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G = (C2×C4).SD16  order 128 = 27

7th non-split extension by C2×C4 of SD16 acting via SD16/C2=D4

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

Aliases: C4⋊C4.14D4, (C2×Q8).17D4, (C2×C4).7SD16, (C22×C4).51D4, C2.9(Q8⋊D4), C23.528(C2×D4), (C22×C4).17C23, C22.15(C2×SD16), C22⋊Q8.11C22, C22.138C22≀C2, C22⋊C8.116C22, C23.31D4.4C2, C23.47D4.4C2, C2.8(C23.7D4), C2.11(D4.10D4), C22.32(C8.C22), C2.C42.24C22, C23.81C23.2C2, C23.41C23.2C2, C22.M4(2).9C2, (C2×C4).206(C2×D4), (C2×C4⋊C4).22C22, SmallGroup(128,343)

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

Derived series
C1 — C22×C4 — (C2×C4).SD16
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C2×C4⋊C4 — C23.41C23 — (C2×C4).SD16
Lower central
C1 — C22 — C22×C4 — (C2×C4).SD16
Upper central
C1 — C22 — C22×C4 — (C2×C4).SD16
Jennings
C1 — C2 — C22 — C22×C4 — (C2×C4).SD16

Generators and relations for (C2×C4).SD16
 G = < a,b,c,d | a2=b4=c8=1, d2=b2, dbd-1=ab=ba, cac-1=ab2, ad=da, cbc-1=ab-1, dcd-1=b2c3 >

Subgroups: 228 in 105 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, C22×C4, C22×C4, C2×Q8, C2×Q8, C2.C42, C2.C42, C22⋊C8, Q8⋊C4, C4.Q8, C2×C4⋊C4, C2×C4⋊C4, C42⋊C2, C22⋊Q8, C22⋊Q8, C42.C2, C4⋊Q8, C22.M4(2), C23.31D4, C23.81C23, C23.47D4, C23.41C23, (C2×C4).SD16
Quotients: C1, C2, C22, D4, C23, SD16, C2×D4, C22≀C2, C2×SD16, C8.C22, Q8⋊D4, D4.10D4, C23.7D4, (C2×C4).SD16

Character table of (C2×C4).SD16

 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
ρ1522-2-2-222-200000000000√-2√-2-√-2-√-2    complex lifted from SD16
ρ1622-2-2-22-2200000000000-√-2√-2√-2-√-2    complex lifted from SD16
ρ1722-2-2-222-200000000000-√-2-√-2√-2√-2    complex lifted from SD16
ρ1822-2-2-22-2200000000000√-2-√-2-√-2√-2    complex lifted from SD16
ρ194-4-4400000020000000-20000    symplectic lifted from D4.10D4, Schur index 2
ρ2044-4-44-400000000000000000    symplectic lifted from C8.C22, Schur index 2
ρ214-4-44000000-2000000020000    symplectic lifted from D4.10D4, Schur index 2
ρ224-44-4000000000002i0-2i00000    complex lifted from C23.7D4
ρ234-44-400000000000-2i02i00000    complex lifted from C23.7D4

Smallest permutation representation of (C2×C4).SD16
►On 32 points
Generators in S32
(2 28)(4 30)(6 32)(8 26)(10 20)(12 22)(14 24)(16 18)

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

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

Matrix representation of (C2×C4).SD16 ►in GL6(𝔽17)

100000
010000
001000
000100
0000160
001616016
,
100000
010000
0071600
00161000
001182
00100109
,
550000
1250000
001616915
001010113
001700
000178
,
100000
0160000
0010100
001700
0077614
00010111

G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,16,0,0,0,1,0,16,0,0,0,0,16,0,0,0,0,0,0,16],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,7,16,1,10,0,0,16,10,1,0,0,0,0,0,8,10,0,0,0,0,2,9],[5,12,0,0,0,0,5,5,0,0,0,0,0,0,16,10,1,0,0,0,16,10,7,1,0,0,9,11,0,7,0,0,15,3,0,8],[1,0,0,0,0,0,0,16,0,0,0,0,0,0,10,1,7,0,0,0,1,7,7,10,0,0,0,0,6,1,0,0,0,0,14,11] >;
 

(C2×C4).SD16 in GAP, Magma, Sage, TeX 

(C_2\times C_4).{\rm SD}_{16}
 
% in TeX
 
G:=Group("(C2xC4).SD16");
 
// GroupNames label
 
G:=SmallGroup(128,343);
 
// by ID
 
G=gap.SmallGroup(128,343);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,-2,2,448,141,232,422,352,1123,570,521,136,1411]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^2=b^4=c^8=1,d^2=b^2,d*b*d^-1=a*b=b*a,c*a*c^-1=a*b^2,a*d=d*a,c*b*c^-1=a*b^-1,d*c*d^-1=b^2*c^3>;
 
// generators/relations
 

Export

Character table of (C2×C4).SD16 in TeX

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