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G = (C2×C4)⋊Q16  order 128 = 27

The semidirect product of C2×C4 and Q16 acting via Q16/C2=D4

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

Aliases: (C2×C4)⋊Q16, C4⋊C4.8D4, (C2×Q8).13D4, C2.14C2≀C22, (C22×C4).47D4, C23.522(C2×D4), C22⋊C8.4C22, C22⋊Q16.1C2, C22.12(C2×Q16), C22⋊Q8.7C22, (C22×C4).11C23, C2.7(C22⋊Q16), C2.8(D4.10D4), C22.132C22≀C2, (C22×Q8).5C22, C23.31D4.2C2, C22.30(C8.C22), C2.C42.19C22, C23.78C23.2C2, C23.41C23.1C2, C22.M4(2).7C2, (C2×C4).200(C2×D4), (C2×C4⋊C4).18C22, SmallGroup(128,337)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series
C1 — C22×C4 — (C2×C4)⋊Q16
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C2×C4⋊C4 — C23.41C23 — (C2×C4)⋊Q16
Lower central
C1 — C22 — C22×C4 — (C2×C4)⋊Q16
Upper central
C1 — C22 — C22×C4 — (C2×C4)⋊Q16
Jennings
C1 — C2 — C22 — C22×C4 — (C2×C4)⋊Q16

Generators and relations for (C2×C4)⋊Q16
 G = < a,b,c,d | a2=b4=c8=1, d2=c4, cbc-1=ab=ba, cac-1=dad-1=ab2, bd=db, dcd-1=c-1 >

Subgroups: 268 in 119 conjugacy classes, 34 normal (18 characteristic)
C1, C2, C2, C4, C22, C22, C8, C2×C4, C2×C4, Q8, C23, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, Q16, C22×C4, C22×C4, C2×Q8, C2×Q8, C2.C42, C2.C42, C22⋊C8, Q8⋊C4, C2×C4⋊C4, C2×C4⋊C4, C42⋊C2, C22⋊Q8, C22⋊Q8, C42.C2, C4⋊Q8, C2×Q16, C22×Q8, C22.M4(2), C23.31D4, C23.78C23, C22⋊Q16, C23.41C23, (C2×C4)⋊Q16
Quotients: C1, C2, C22, D4, C23, Q16, C2×D4, C22≀C2, C2×Q16, C8.C22, C22⋊Q16, D4.10D4, C2≀C22, (C2×C4)⋊Q16

Character table of (C2×C4)⋊Q16

 class 12A2B2C2D2E4A4B4C4D4E4F4G4H4I4J4K4L4M8A8B8C8D
 size 11112244448888888888888
ρ111111111111111111111111    trivial
ρ2111111-1-111-11-11-11-11-1-11-11    linear of order 2
ρ311111111111-1-1-1-11111-1-1-1-1    linear of order 2
ρ4111111-1-111-1-11-111-11-11-11-1    linear of order 2
ρ51111111111-11111-11-1-1-1-1-1-1    linear of order 2
ρ6111111-1-11111-11-1-1-1-111-11-1    linear of order 2
ρ71111111111-1-1-1-1-1-11-1-11111    linear of order 2
ρ8111111-1-1111-11-11-1-1-11-11-11    linear of order 2
ρ92222-2-200-2200-20200000000    orthogonal lifted from D4
ρ1022222222-2-2000000-2000000    orthogonal lifted from D4
ρ112222-2-2002-2020-2000000000    orthogonal lifted from D4
ρ122222-2-2002-20-202000000000    orthogonal lifted from D4
ρ13222222-2-2-2-20000002000000    orthogonal lifted from D4
ρ142222-2-200-220020-200000000    orthogonal lifted from D4
ρ1522-2-2-22-2200000000000√2-√2-√2√2    symplectic lifted from Q16, Schur index 2
ρ1622-2-2-222-200000000000√2√2-√2-√2    symplectic lifted from Q16, Schur index 2
ρ1722-2-2-22-2200000000000-√2√2√2-√2    symplectic lifted from Q16, Schur index 2
ρ1822-2-2-222-200000000000-√2-√2√2√2    symplectic lifted from Q16, Schur index 2
ρ194-44-40000000000020-200000    orthogonal lifted from C2≀C22
ρ204-44-400000000000-20200000    orthogonal lifted from C2≀C22
ρ214-4-4400000020000000-20000    symplectic lifted from D4.10D4, Schur index 2
ρ2244-4-44-400000000000000000    symplectic lifted from C8.C22, Schur index 2
ρ234-4-44000000-2000000020000    symplectic lifted from D4.10D4, Schur index 2

Smallest permutation representation of (C2×C4)⋊Q16
►On 32 points
Generators in S32
(1 28)(2 6)(3 30)(4 8)(5 32)(7 26)(9 13)(10 19)(11 15)(12 21)(14 23)(16 17)(18 22)(20 24)(25 29)(27 31)

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

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

Matrix representation of (C2×C4)⋊Q16 ►in GL6(𝔽17)

100000
010000
001000
000100
0000160
0000016
,
1600000
0160000
0010100
001700
0000101
000017
,
640000
400000
000017
0000716
0010100
001700
,
4110000
0130000
000010
000001
0016000
0001600

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

(C2×C4)⋊Q16 in GAP, Magma, Sage, TeX 

(C_2\times C_4)\rtimes Q_{16}
 
% in TeX
 
G:=Group("(C2xC4):Q16");
 
// GroupNames label
 
G:=SmallGroup(128,337);
 
// by ID
 
G=gap.SmallGroup(128,337);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,-2,2,224,141,232,422,352,1123,570,521,136,1411]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^2=b^4=c^8=1,d^2=c^4,c*b*c^-1=a*b=b*a,c*a*c^-1=d*a*d^-1=a*b^2,b*d=d*b,d*c*d^-1=c^-1>;
 
// generators/relations
 

Export

Character table of (C2×C4)⋊Q16 in TeX

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