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G = C4⋊Q8.C4order 128 = 27

5th non-split extension by C4⋊Q8 of C4 acting faithfully

p-group, metabelian, nilpotent (class 4), monomial

Aliases: C4⋊Q8.5C4, C42.12(C2×C4), (C2×Q8).119D4, C4.4D4.1C4, C4⋊Q8.95C22, C42.3C46C2, C42.C46C2, (C22×C4).100D4, (C2×Q8).13C23, C42⋊C2.10C4, C23.85(C22⋊C4), C22.10(C23⋊C4), C4.10D4.7C22, C4.4D4.16C22, (C22×Q8).86C22, C23.38C23.8C2, (C2×C4).10(C2×D4), (C2×C4○D4).10C4, (C2×D4).41(C2×C4), C2.45(C2×C23⋊C4), (C22×C4).34(C2×C4), (C2×Q8).109(C2×C4), (C2×C4.10D4)⋊27C2, (C2×C4).31(C22⋊C4), (C2×C4).102(C22×C4), C22.69(C2×C22⋊C4), SmallGroup(128,865)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C4⋊Q8.C4
C1C2C22C2×C4C2×Q8C22×Q8C23.38C23 — C4⋊Q8.C4
C1C2C22C2×C4 — C4⋊Q8.C4
C1C2C23C22×Q8 — C4⋊Q8.C4
C1C2C22C2×Q8 — C4⋊Q8.C4

Generators and relations for C4⋊Q8.C4
 G = < a,b,c,d | a4=b4=1, c2=d4=b2, ab=ba, cac-1=a-1, dad-1=ab-1, cbc-1=b-1, dbd-1=a2b, cd=dc >

Subgroups: 252 in 115 conjugacy classes, 42 normal (22 characteristic)
C1, C2, C2 [×4], C4 [×9], C22, C22 [×2], C22 [×4], C8 [×4], C2×C4 [×2], C2×C4 [×4], C2×C4 [×11], D4 [×3], Q8 [×5], C23, C23, C42 [×2], C22⋊C4 [×5], C4⋊C4 [×5], C2×C8 [×2], M4(2) [×6], C22×C4, C22×C4 [×2], C22×C4, C2×D4, C2×D4, C2×Q8 [×3], C2×Q8 [×2], C2×Q8 [×2], C4○D4 [×2], C4.10D4 [×4], C4.10D4 [×2], C42⋊C2, C22⋊Q8 [×2], C22.D4 [×2], C4.4D4 [×2], C4⋊Q8 [×2], C2×M4(2) [×2], C22×Q8, C2×C4○D4, C42.C4 [×2], C42.3C4 [×2], C2×C4.10D4 [×2], C23.38C23, C4⋊Q8.C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×2], C23⋊C4 [×2], C2×C22⋊C4, C2×C23⋊C4, C4⋊Q8.C4

Character table of C4⋊Q8.C4

 class 12A2B2C2D2E4A4B4C4D4E4F4G4H4I8A8B8C8D8E8F8G8H
 size 11222844444488888888888
ρ111111111111111111111111    trivial
ρ211-11-1-1-1111-1-11-11-11-1-1111-1    linear of order 2
ρ311-11-11-1111-1-1-11-111-11-1-11-1    linear of order 2
ρ411111-1111111-1-1-1-111-1-1-111    linear of order 2
ρ511-11-11-1111-1-1-11-1-1-11-111-11    linear of order 2
ρ611111-1111111-1-1-11-1-1111-1-1    linear of order 2
ρ7111111111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ811-11-1-1-1111-1-11-111-111-1-1-11    linear of order 2
ρ911-11-111-11-11-1-1-11i-ii-ii-ii-i    linear of order 4
ρ1011-11-1-11-11-11-111-1-i-iii-iii-i    linear of order 4
ρ11111111-1-11-1-111-1-1i-i-i-i-iiii    linear of order 4
ρ1211111-1-1-11-1-11-111-i-i-iii-iii    linear of order 4
ρ1311-11-111-11-11-1-1-11-ii-ii-ii-ii    linear of order 4
ρ1411-11-1-11-11-11-111-1ii-i-ii-i-ii    linear of order 4
ρ15111111-1-11-1-111-1-1-iiiii-i-i-i    linear of order 4
ρ1611111-1-1-11-1-11-111iii-i-ii-i-i    linear of order 4
ρ1722-22-2022-2-2-2200000000000    orthogonal lifted from D4
ρ182222202-2-22-2-200000000000    orthogonal lifted from D4
ρ19222220-22-2-22-200000000000    orthogonal lifted from D4
ρ2022-22-20-2-2-222200000000000    orthogonal lifted from D4
ρ21444-4-4000000000000000000    orthogonal lifted from C23⋊C4
ρ2244-4-44000000000000000000    orthogonal lifted from C23⋊C4
ρ238-8000000000000000000000    symplectic faithful, Schur index 2

Smallest permutation representation of C4⋊Q8.C4
On 32 points
Generators in S32
(1 25)(2 8 6 4)(3 31)(5 29)(7 27)(9 17)(10 12 14 16)(11 23)(13 21)(15 19)(18 24 22 20)(26 28 30 32)
(1 31 5 27)(2 32 6 28)(3 29 7 25)(4 30 8 26)(9 19 13 23)(10 20 14 24)(11 17 15 21)(12 18 16 22)
(1 11 5 15)(2 12 6 16)(3 13 7 9)(4 14 8 10)(17 31 21 27)(18 32 22 28)(19 25 23 29)(20 26 24 30)
(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)

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

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

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

Matrix representation of C4⋊Q8.C4 in GL8(𝔽17)

000160000
00100000
01000000
160000000
000001600
00001000
161611161162
01610160161
,
00100000
00010000
160000000
016000000
000000160
161611161162
00001000
1100116016
,
000130000
001300000
013000000
130000000
131344134138
000000130
000001300
00000004
,
00001000
00000100
00000010
111616116115
01000000
160000000
000160000
000001161

G:=sub<GL(8,GF(17))| [0,0,0,16,0,0,16,0,0,0,1,0,0,0,16,16,0,1,0,0,0,0,1,1,16,0,0,0,0,0,1,0,0,0,0,0,0,1,16,16,0,0,0,0,16,0,1,0,0,0,0,0,0,0,16,16,0,0,0,0,0,0,2,1],[0,0,16,0,0,16,0,1,0,0,0,16,0,16,0,1,1,0,0,0,0,1,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,16,1,1,0,0,0,0,0,1,0,16,0,0,0,0,16,16,0,0,0,0,0,0,0,2,0,16],[0,0,0,13,13,0,0,0,0,0,13,0,13,0,0,0,0,13,0,0,4,0,0,0,13,0,0,0,4,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,4,0,13,0,0,0,0,0,13,13,0,0,0,0,0,0,8,0,0,4],[0,0,0,1,0,16,0,0,0,0,0,1,1,0,0,0,0,0,0,16,0,0,0,0,0,0,0,16,0,0,16,0,1,0,0,1,0,0,0,0,0,1,0,16,0,0,0,1,0,0,1,1,0,0,0,16,0,0,0,15,0,0,0,1] >;

C4⋊Q8.C4 in GAP, Magma, Sage, TeX

C_4\rtimes Q_8.C_4
% in TeX

G:=Group("C4:Q8.C4");
// GroupNames label

G:=SmallGroup(128,865);
// by ID

G=gap.SmallGroup(128,865);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,-2,112,141,723,1123,1018,248,1971,375,172,4037]);
// Polycyclic

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

Export

Character table of C4⋊Q8.C4 in TeX

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