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G = C42.275D4order 128 = 27

257th non-split extension by C42 of D4 acting via D4/C2=C22

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

Aliases: C42.275D4, C42.403C23, C4.1092+ 1+4, C4⋊D826C2, C83D411C2, C4⋊C819C22, C22⋊D822C2, C4⋊SD1610C2, (C2×D8)⋊25C22, (C4×D4)⋊14C22, (C2×C8).65C23, (C4×Q8)⋊14C22, C8⋊C410C22, C22⋊SD1611C2, C4⋊C4.156C23, (C2×C4).415C24, C23.695(C2×D4), (C22×C4).504D4, D4⋊C431C22, C4.126(C8⋊C22), C42.6C413C2, (C2×SD16)⋊23C22, (C2×D4).164C23, C4.4D461C22, C22⋊C8.50C22, (C2×Q8).152C23, C42.C238C22, C41D4.167C22, C4⋊D4.193C22, C22.32(C8⋊C22), (C2×C42).882C22, C22.675(C22×D4), C22⋊Q8.198C22, C42.29C221C2, (C22×C4).1086C23, (C22×D4).391C22, C23.36C2311C2, C2.86(C22.29C24), (C2×C41D4)⋊21C2, (C2×C4).544(C2×D4), C2.58(C2×C8⋊C22), SmallGroup(128,1949)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.275D4
C1C2C4C2×C4C22×C4C22×D4C2×C41D4 — C42.275D4
C1C2C2×C4 — C42.275D4
C1C22C2×C42 — C42.275D4
C1C2C2C2×C4 — C42.275D4

Generators and relations for C42.275D4
 G = < a,b,c,d | a4=b4=d2=1, c4=b2, ab=ba, cac-1=ab2, dad=a-1b2, cbc-1=a2b, dbd=a2b-1, dcd=b2c3 >

Subgroups: 628 in 245 conjugacy classes, 88 normal (34 characteristic)
C1, C2 [×3], C2 [×7], C4 [×4], C4 [×7], C22, C22 [×2], C22 [×25], C8 [×4], C2×C4 [×6], C2×C4 [×11], D4 [×32], Q8 [×2], C23, C23 [×17], C42 [×4], C42, C22⋊C4 [×5], C4⋊C4 [×2], C4⋊C4 [×4], C2×C8 [×4], D8 [×4], SD16 [×4], C22×C4 [×3], C22×C4, C2×D4, C2×D4 [×4], C2×D4 [×23], C2×Q8, C24 [×2], C8⋊C4 [×2], C22⋊C8 [×2], D4⋊C4 [×8], C4⋊C8 [×2], C2×C42, C42⋊C2, C4×D4, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C42.C2, C422C2, C41D4 [×4], C41D4 [×2], C2×D8 [×4], C2×SD16 [×4], C22×D4 [×2], C22×D4 [×2], C42.6C4, C22⋊D8 [×2], C22⋊SD16 [×2], C4⋊D8 [×2], C4⋊SD16 [×2], C42.29C22 [×2], C83D4 [×2], C23.36C23, C2×C41D4, C42.275D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C24, C8⋊C22 [×4], C22×D4, 2+ 1+4 [×2], C22.29C24, C2×C8⋊C22 [×2], C42.275D4

