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G = C4⋊C4.97D4order 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.C4214C2, C4.70(C4.4D4), C2.25(D4.3D4), C23.276(C4○D4), C23.38D429C2, (C22×C8).322C22, (C22×C4).727C23, (C22×SD16).11C2, C23.37D4.8C2, (C22×D4).83C22, (C22×Q8).68C22, C22.234(C4⋊D4), C42.6C2219C2, 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

C1C22×C4 — C4⋊C4.97D4
C1C2C22C23C22×C4C2×M4(2)C2×C4.D4 — C4⋊C4.97D4
C1C2C22×C4 — C4⋊C4.97D4
C1C22C22×C4 — C4⋊C4.97D4
C1C2C2C22×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 [×2], C2 [×4], C4 [×4], C4 [×4], C22 [×3], C22 [×10], C8 [×5], C2×C4 [×6], C2×C4 [×6], D4 [×6], Q8 [×6], C23, C23 [×6], C42, C22⋊C4, C4⋊C4 [×2], C2×C8 [×2], C2×C8 [×5], M4(2) [×6], SD16 [×8], C22×C4, C22×C4, C2×D4 [×2], C2×D4 [×5], C2×Q8 [×2], C2×Q8 [×5], C24, C4.D4 [×2], C4.10D4 [×2], D4⋊C4 [×2], Q8⋊C4 [×2], C4⋊C8 [×2], C42⋊C2, C22×C8, C2×M4(2) [×3], C2×SD16 [×6], 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 [×7], C22 [×7], D4 [×8], C23, C2×D4 [×4], C4○D4 [×3], C22≀C2, C4⋊D4 [×3], C22.D4 [×2], C4.4D4, C23.10D4, D4.3D4 [×2], 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 12 5 16)(2 13 6 9)(3 14 7 10)(4 15 8 11)(17 30 21 26)(18 31 22 27)(19 32 23 28)(20 25 24 29)
(1 32 10 17)(2 18 11 25)(3 26 12 19)(4 20 13 27)(5 28 14 21)(6 22 15 29)(7 30 16 23)(8 24 9 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 13 5 9)(2 16 6 12)(3 11 7 15)(4 14 8 10)(17 25 21 29)(18 28 22 32)(19 31 23 27)(20 26 24 30)

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

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

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

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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