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G = C4.(C4×D4)  order 128 = 27

5th non-split extension by C4 of C4×D4 acting via C4×D4/C42=C2

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

Aliases: C4⋊Q83C4, C4.5(C4×D4), (C2×C8).30D4, (C2×D4).79D4, C4.4D41C4, C42.7(C2×C4), C4.9C429C2, C4.99C22≀C2, (C22×C4).67D4, C23.132(C2×D4), Q8○M4(2).5C2, (C22×C4).33C23, C23.38D425C2, C22.53(C4⋊D4), C23.13(C22⋊C4), (C22×Q8).24C22, C42⋊C2.31C22, C4.13(C22.D4), C23.C23.7C2, C2.48(C23.23D4), (C2×M4(2)).192C22, C23.38C23.1C2, (C2×D4).88(C2×C4), (C2×C4).244(C2×D4), (C2×Q8).76(C2×C4), (C2×C4).328(C4○D4), (C2×C4).17(C22⋊C4), (C2×C4).193(C22×C4), (C2×C4○D4).27C22, C22.47(C2×C22⋊C4), SmallGroup(128,641)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C4.(C4×D4)
C1C2C22C23C22×C4C2×C4○D4Q8○M4(2) — C4.(C4×D4)
C1C2C2×C4 — C4.(C4×D4)
C1C2C22×C4 — C4.(C4×D4)
C1C2C2C22×C4 — C4.(C4×D4)

Generators and relations for C4.(C4×D4)
 G = < a,b,c,d | a4=c4=1, b4=d2=a2, ab=ba, ac=ca, dad-1=a-1, cbc-1=a-1b, bd=db, dcd-1=c-1 >

Subgroups: 300 in 156 conjugacy classes, 54 normal (18 characteristic)
C1, C2, C2 [×5], C4 [×2], C4 [×2], C4 [×8], C22, C22 [×2], C22 [×5], C8 [×4], C2×C4 [×2], C2×C4 [×6], C2×C4 [×12], D4 [×6], Q8 [×8], C23, C23 [×2], C42 [×2], C42 [×2], C22⋊C4 [×7], C4⋊C4 [×7], C2×C8 [×4], C2×C8 [×4], M4(2) [×8], C22×C4, C22×C4 [×2], C22×C4, C2×D4, C2×D4 [×2], C2×Q8, C2×Q8 [×2], C2×Q8 [×5], C4○D4 [×4], C23⋊C4 [×2], Q8⋊C4 [×4], C42⋊C2, C42⋊C2 [×2], C22⋊Q8 [×2], C22.D4 [×2], C4.4D4 [×2], C4⋊Q8 [×2], C2×M4(2) [×2], C2×M4(2) [×2], C8○D4 [×4], C22×Q8, C2×C4○D4, C4.9C42 [×2], C23.C23, C23.38D4 [×2], C23.38C23, Q8○M4(2), C4.(C4×D4)
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×8], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×4], C4○D4 [×2], C2×C22⋊C4, C4×D4 [×2], C22≀C2, C4⋊D4 [×2], C22.D4, C23.23D4, C4.(C4×D4)

Character table of C4.(C4×D4)