Character table of C42.275D4

 class 12A2B2C2D2E2F2G2H2I2J4A4B4C4D4E4F4G4H4I4J4K8A8B8C8D
 size 11112288888222244448888888
ρ111111111111111111111111111    trivial
ρ21111-1-1-1-1-1111111-1-11-11-1111-1-1    linear of order 2
ρ31111-1-11-111-1-11-1111-1-11-1-1-111-1    linear of order 2
ρ4111111-11-11-1-11-11-1-1-1111-1-11-11    linear of order 2
ρ51111-1-111-1-11-11-1111-1-11-1-11-1-11    linear of order 2
ρ6111111-1-11-11-11-11-1-1-1111-11-11-1    linear of order 2
ρ71111111-1-1-1-111111111111-1-1-1-1    linear of order 2
ρ81111-1-1-111-1-11111-1-11-11-11-1-111    linear of order 2
ρ91111-1-1-1-111-1-11-1111-1-1-1111-1-11    linear of order 2
ρ1011111111-11-1-11-11-1-1-11-1-111-11-1    linear of order 2
ρ11111111-1111111111111-1-1-1-1-1-1-1    linear of order 2
ρ121111-1-11-1-1111111-1-11-1-11-1-1-111    linear of order 2
ρ13111111-1-1-1-1-111111111-1-1-11111    linear of order 2
ρ141111-1-1111-1-11111-1-11-1-11-111-1-1    linear of order 2
ρ151111-1-1-11-1-11-11-1111-1-1-111-111-1    linear of order 2
ρ161111111-11-11-11-11-1-1-11-1-11-11-11    linear of order 2
ρ172222-2-2000002-22-2-22-220000000    orthogonal lifted from D4
ρ182222-2-200000-2-2-2-22-2220000000    orthogonal lifted from D4
ρ19222222000002-22-22-2-2-20000000    orthogonal lifted from D4
ρ2022222200000-2-2-2-2-222-20000000    orthogonal lifted from D4
ρ2144-4-4000000040-4000000000000    orthogonal lifted from C8⋊C22
ρ224-44-400000000-40400000000000    orthogonal lifted from 2+ 1+4
ρ234-4-444-400000000000000000000    orthogonal lifted from C8⋊C22
ρ2444-4-40000000-404000000000000    orthogonal lifted from C8⋊C22
ρ254-44-40000000040-400000000000    orthogonal lifted from 2+ 1+4
ρ264-4-44-4400000000000000000000    orthogonal lifted from C8⋊C22

Smallest permutation representation of C42.275D4
On 32 points
Generators in S32
(1 10 30 19)(2 15 31 24)(3 12 32 21)(4 9 25 18)(5 14 26 23)(6 11 27 20)(7 16 28 17)(8 13 29 22)
(1 7 5 3)(2 29 6 25)(4 31 8 27)(9 24 13 20)(10 16 14 12)(11 18 15 22)(17 23 21 19)(26 32 30 28)
(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 13)(2 12)(3 11)(4 10)(5 9)(6 16)(7 15)(8 14)(17 27)(18 26)(19 25)(20 32)(21 31)(22 30)(23 29)(24 28)

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

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

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

Matrix representation of C42.275D4 in GL8(𝔽17)

016000000
10000000
000160000
00100000
000016100
000015100
000016919
0000201316
,
01000000
160000000
00010000
001600000
000016000
000001600
000016010
00000001
,
000160000
001600000
160000000
01000000
0000101516
000000016
000018168
00000100
,
00100000
00010000
10000000
01000000
000010150
00000001
000000160
00000100

G:=sub<GL(8,GF(17))| [0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,16,15,16,2,0,0,0,0,1,1,9,0,0,0,0,0,0,0,1,13,0,0,0,0,0,0,9,16],[0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,16,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,0,0,1,0,1,0,0,0,0,0,0,0,8,1,0,0,0,0,15,0,16,0,0,0,0,0,16,16,8,0],[0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,15,0,16,0,0,0,0,0,0,1,0,0] >;

C42.275D4 in GAP, Magma, Sage, TeX

C_4^2._{275}D_4
% in TeX

G:=Group("C4^2.275D4");
// GroupNames label

G:=SmallGroup(128,1949);
// by ID

G=gap.SmallGroup(128,1949);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,568,758,891,675,4037,1027,124]);
// 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*b^2,c*b*c^-1=a^2*b,d*b*d=a^2*b^-1,d*c*d=b^2*c^3>;
// generators/relations

Export

Character table of C42.275D4 in TeX

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