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G = C23.Q16order 128 = 27

1st non-split extension by C23 of Q16 acting via Q16/C2=D4

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

Aliases: C24.3D4, C23.1Q16, C23.5SD16, C2.6C2≀C4, C23⋊C8.4C2, (C22×Q8)⋊1C4, C22.17C4≀C2, (C22×C4).10D4, C2.C425C4, C23⋊Q8.1C2, C2.5(C423C4), C23.9D4.4C2, C22.58(C23⋊C4), C23.158(C22⋊C4), C22.16(Q8⋊C4), C2.4(C23.31D4), (C22×C4).5(C2×C4), (C2×C22⋊C4).84C22, SmallGroup(128,83)

Series: Derived Chief Lower central Upper central Jennings

C1C22×C4 — C23.Q16
C1C2C22C23C24C2×C22⋊C4C23⋊Q8 — C23.Q16
C1C2C23C22×C4 — C23.Q16
C1C22C23C2×C22⋊C4 — C23.Q16
C1C22C23C2×C22⋊C4 — C23.Q16

Generators and relations for C23.Q16
 G = < a,b,c,d,e | a2=b2=c2=d8=1, e2=bd4, dad-1=ab=ba, ac=ca, eae-1=abc, dbd-1=bc=cb, be=eb, cd=dc, ce=ec, ede-1=acd-1 >

Subgroups: 252 in 81 conjugacy classes, 20 normal (all characteristic)
C1, C2 [×3], C2 [×4], C4 [×7], C22 [×3], C22 [×9], C8, C2×C4 [×15], Q8 [×4], C23 [×3], C23 [×4], C22⋊C4 [×6], C2×C8, C22×C4 [×2], C22×C4 [×3], C2×Q8 [×3], C24, C2.C42, C2.C42, C22⋊C8, C2×C22⋊C4, C2×C22⋊C4 [×2], C22×Q8, C23⋊C8, C23.9D4, C23⋊Q8, C23.Q16
Quotients: C1, C2 [×3], C4 [×2], C22, C2×C4, D4 [×2], C22⋊C4, SD16, Q16, C23⋊C4, Q8⋊C4, C4≀C2, C23.31D4, C2≀C4, C423C4, C23.Q16

Character table of C23.Q16

 class 12A2B2C2D2E2F2G4A4B4C4D4E4F4G4H4I4J4K8A8B8C8D
 size 11112244448888888888888
ρ111111111111111111111111    trivial
ρ21111111111-1-1-1-11-1-1-1-11111    linear of order 2
ρ31111111111-11-111-1-111-1-1-1-1    linear of order 2
ρ411111111111-11-1111-1-1-1-1-1-1    linear of order 2
ρ5111111-1-1-1-1-i-1i11i-i1-1-iii-i    linear of order 4
ρ6111111-1-1-1-1i1-i-11-ii-11-iii-i    linear of order 4
ρ7111111-1-1-1-1-i1i-11i-i-11i-i-ii    linear of order 4
ρ8111111-1-1-1-1i-1-i11-ii1-1i-i-ii    linear of order 4
ρ922222222-2-20000-200000000    orthogonal lifted from D4
ρ10222222-2-2220000-200000000    orthogonal lifted from D4
ρ112-22-22-22-200000000000-2-222    symplectic lifted from Q16, Schur index 2
ρ122-22-22-22-20000000000022-2-2    symplectic lifted from Q16, Schur index 2
ρ132-22-2-22002i-2i1-i0-1-i001+i-1+i000000    complex lifted from C4≀C2
ρ142-22-2-2200-2i2i-1-i01-i00-1+i1+i000000    complex lifted from C4≀C2
ρ152-22-2-22002i-2i-1+i01+i00-1-i1-i000000    complex lifted from C4≀C2
ρ162-22-22-2-2200000000000-2--2-2--2    complex lifted from SD16
ρ172-22-2-2200-2i2i1+i0-1+i001-i-1-i000000    complex lifted from C4≀C2
ρ182-22-22-2-2200000000000--2-2--2-2    complex lifted from SD16
ρ1944-4-40000000-200000020000    orthogonal lifted from C2≀C4
ρ204444-4-400000000000000000    orthogonal lifted from C23⋊C4
ρ2144-4-400000002000000-20000    orthogonal lifted from C2≀C4
ρ224-4-440000000002i000-2i00000    complex lifted from C423C4
ρ234-4-44000000000-2i0002i00000    complex lifted from C423C4

Smallest permutation representation of C23.Q16
On 32 points
Generators in S32
(1 5)(2 10)(3 18)(4 31)(6 14)(7 22)(8 27)(9 13)(11 26)(12 23)(15 30)(16 19)(17 25)(20 24)(21 29)(28 32)
(1 13)(2 25)(3 15)(4 27)(5 9)(6 29)(7 11)(8 31)(10 17)(12 19)(14 21)(16 23)(18 30)(20 32)(22 26)(24 28)
(1 20)(2 21)(3 22)(4 23)(5 24)(6 17)(7 18)(8 19)(9 28)(10 29)(11 30)(12 31)(13 32)(14 25)(15 26)(16 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 6 9 25)(2 13 29 5)(3 16 11 19)(4 18 31 26)(7 12 15 23)(8 22 27 30)(10 24 21 32)(14 20 17 28)

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

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

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

Matrix representation of C23.Q16 in GL6(𝔽17)

100000
010000
0016000
0001600
000010
000001
,
100000
010000
001200
0001600
000012
0000016
,
100000
010000
0016000
0001600
0000160
0000016
,
1430000
14140000
0011016
00161600
000161616
000011
,
12120000
1250000
000111
0001600
001616016
000001

G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,2,16,0,0,0,0,0,0,1,0,0,0,0,0,2,16],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[14,14,0,0,0,0,3,14,0,0,0,0,0,0,1,16,0,0,0,0,1,16,16,0,0,0,0,0,16,1,0,0,16,0,16,1],[12,12,0,0,0,0,12,5,0,0,0,0,0,0,0,0,16,0,0,0,1,16,16,0,0,0,1,0,0,0,0,0,1,0,16,1] >;

C23.Q16 in GAP, Magma, Sage, TeX

C_2^3.Q_{16}
% in TeX

G:=Group("C2^3.Q16");
// GroupNames label

G:=SmallGroup(128,83);
// by ID

G=gap.SmallGroup(128,83);
# by ID

G:=PCGroup([7,-2,2,-2,2,-2,2,-2,56,85,120,422,387,520,1690,521,2804]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^8=1,e^2=b*d^4,d*a*d^-1=a*b=b*a,a*c=c*a,e*a*e^-1=a*b*c,d*b*d^-1=b*c=c*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=a*c*d^-1>;
// generators/relations

Export

Character table of C23.Q16 in TeX

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