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G = C42.70D4order 128 = 27

52nd non-split extension by C42 of D4 acting via D4/C2=C22

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

Aliases: C42.70D4, C42.151C23, C42.92(C2×C4), C4.4D4.9C4, (C22×D4).9C4, C8⋊C427C22, (C22×Q8).8C4, (C22×C4).228D4, C42.6C437C2, (C2×C42).195C22, C23.179(C22⋊C4), C42.C2211C2, C4.4D4.117C22, C22.17(C4.D4), C2.33(C42⋊C22), (C2×D4).24(C2×C4), (C2×Q8).24(C2×C4), (C2×C4).1179(C2×D4), (C2×C4.4D4).4C2, C2.13(C2×C4.D4), (C2×C4).96(C22⋊C4), (C2×C4).145(C22×C4), (C22×C4).217(C2×C4), C22.209(C2×C22⋊C4), SmallGroup(128,265)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.70D4
C1C2C22C2×C4C42C2×C42C2×C4.4D4 — C42.70D4
C1C22C2×C4 — C42.70D4
C1C22C2×C42 — C42.70D4
C1C22C22C42 — C42.70D4

Generators and relations for C42.70D4
 G = < a,b,c,d | a4=b4=1, c4=a2b2, d2=b, ab=ba, cac-1=ab2, dad-1=a-1b2, cbc-1=a2b-1, bd=db, dcd-1=a2b-1c3 >

Subgroups: 308 in 124 conjugacy classes, 44 normal (14 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×8], C22, C22 [×2], C22 [×12], C8 [×4], C2×C4 [×2], C2×C4 [×4], C2×C4 [×8], D4 [×4], Q8 [×4], C23, C23 [×8], C42 [×2], C42 [×2], C22⋊C4 [×8], C2×C8 [×4], C22×C4, C22×C4 [×2], C22×C4, C2×D4 [×2], C2×D4 [×3], C2×Q8 [×2], C2×Q8 [×3], C24, C8⋊C4 [×4], C22⋊C8 [×2], C4⋊C8 [×2], C2×C42, C2×C22⋊C4 [×2], C4.4D4 [×4], C4.4D4 [×2], C22×D4, C22×Q8, C42.C22 [×4], C42.6C4 [×2], C2×C4.4D4, C42.70D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×2], C4.D4 [×2], C2×C22⋊C4, C2×C4.D4, C42⋊C22 [×2], C42.70D4

Character table of C42.70D4

 class 12A2B2C2D2E2F2G4A4B4C4D4E4F4G4H4I4J8A8B8C8D8E8F8G8H
 size 11112288222244448888888888
ρ111111111111111111111111111    trivial
ρ21111-1-1-111111-1-11-1-11-1-1111-1-11    linear of order 2
ρ3111111111111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ41111-1-1-111111-1-11-1-1111-1-1-111-1    linear of order 2
ρ51111-1-11-11111-1-11-11-1-1-11-1-1111    linear of order 2
ρ6111111-1-111111111-1-1111-1-1-1-11    linear of order 2
ρ71111-1-11-11111-1-11-11-111-111-1-1-1    linear of order 2
ρ8111111-1-111111111-1-1-1-1-11111-1    linear of order 2
ρ91111-1-1-11-1-1-1-1-11111-1-iii-iii-i-i    linear of order 4
ρ1011111111-1-1-1-11-11-1-1-1i-ii-ii-ii-i    linear of order 4
ρ111111-1-1-11-1-1-1-1-11111-1i-i-ii-i-iii    linear of order 4
ρ1211111111-1-1-1-11-11-1-1-1-ii-ii-ii-ii    linear of order 4
ρ13111111-1-1-1-1-1-11-11-111i-iii-ii-i-i    linear of order 4
ρ141111-1-11-1-1-1-1-1-1111-11-iiii-i-ii-i    linear of order 4
ρ15111111-1-1-1-1-1-11-11-111-ii-i-ii-iii    linear of order 4
ρ161111-1-11-1-1-1-1-1-1111-11i-i-i-iii-ii    linear of order 4
ρ172222-2-20022-2-22-2-220000000000    orthogonal lifted from D4
ρ182222-2-200-2-22222-2-20000000000    orthogonal lifted from D4
ρ1922222200-2-222-2-2-220000000000    orthogonal lifted from D4
ρ202222220022-2-2-22-2-20000000000    orthogonal lifted from D4
ρ2144-4-4-4400000000000000000000    orthogonal lifted from C4.D4
ρ2244-4-44-400000000000000000000    orthogonal lifted from C4.D4
ρ234-4-44000000-4i4i00000000000000    complex lifted from C42⋊C22
ρ244-44-40000-4i4i0000000000000000    complex lifted from C42⋊C22
ρ254-4-440000004i-4i00000000000000    complex lifted from C42⋊C22
ρ264-44-400004i-4i0000000000000000    complex lifted from C42⋊C22

Smallest permutation representation of C42.70D4
On 32 points
Generators in S32
(1 9 23 30)(2 27 24 14)(3 11 17 32)(4 29 18 16)(5 13 19 26)(6 31 20 10)(7 15 21 28)(8 25 22 12)
(1 32 19 15)(2 29 20 12)(3 26 21 9)(4 31 22 14)(5 28 23 11)(6 25 24 16)(7 30 17 13)(8 27 18 10)
(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 27 32 18 19 10 15 8)(2 21 29 9 20 3 12 26)(4 5 31 28 22 23 14 11)(6 17 25 13 24 7 16 30)

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

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

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

Matrix representation of C42.70D4 in GL8(𝔽17)

00100000
00010000
10000000
01000000
000001300
000013000
00000004
00000040
,
01000000
160000000
00010000
001600000
00000100
00001000
00000001
00000010
,
31014100000
10141030000
371470000
714730000
00000010
000000016
000013000
00000400
,
31014100000
737140000
371470000
10310140000
00000010
00000001
00000100
00001000

G:=sub<GL(8,GF(17))| [0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0],[0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[3,10,3,7,0,0,0,0,10,14,7,14,0,0,0,0,14,10,14,7,0,0,0,0,10,3,7,3,0,0,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,4,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0],[3,7,3,10,0,0,0,0,10,3,7,3,0,0,0,0,14,7,14,10,0,0,0,0,10,14,7,14,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0] >;

C42.70D4 in GAP, Magma, Sage, TeX

C_4^2._{70}D_4
% in TeX

G:=Group("C4^2.70D4");
// GroupNames label

G:=SmallGroup(128,265);
// by ID

G=gap.SmallGroup(128,265);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,1430,1123,1018,248,1971,102]);
// Polycyclic

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

Export

Character table of C42.70D4 in TeX

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