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G = C4⋊Q8⋊C4order 128 = 27

5th semidirect product of C4⋊Q8 and C4 acting faithfully

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

Aliases: C4⋊Q85C4, C423C44C2, (C2×D4).134D4, C42.11(C2×C4), (C22×Q8)⋊10C4, (C22×C4).96D4, C23.12(C2×D4), C42⋊C211C4, C4.19(C23⋊C4), (C2×D4).23C23, C23⋊C4.14C22, C23.38(C22⋊C4), C4.4D4.15C22, C23.C23.10C2, C23.38C23.7C2, C2.41(C2×C23⋊C4), (C2×Q8).37(C2×C4), (C22×C4).33(C2×C4), (C2×C4).98(C22×C4), (C2×C4).28(C22⋊C4), (C2×C4○D4).76C22, C22.65(C2×C22⋊C4), SmallGroup(128,861)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C4⋊Q8⋊C4
C1C2C22C23C2×D4C2×C4○D4C23.38C23 — C4⋊Q8⋊C4
C1C2C22C2×C4 — C4⋊Q8⋊C4
C1C2C2×C4C2×C4○D4 — C4⋊Q8⋊C4
C1C2C22C2×D4 — C4⋊Q8⋊C4

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

Subgroups: 284 in 119 conjugacy classes, 42 normal (14 characteristic)
C1, C2, C2 [×4], C4 [×2], C4 [×11], C22, C22 [×6], C2×C4 [×2], C2×C4 [×2], C2×C4 [×16], D4 [×3], Q8 [×5], C23, C23 [×2], C42 [×2], C42 [×2], C22⋊C4 [×9], C4⋊C4 [×7], C22×C4, C22×C4 [×2], C22×C4, C2×D4, C2×D4 [×2], C2×Q8, C2×Q8 [×2], C2×Q8 [×3], C4○D4 [×2], C23⋊C4 [×4], C23⋊C4 [×2], C42⋊C2, C42⋊C2 [×2], C22⋊Q8 [×2], C22.D4 [×2], C4.4D4 [×2], C4⋊Q8 [×2], C22×Q8, C2×C4○D4, C423C4 [×4], C23.C23 [×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 12A2B2C2D2E4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O4P4Q
 size 11244422444888888888888
ρ111111111111111111111111    trivial
ρ211111111111-111-1-111-1-1-1-1-1    linear of order 2
ρ311111-1-1-1-11-1-1-111-11-11-11-11    linear of order 2
ρ411111-1-1-1-11-11-11-111-1-11-11-1    linear of order 2
ρ5111111111111-1-111-1-1-1-1-1-11    linear of order 2
ρ611111111111-1-1-1-1-1-1-11111-1    linear of order 2
ρ711111-1-1-1-11-1-11-11-1-11-11-111    linear of order 2
ρ811111-1-1-1-11-111-1-11-111-11-1-1    linear of order 2
ρ9111-1-1-1-1-1111-1-ii-11-ii-iii-i1    linear of order 4
ρ10111-1-1-1-1-11111-ii1-1-iii-i-ii-1    linear of order 4
ρ11111-1-1111-11-11ii-1-1-i-i-i-iii1    linear of order 4
ρ12111-1-1111-11-1-1ii11-i-iii-i-i-1    linear of order 4
ρ13111-1-1-1-1-1111-1i-i-11i-ii-i-ii1    linear of order 4
ρ14111-1-1-1-1-11111i-i1-1i-i-iii-i-1    linear of order 4
ρ15111-1-1111-11-11-i-i-1-1iiii-i-i1    linear of order 4
ρ16111-1-1111-11-1-1-i-i11ii-i-iii-1    linear of order 4
ρ172222-2-222-2-22000000000000    orthogonal lifted from D4
ρ18222-222-2-2-2-22000000000000    orthogonal lifted from D4
ρ192222-22-2-22-2-2000000000000    orthogonal lifted from D4
ρ20222-22-2222-2-2000000000000    orthogonal lifted from D4
ρ2144-40004-4000000000000000    orthogonal lifted from C23⋊C4
ρ2244-4000-44000000000000000    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 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 5 19 25)(2 6 20 26)(3 7 17 27)(4 8 18 28)(9 15 30 23)(10 16 31 24)(11 13 32 21)(12 14 29 22)
(1 14 19 22)(2 13 20 21)(3 16 17 24)(4 15 18 23)(5 12 25 29)(6 11 26 32)(7 10 27 31)(8 9 28 30)
(2 28 18 26)(3 17)(4 6 20 8)(5 27 25 7)(9 15 11 21)(10 12 31 29)(13 30 23 32)(16 24)

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

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

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

Matrix representation of C4⋊Q8⋊C4 in GL8(𝔽5)

11412223
11142232
14113222
41112322
00004414
00004441
00004144
00001444
,
00100000
00010000
40000000
04000000
00000010
00000001
00004000
00000400
,
04003000
40000300
00010020
00100002
10000100
01001000
00400004
00040040
,
10000000
04000000
00040000
00100000
01001000
40000400
00400004
00010010

G:=sub<GL(8,GF(5))| [1,1,1,4,0,0,0,0,1,1,4,1,0,0,0,0,4,1,1,1,0,0,0,0,1,4,1,1,0,0,0,0,2,2,3,2,4,4,4,1,2,2,2,3,4,4,1,4,2,3,2,2,1,4,4,4,3,2,2,2,4,1,4,4],[0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[0,4,0,0,1,0,0,0,4,0,0,0,0,1,0,0,0,0,0,1,0,0,4,0,0,0,1,0,0,0,0,4,3,0,0,0,0,1,0,0,0,3,0,0,1,0,0,0,0,0,2,0,0,0,0,4,0,0,0,2,0,0,4,0],[1,0,0,0,0,4,0,0,0,4,0,0,1,0,0,0,0,0,0,1,0,0,4,0,0,0,4,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0] >;

C4⋊Q8⋊C4 in GAP, Magma, Sage, TeX

C_4\rtimes Q_8\rtimes C_4
% in TeX

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

G:=SmallGroup(128,861);
// by ID

G=gap.SmallGroup(128,861);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,-2,112,141,456,723,352,1123,1018,248,1971,375,4037]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^4=d^4=1,c^2=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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