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G = C2×D4.3D4order 128 = 27

Direct product of C2 and D4.3D4

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

Aliases: C2×D4.3D4, M4(2).7C23, C4○D4.31D4, D4.11(C2×D4), C8.121(C2×D4), (C2×C8).370D4, Q8.11(C2×D4), (C2×D4).223D4, C8○D411C22, (C2×C4).15C24, (C2×Q8).178D4, (C2×C8).260C23, C4○D4.27C23, (C22×SD16)⋊3C2, (C2×D4).70C23, C4.162(C22×D4), C8⋊C22.3C22, (C2×Q8).58C23, C4.113(C4⋊D4), C8.C415C22, (C2×SD16)⋊56C22, C4.D412C22, C8.C2211C22, C23.316(C4○D4), C4.10D412C22, C22.89(C4⋊D4), (C22×C4).990C23, (C22×C8).264C22, (C22×D4).352C22, (C2×M4(2)).58C22, (C22×Q8).285C22, (C2×C8○D4)⋊7C2, C2.86(C2×C4⋊D4), (C2×C8.C4)⋊25C2, (C2×C4).1427(C2×D4), (C2×C4.D4)⋊10C2, (C2×C8⋊C22).12C2, (C2×C8.C22)⋊20C2, C22.18(C2×C4○D4), (C2×C4).289(C4○D4), (C2×C4.10D4)⋊10C2, (C2×C4○D4).304C22, SmallGroup(128,1796)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C2×D4.3D4
C1C2C4C2×C4C22×C4C2×C4○D4C2×C8○D4 — C2×D4.3D4
C1C2C2×C4 — C2×D4.3D4
C1C22C22×C4 — C2×D4.3D4
C1C2C2C2×C4 — C2×D4.3D4

Generators and relations for C2×D4.3D4
 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, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, C2×C8, C2×C8, C2×C8, M4(2), M4(2), D8, SD16, Q16, C22×C4, C22×C4, C2×D4, C2×D4, C2×D4, C2×Q8, C2×Q8, C2×Q8, C4○D4, C4○D4, C24, C4.D4, C4.10D4, C8.C4, C22×C8, C22×C8, C2×M4(2), C2×M4(2), C8○D4, C8○D4, C2×D8, C2×SD16, C2×SD16, C2×Q16, C8⋊C22, C8⋊C22, C8.C22, C8.C22, C22×D4, C22×Q8, C2×C4○D4, C2×C4.D4, C2×C4.10D4, C2×C8.C4, D4.3D4, C2×C8○D4, C22×SD16, C2×C8⋊C22, C2×C8.C22, C2×D4.3D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C4⋊D4, C22×D4, C2×C4○D4, D4.3D4, C2×C4⋊D4, C2×D4.3D4

Smallest permutation representation of C2×D4.3D4
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+++++++++++++
imageC1C2C2C2C2C2C2C2C2D4D4D4D4C4○D4C4○D4D4.3D4
kernelC2×D4.3D4C2×C4.D4C2×C4.10D4C2×C8.C4D4.3D4C2×C8○D4C22×SD16C2×C8⋊C22C2×C8.C22C2×C8C2×D4C2×Q8C4○D4C2×C4C23C2
# reps1111811114112224

Matrix representation of C2×D4.3D4 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] >;

C2×D4.3D4 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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