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G = C2xD4.3D4order 128 = 27

Direct product of C2 and D4.3D4

direct product, p-group, metabelian, nilpotent (class 3), monomial

Aliases: C2xD4.3D4, M4(2).7C23, C4oD4.31D4, D4.11(C2xD4), C8.121(C2xD4), (C2xC8).370D4, Q8.11(C2xD4), (C2xD4).223D4, C8oD4:11C22, (C2xC4).15C24, (C2xQ8).178D4, (C2xC8).260C23, C4oD4.27C23, (C22xSD16):3C2, (C2xD4).70C23, C4.162(C22xD4), C8:C22.3C22, (C2xQ8).58C23, C4.113(C4:D4), C8.C4:15C22, (C2xSD16):56C22, C4.D4:12C22, C8.C22:11C22, C23.316(C4oD4), C4.10D4:12C22, C22.89(C4:D4), (C22xC4).990C23, (C22xC8).264C22, (C22xD4).352C22, (C2xM4(2)).58C22, (C22xQ8).285C22, (C2xC8oD4):7C2, C2.86(C2xC4:D4), (C2xC8.C4):25C2, (C2xC4).1427(C2xD4), (C2xC4.D4):10C2, (C2xC8:C22).12C2, (C2xC8.C22):20C2, C22.18(C2xC4oD4), (C2xC4).289(C4oD4), (C2xC4.10D4):10C2, (C2xC4oD4).304C22, SmallGroup(128,1796)

Series: Derived Chief Lower central Upper central Jennings

C1C2xC4 — C2xD4.3D4
C1C2C4C2xC4C22xC4C2xC4oD4C2xC8oD4 — C2xD4.3D4
C1C2C2xC4 — C2xD4.3D4
C1C22C22xC4 — C2xD4.3D4
C1C2C2C2xC4 — C2xD4.3D4

Generators and relations for C2xD4.3D4
 G = < a,b,c,d,e | a2=b4=c2=1, d4=e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe-1=b-1, bd=db, cd=dc, ece-1=bc, ede-1=d3 >

Subgroups: 460 in 234 conjugacy classes, 100 normal (36 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C8, C8, C2xC4, C2xC4, D4, D4, Q8, Q8, C23, C23, C2xC8, C2xC8, C2xC8, M4(2), M4(2), D8, SD16, Q16, C22xC4, C22xC4, C2xD4, C2xD4, C2xD4, C2xQ8, C2xQ8, C2xQ8, C4oD4, C4oD4, C24, C4.D4, C4.10D4, C8.C4, C22xC8, C22xC8, C2xM4(2), C2xM4(2), C8oD4, C8oD4, C2xD8, C2xSD16, C2xSD16, C2xQ16, C8:C22, C8:C22, C8.C22, C8.C22, C22xD4, C22xQ8, C2xC4oD4, C2xC4.D4, C2xC4.10D4, C2xC8.C4, D4.3D4, C2xC8oD4, C22xSD16, C2xC8:C22, C2xC8.C22, C2xD4.3D4
Quotients: C1, C2, C22, D4, C23, C2xD4, C4oD4, C24, C4:D4, C22xD4, C2xC4oD4, D4.3D4, C2xC4:D4, C2xD4.3D4

Smallest permutation representation of C2xD4.3D4
On 32 points
Generators in S32
(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 9)(8 10)(17 32)(18 25)(19 26)(20 27)(21 28)(22 29)(23 30)(24 31)
(1 13 5 9)(2 14 6 10)(3 15 7 11)(4 16 8 12)(17 30 21 26)(18 31 22 27)(19 32 23 28)(20 25 24 29)
(1 22)(2 23)(3 24)(4 17)(5 18)(6 19)(7 20)(8 21)(9 27)(10 28)(11 29)(12 30)(13 31)(14 32)(15 25)(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 10 5 14)(2 13 6 9)(3 16 7 12)(4 11 8 15)(17 24 21 20)(18 19 22 23)(25 26 29 30)(27 32 31 28)

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

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

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

32 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I4A4B4C4D4E4F4G4H8A8B8C8D8E···8J8K8L8M8N
order12222222224444444488888···88888
size11112244882222448822224···48888

32 irreducible representations

dim1111111112222224
type+++++++++++++
imageC1C2C2C2C2C2C2C2C2D4D4D4D4C4oD4C4oD4D4.3D4
kernelC2xD4.3D4C2xC4.D4C2xC4.10D4C2xC8.C4D4.3D4C2xC8oD4C22xSD16C2xC8:C22C2xC8.C22C2xC8C2xD4C2xQ8C4oD4C2xC4C23C2
# reps1111811114112224

Matrix representation of C2xD4.3D4 in GL6(F17)

1600000
0160000
001000
000100
000010
000001
,
1600000
0160000
0011500
0011600
0000162
0000161
,
010000
100000
0000115
0000116
0016200
0016100
,
040000
400000
0001000
00121000
0000010
00001210
,
0130000
400000
0001000
005000
0000107
000057

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,1,0,0,0,0,15,16,0,0,0,0,0,0,16,16,0,0,0,0,2,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,16,16,0,0,0,0,2,1,0,0,1,1,0,0,0,0,15,16,0,0],[0,4,0,0,0,0,4,0,0,0,0,0,0,0,0,12,0,0,0,0,10,10,0,0,0,0,0,0,0,12,0,0,0,0,10,10],[0,4,0,0,0,0,13,0,0,0,0,0,0,0,0,5,0,0,0,0,10,0,0,0,0,0,0,0,10,5,0,0,0,0,7,7] >;

C2xD4.3D4 in GAP, Magma, Sage, TeX

C_2\times D_4._3D_4
% in TeX

G:=Group("C2xD4.3D4");
// GroupNames label

G:=SmallGroup(128,1796);
// by ID

G=gap.SmallGroup(128,1796);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,120,758,2804,172,4037,1027,124]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^4=c^2=1,d^4=e^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=e*b*e^-1=b^-1,b*d=d*b,c*d=d*c,e*c*e^-1=b*c,e*d*e^-1=d^3>;
// generators/relations

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