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G = C4⋊C4.97D4  order 128 = 27

52nd non-split extension by C4⋊C4 of D4 acting via D4/C2=C22

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

Aliases: C4⋊C4.97D4, (C2×C8).157D4, (C2×Q8).98D4, C4.68C22≀C2, (C2×D4).108D4, C4.26(C4⋊D4), C4.C42⋊14C2, C4.70(C4.4D4), C2.25(D4.3D4), C23.276(C4○D4), C23.38D4⋊29C2, (C22×C8).322C22, (C22×C4).727C23, (C22×SD16).11C2, C23.37D4.8C2, (C22×D4).83C22, (C22×Q8).68C22, C22.234(C4⋊D4), C42.6C22⋊19C2, C42⋊C2.59C22, C2.19(C23.10D4), (C2×M4(2)).228C22, C22.55(C22.D4), (C2×C4).81(C4○D4), (C2×C4).1043(C2×D4), (C2×C4.10D4)⋊24C2, (C2×C4.D4).11C2, SmallGroup(128,778)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series
C1 — C22×C4 — C4⋊C4.97D4
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C2×M4(2) — C2×C4.D4 — C4⋊C4.97D4
Lower central
C1 — C2 — C22×C4 — C4⋊C4.97D4
Upper central
C1 — C22 — C22×C4 — C4⋊C4.97D4
Jennings
C1 — C2 — C2 — C22×C4 — C4⋊C4.97D4

Generators and relations for C4⋊C4.97D4
 G = < a,b,c,d | a4=b4=1, c4=d2=a2, bab-1=dad-1=a-1, ac=ca, cbc-1=b-1, dbd-1=a-1b, dcd-1=c3 >

Subgroups: 328 in 138 conjugacy classes, 42 normal (32 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), SD16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C24, C4.D4, C4.10D4, D4⋊C4, Q8⋊C4, C4⋊C8, C42⋊C2, C22×C8, C2×M4(2), C2×SD16, C22×D4, C22×Q8, C4.C42, C2×C4.D4, C2×C4.10D4, C23.37D4, C23.38D4, C42.6C22, C22×SD16, C4⋊C4.97D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C22≀C2, C4⋊D4, C22.D4, C4.4D4, C23.10D4, D4.3D4, C4⋊C4.97D4

Character table of C4⋊C4.97D4

 class 12A2B2C2D2E2F2G4A4B4C4D4E4F4G4H8A8B8C8D8E8F8G8H8I8J
 size 11112288222288884444888888
ρ111111111111111111111111111    trivial
ρ2111111111111-11-11-1-1-1-1-1-1-1-111    linear of order 2
ρ3111111-1-111111111-1-1-1-1-111-1-1-1    linear of order 2
ρ4111111-1-11111-11-1111111-1-11-1-1    linear of order 2
ρ5111111111111-1-1-1-1-1-1-1-11111-1-1    linear of order 2
ρ61111111111111-11-11111-1-1-1-1-1-1    linear of order 2
ρ7111111-1-111111-11-1-1-1-1-11-1-1111    linear of order 2
ρ8111111-1-11111-1-1-1-11111-111-111    linear of order 2
ρ92222-2-20022-2-20-2020000000000    orthogonal lifted from D4
ρ102-2-22-22002-22-220-200000000000    orthogonal lifted from D4
ρ112222-2-200-2-22200002-22-2000000    orthogonal lifted from D4
ρ122-2-22-22002-22-2-20200000000000    orthogonal lifted from D4
ρ132-2-222-2-22-222-200000000000000    orthogonal lifted from D4
ρ142-2-222-22-2-222-200000000000000    orthogonal lifted from D4
ρ152222-2-200-2-2220000-22-22000000    orthogonal lifted from D4
ρ162222-2-20022-2-2020-20000000000    orthogonal lifted from D4
ρ172-2-222-2002-2-22000000000-2i2i000    complex lifted from C4○D4
ρ182-2-222-2002-2-220000000002i-2i000    complex lifted from C4○D4
ρ192-2-22-2200-22-220000000000002i-2i    complex lifted from C4○D4
ρ2022222200-2-2-2-200000000-2i002i00    complex lifted from C4○D4
ρ2122222200-2-2-2-2000000002i00-2i00    complex lifted from C4○D4
ρ222-2-22-2200-22-22000000000000-2i2i    complex lifted from C4○D4
ρ2344-4-40000000000000-2√-202√-2000000    complex lifted from D4.3D4
ρ244-44-40000000000002√-20-2√-20000000    complex lifted from D4.3D4
ρ2544-4-400000000000002√-20-2√-2000000    complex lifted from D4.3D4
ρ264-44-4000000000000-2√-202√-20000000    complex lifted from D4.3D4

Smallest permutation representation of C4⋊C4.97D4
►On 32 points
Generators in S32
(1 10 5 14)(2 11 6 15)(3 12 7 16)(4 13 8 9)(17 26 21 30)(18 27 22 31)(19 28 23 32)(20 29 24 25)

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

Matrix representation of C4⋊C4.97D4 ►in GL6(𝔽17)

1600000
0160000
00161600
002100
000011
00001516
,
0130000
1300000
0000105
000070
000500
007700
,
010000
1600000
0010500
007000
0000105
000070
,
1600000
010000
000500
0010000
000005
0000100

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,2,0,0,0,0,16,1,0,0,0,0,0,0,1,15,0,0,0,0,1,16],[0,13,0,0,0,0,13,0,0,0,0,0,0,0,0,0,0,7,0,0,0,0,5,7,0,0,10,7,0,0,0,0,5,0,0,0],[0,16,0,0,0,0,1,0,0,0,0,0,0,0,10,7,0,0,0,0,5,0,0,0,0,0,0,0,10,7,0,0,0,0,5,0],[16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,10,0,0,0,0,5,0,0,0,0,0,0,0,0,10,0,0,0,0,5,0] >;
 

C4⋊C4.97D4 in GAP, Magma, Sage, TeX 

C_4\rtimes C_4._{97}D_4
 
% in TeX
 
G:=Group("C4:C4.97D4");
 
// GroupNames label
 
G:=SmallGroup(128,778);
 
// by ID
 
G=gap.SmallGroup(128,778);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,2,-2,448,141,422,387,58,2019,1018,248,718,1027]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^4=b^4=1,c^4=d^2=a^2,b*a*b^-1=d*a*d^-1=a^-1,a*c=c*a,c*b*c^-1=b^-1,d*b*d^-1=a^-1*b,d*c*d^-1=c^3>;
 
// generators/relations
 

Export

Character table of C4⋊C4.97D4 in TeX

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