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G = C42.70D4  order 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⋊C4⋊27C22, (C22×Q8).8C4, (C22×C4).228D4, C42.6C4⋊37C2, (C2×C42).195C22, C23.179(C22⋊C4), C42.C22⋊11C2, 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

Derived series
C1 — C2×C4 — C42.70D4
Chief series
C1 — C2 — C22 — C2×C4 — C42 — C2×C42 — C2×C4.4D4 — C42.70D4
Lower central
C1 — C22 — C2×C4 — C42.70D4
Upper central
C1 — C22 — C2×C42 — C42.70D4
Jennings
C1 — C22 — C22 — C42 — 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, C2, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C42, C22⋊C4, C2×C8, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C24, C8⋊C4, C22⋊C8, C4⋊C8, C2×C42, C2×C22⋊C4, C4.4D4, C4.4D4, C22×D4, C22×Q8, C42.C22, C42.6C4, C2×C4.4D4, C42.70D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C4.D4, C2×C22⋊C4, C2×C4.D4, C42⋊C22, 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 30 24 10)(2 15 17 27)(3 32 18 12)(4 9 19 29)(5 26 20 14)(6 11 21 31)(7 28 22 16)(8 13 23 25)

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

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

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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