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G = C42.271D4order 128 = 27

253rd non-split extension by C42 of D4 acting via D4/C2=C22

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

Aliases: C42.271D4, C42.399C23, C4.1052+ 1+4, C83D49C2, C4⋊C817C22, C4⋊Q871C22, C22⋊D821C2, C8⋊C48C22, Q8⋊D411C2, (C4×D4)⋊13C22, (C2×C8).61C23, D4.2D423C2, C4⋊C4.152C23, C41D441C22, (C2×C4).411C24, C42.6C49C2, (C2×D8).70C22, (C22×C4).500D4, C23.693(C2×D4), Q8⋊C433C22, (C2×SD16)⋊22C22, (C2×D4).160C23, C22⋊C8.46C22, (C2×Q8).148C23, D4⋊C4.42C22, C4⋊D4.191C22, C22.51(C8⋊C22), (C2×C42).878C22, C22.671(C22×D4), C2.56(D8⋊C22), C22.26C2419C2, C42.28C221C2, (C22×C4).1082C23, C4.4D4.152C22, (C22×D4).389C22, (C22×Q8).322C22, C2.82(C22.29C24), (C2×C4).540(C2×D4), C2.56(C2×C8⋊C22), (C2×C4.4D4)⋊44C2, SmallGroup(128,1945)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.271D4
C1C2C4C2×C4C22×C4C22×D4C2×C4.4D4 — C42.271D4
C1C2C2×C4 — C42.271D4
C1C22C2×C42 — C42.271D4
C1C2C2C2×C4 — C42.271D4

Generators and relations for C42.271D4
 G = < a,b,c,d | a4=b4=d2=1, c4=b2, ab=ba, cac-1=ab2, dad=a-1, cbc-1=a2b-1, dbd=a2b, dcd=b2c3 >

Subgroups: 524 in 222 conjugacy classes, 86 normal (28 characteristic)
C1, C2 [×3], C2 [×6], C4 [×2], C4 [×9], C22, C22 [×2], C22 [×18], C8 [×4], C2×C4 [×6], C2×C4 [×15], D4 [×18], Q8 [×8], C23, C23 [×10], C42 [×4], C22⋊C4 [×12], C4⋊C4 [×2], C4⋊C4, C2×C8 [×4], D8 [×4], SD16 [×4], C22×C4 [×3], C22×C4 [×3], C2×D4 [×4], C2×D4 [×7], C2×Q8 [×2], C2×Q8 [×4], C4○D4 [×4], C24, C8⋊C4 [×2], C22⋊C8 [×2], D4⋊C4 [×4], Q8⋊C4 [×4], C4⋊C8 [×2], C2×C42, C2×C22⋊C4 [×2], C4×D4 [×2], C4×D4, C4⋊D4 [×2], C4⋊D4, C4.4D4 [×4], C4.4D4 [×3], C41D4, C4⋊Q8, C2×D8 [×4], C2×SD16 [×4], C22×D4, C22×Q8, C2×C4○D4, C42.6C4, C22⋊D8 [×2], Q8⋊D4 [×2], D4.2D4 [×4], C42.28C22 [×2], C83D4 [×2], C2×C4.4D4, C22.26C24, C42.271D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C24, C8⋊C22 [×2], C22×D4, 2+ 1+4 [×2], C22.29C24, C2×C8⋊C22, D8⋊C22, C42.271D4

Character table of C42.271D4

 class 12A2B2C2D2E2F2G2H2I4A4B4C4D4E4F4G4H4I4J4K4L8A8B8C8D
 size 11112288882222444488888888
ρ111111111111111111111111111    trivial
ρ21111-1-1-111111-1-111-1-1-1-1-111-1-11    linear of order 2
ρ3111111-1-1-1-111111111-1-1-1-11111    linear of order 2
ρ41111-1-11-1-1-111-1-111-1-1111-11-1-11    linear of order 2
ρ51111111-11111-1-1-1-1-11-1-11-111-1-1    linear of order 2
ρ61111-1-1-1-1111111-1-11-111-1-11-11-1    linear of order 2
ρ7111111-11-1-111-1-1-1-1-1111-1111-1-1    linear of order 2
ρ81111-1-111-1-11111-1-11-1-1-1111-11-1    linear of order 2
ρ9111111-1-1-1111-1-1-1-1-11-1111-1-111    linear of order 2
ρ101111-1-11-1-111111-1-11-11-1-11-11-11    linear of order 2
ρ11111111111-111-1-1-1-1-111-1-1-1-1-111    linear of order 2
ρ121111-1-1-111-11111-1-11-1-111-1-11-11    linear of order 2
ρ13111111-11-11111111111-11-1-1-1-1-1    linear of order 2
ρ141111-1-111-1111-1-111-1-1-11-1-1-111-1    linear of order 2
ρ151111111-11-111111111-11-11-1-1-1-1    linear of order 2
ρ161111-1-1-1-11-111-1-111-1-11-111-111-1    linear of order 2
ρ172222220000-2-2-2-2-222-200000000    orthogonal lifted from D4
ρ182222-2-20000-2-2-2-22-22200000000    orthogonal lifted from D4
ρ192222-2-20000-2-222-22-2200000000    orthogonal lifted from D4
ρ202222220000-2-2222-2-2-200000000    orthogonal lifted from D4
ρ214-44-40000004-400000000000000    orthogonal lifted from 2+ 1+4
ρ224-44-4000000-4400000000000000    orthogonal lifted from 2+ 1+4
ρ234-4-444-400000000000000000000    orthogonal lifted from C8⋊C22
ρ244-4-44-4400000000000000000000    orthogonal lifted from C8⋊C22
ρ2544-4-4000000004i-4i000000000000    complex lifted from D8⋊C22
ρ2644-4-400000000-4i4i000000000000    complex lifted from D8⋊C22

Smallest permutation representation of C42.271D4
On 32 points
Generators in S32
(1 9 25 20)(2 14 26 17)(3 11 27 22)(4 16 28 19)(5 13 29 24)(6 10 30 21)(7 15 31 18)(8 12 32 23)
(1 7 5 3)(2 28 6 32)(4 30 8 26)(9 15 13 11)(10 23 14 19)(12 17 16 21)(18 24 22 20)(25 31 29 27)
(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 2)(3 8)(4 7)(5 6)(9 17)(10 24)(11 23)(12 22)(13 21)(14 20)(15 19)(16 18)(25 26)(27 32)(28 31)(29 30)

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

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

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

Matrix representation of C42.271D4 in GL8(𝔽17)

04000000
130000000
00040000
001300000
00000100
000016000
000000016
00000010
,
01000000
160000000
00010000
001600000
00001000
00000100
000000160
000000016
,
00010000
00100000
10000000
016000000
00000010
000000016
00001000
000001600
,
00100000
00010000
10000000
01000000
00000010
00000001
00001000
00000100

G:=sub<GL(8,GF(17))| [0,13,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0],[0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16],[0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0],[0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0] >;

C42.271D4 in GAP, Magma, Sage, TeX

C_4^2._{271}D_4
% in TeX

G:=Group("C4^2.271D4");
// GroupNames label

G:=SmallGroup(128,1945);
// by ID

G=gap.SmallGroup(128,1945);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,120,758,891,675,4037,1027,124]);
// Polycyclic

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

Export

Character table of C42.271D4 in TeX

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