 class 12A2B2C2D2E2F4A4B4C4D4E4F4G4H4I4J4K4L4M4N8A8B8C8D8E8F8G8H
 size 11222442222448888888844444444
ρ111111111111111111111111111111    trivial
ρ211111-1-11111-1-111-1-11-1-11111-1-1-1-11    linear of order 2
ρ311111-1-11111-1-1-1-111-111-1111-1-1-1-11    linear of order 2
ρ41111111111111-1-1-1-1-1-1-1-111111111    linear of order 2
ρ51111111111111-111-111-1-1-1-1-1-1-1-1-1-1    linear of order 2
ρ611111-1-11111-1-1-11-111-11-1-1-1-11111-1    linear of order 2
ρ711111-1-11111-1-11-11-1-11-11-1-1-11111-1    linear of order 2
ρ811111111111111-1-11-1-111-1-1-1-1-1-1-1-1    linear of order 2
ρ911-1-1111-11-11-1-11ii1-i-i-1-1i-ii-ii-ii-i    linear of order 4
ρ1011-1-11-1-1-11-11111i-i-1-ii1-1i-iii-ii-i-i    linear of order 4
ρ1111-1-11-1-1-11-1111-1-ii1i-i-11i-iii-ii-i-i    linear of order 4
ρ1211-1-1111-11-11-1-1-1-i-i-1ii11i-ii-ii-ii-i    linear of order 4
ρ1311-1-11-1-1-11-1111-1i-i1-ii-11-ii-i-ii-iii    linear of order 4
ρ1411-1-1111-11-11-1-1-1ii-1-i-i11-ii-ii-ii-ii    linear of order 4
ρ1511-1-1111-11-11-1-11-i-i1ii-1-1-ii-ii-ii-ii    linear of order 4
ρ1611-1-11-1-1-11-11111-ii-1i-i1-1-ii-i-ii-iii    linear of order 4
ρ17222222-2-2-2-2-22-20000000000000000    orthogonal lifted from D4
ρ18222-2-20022-2-2000000000022-20000-2    orthogonal lifted from D4
ρ19222-2-20022-2-20000000000-2-2200002    orthogonal lifted from D4
ρ2022-2-22-222-22-22-20000000000000000    orthogonal lifted from D4
ρ2122-22-2002-2-220000000000000-2-2220    orthogonal lifted from D4
ρ2222-2-222-22-22-2-220000000000000000    orthogonal lifted from D4
ρ2322-22-2002-2-22000000000000022-2-20    orthogonal lifted from D4
ρ2422222-22-2-2-2-2-220000000000000000    orthogonal lifted from D4
ρ25222-2-200-2-22200000000000002i-2i-2i2i0    complex lifted from C4○D4
ρ2622-22-200-222-200000000002i-2i-2i00002i    complex lifted from C4○D4
ρ2722-22-200-222-20000000000-2i2i2i0000-2i    complex lifted from C4○D4
ρ28222-2-200-2-2220000000000000-2i2i2i-2i0    complex lifted from C4○D4
ρ298-8000000000000000000000000000    symplectic faithful, Schur index 2

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

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

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

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

Matrix representation of C4.(C4×D4) in GL8(𝔽17)

00100000
00010000
160000000
016000000
000000160
000000016
00001000
00000100
,
16117110000
1617100000
10616110000
1071610000
7140313343
014131441413
031031314133
130334144
,
107086811
0000116710
10010811911
0000710161
71301313343
7140313343
0131041314133
031031314133
,
0016015000
000016100
1600000150
000000161
10000010
116000010
00101000
001161000

G:=sub<GL(8,GF(17))| [0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0],[16,16,10,10,7,0,0,1,11,1,6,7,14,14,3,3,7,7,16,16,0,1,10,0,11,10,11,1,3,3,3,3,0,0,0,0,13,14,13,3,0,0,0,0,3,4,14,4,0,0,0,0,4,14,13,14,0,0,0,0,3,13,3,4],[1,0,10,0,7,7,0,0,0,0,0,0,13,14,13,3,7,0,1,0,0,0,10,10,0,0,0,0,13,3,4,3,8,1,8,7,13,13,13,13,6,16,11,10,3,3,14,14,8,7,9,16,4,4,13,13,11,10,11,1,3,3,3,3],[0,0,16,0,1,1,0,0,0,0,0,0,0,16,0,0,16,0,0,0,0,0,1,1,0,0,0,0,0,0,0,16,15,16,0,0,0,0,1,1,0,1,0,0,0,0,0,0,0,0,15,16,1,1,0,0,0,0,0,1,0,0,0,0] >;

C4.(C4×D4) in GAP, Magma, Sage, TeX

C_4.(C_4\times D_4)
% in TeX

G:=Group("C4.(C4xD4)");
// GroupNames label

G:=SmallGroup(128,641);
// by ID

G=gap.SmallGroup(128,641);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,288,422,2019,521,248,2804,1411,1027]);
// Polycyclic

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

Export

Character table of C4.(C4×D4) in TeX

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