D4
On 32 points
Generators in S32
(1 9)(2 17)(3 11)(4 19)(5 25)(6 31)(7 27)(8 29)(10 22)(12 24)(13 28)(14 30)(15 26)(16 32)(18 23)(20 21)
(1 13)(2 24)(3 15)(4 22)(5 16)(6 23)(7 14)(8 21)(9 28)(10 19)(11 26)(12 17)(18 31)(20 29)(25 32)(27 30)
(1 6)(2 7)(3 8)(4 5)(9 31)(10 32)(11 29)(12 30)(13 23)(14 24)(15 21)(16 22)(17 27)(18 28)(19 25)(20 26)
(1 15)(2 16)(3 13)(4 14)(5 24)(6 21)(7 22)(8 23)(9 26)(10 27)(11 28)(12 25)(17 32)(18 29)(19 30)(20 31)
(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 3)(6 8)(9 29)(10 32)(11 31)(12 30)(13 15)(17 27)(18 26)(19 25)(20 28)(21 23)

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

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

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

32 conjugacy classes

class 1 2A···2G2H2I2J2K2L···2S2T4A4B4C4D4E···4K
order12···222222···2244444···4
size11···122224···4844448···8

32 irreducible representations

dim11111111224
type+++++++++++
imageC1C2C2C2C2C2C2C2D4D42+ 1+4
kernelC249D4C243C4C23.34D4C232D4C23.10D4C2×C22≀C2C2×C22.D4D4×C23C22×C4C24C22
# reps12144211484

Matrix representation of C249D4 in GL8(ℤ)

-10000000
01000000
00-100000
00010000
00000010
00000001
00001000
00000100
,
-10000000
0-1000000
00100000
00010000
00001000
00000-100
00000010
0000000-1
,
10000000
01000000
00100000
00010000
0000-1000
00000-100
000000-10
0000000-1
,
-10000000
0-1000000
00-100000
000-10000
00001000
00000100
00000010
00000001
,
0-1000000
10000000
000-10000
00-100000
00000100
00001000
0000000-1
000000-10
,
10000000
0-1000000
00100000
00010000
00001000
00000100
000000-10
0000000-1

G:=sub<GL(8,Integers())| [-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-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],[-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,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0],[1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1] >;

C249D4 in GAP, Magma, Sage, TeX

C_2^4\rtimes_9D_4
% in TeX

G:=Group("C2^4:9D4");
// GroupNames label

G:=SmallGroup(128,1345);
// by ID

G=gap.SmallGroup(128,1345);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,253,758,723,185]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=e^4=f^2=1,a*b=b*a,f*a*f=a*c=c*a,a*d=d*a,e*a*e^-1=a*c*d,e*b*e^-1=b*c=c*b,b*d=d*b,b*f=f*b,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f=e^-1>;
// generators/relations

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