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G = (C2×C4)⋊SD16order 128 = 27

1st semidirect product of C2×C4 and SD16 acting via SD16/C2=D4

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

Aliases: C4⋊C4.3D4, (C2×C4)⋊1SD16, C2.8C2≀C22, (C2×Q8).10D4, (C22×C4).43D4, C2.7(Q8⋊D4), C22⋊SD16.4C2, C23.516(C2×D4), (C22×C4).5C23, C22⋊Q8.3C22, C2.8(D4.8D4), C22.13(C2×SD16), (C22×D4).6C22, C22.126C22≀C2, C23.31D411C2, C22⋊C8.111C22, C23.10D4.2C2, C23.41C231C2, C22.28(C8.C22), C22.M4(2)⋊9C2, C2.C42.14C22, (C2×C4).194(C2×D4), (C2×C4⋊C4).15C22, SmallGroup(128,331)

Series: Derived Chief Lower central Upper central Jennings

C1C22×C4 — (C2×C4)⋊SD16
C1C2C22C23C22×C4C2×C4⋊C4C23.41C23 — (C2×C4)⋊SD16
C1C22C22×C4 — (C2×C4)⋊SD16
C1C22C22×C4 — (C2×C4)⋊SD16
C1C2C22C22×C4 — (C2×C4)⋊SD16

Generators and relations for (C2×C4)⋊SD16
 G = < a,b,c,d | a2=b4=c8=d2=1, dbd=ab=ba, cac-1=ab2, ad=da, cbc-1=ab-1, dcd=c3 >

Subgroups: 356 in 133 conjugacy classes, 34 normal (18 characteristic)
C1, C2 [×3], C2 [×4], C4 [×10], C22 [×3], C22 [×12], C8 [×2], C2×C4 [×4], C2×C4 [×12], D4 [×6], Q8 [×3], C23, C23 [×8], C42 [×2], C22⋊C4 [×6], C4⋊C4 [×2], C4⋊C4 [×9], C2×C8 [×2], SD16 [×4], C22×C4 [×3], C22×C4, C2×D4 [×4], C2×Q8 [×2], C2×Q8, C24, C2.C42, C22⋊C8 [×2], D4⋊C4 [×2], C2×C22⋊C4 [×2], C2×C4⋊C4, C42⋊C2, C22⋊Q8 [×2], C22⋊Q8, C42.C2 [×2], C4⋊Q8 [×2], C2×SD16 [×2], C22×D4, C22.M4(2), C23.31D4 [×2], C23.10D4, C22⋊SD16 [×2], C23.41C23, (C2×C4)⋊SD16
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, SD16 [×2], C2×D4 [×3], C22≀C2, C2×SD16, C8.C22, Q8⋊D4, D4.8D4, C2≀C22, (C2×C4)⋊SD16

Character table of (C2×C4)⋊SD16

 class 12A2B2C2D2E2F2G4A4B4C4D4E4F4G4H4I4J4K8A8B8C8D
 size 11112288444488888888888
ρ111111111111111111111111    trivial
ρ2111111-1-11111-1-1-11-1-1-11111    linear of order 2
ρ3111111-1-1-11-1111-1-11-111-11-1    linear of order 2
ρ411111111-11-11-1-11-1-11-11-11-1    linear of order 2
ρ511111111-11-11-11-1-11-1-1-11-11    linear of order 2
ρ6111111-1-1-11-111-11-1-111-11-11    linear of order 2
ρ7111111-1-11111-111111-1-1-1-1-1    linear of order 2
ρ81111111111111-1-11-1-11-1-1-1-1    linear of order 2
ρ9222222002-22-2000-20000000    orthogonal lifted from D4
ρ102222-2-2000-2020200-2000000    orthogonal lifted from D4
ρ112222-2-200020-200-200200000    orthogonal lifted from D4
ρ1222222200-2-2-2-200020000000    orthogonal lifted from D4
ρ132222-2-2000-2020-2002000000    orthogonal lifted from D4
ρ142222-2-200020-200200-200000    orthogonal lifted from D4
ρ1522-2-2-2200-20200000000--2-2-2--2    complex lifted from SD16
ρ1622-2-2-220020-200000000--2--2-2-2    complex lifted from SD16
ρ1722-2-2-220020-200000000-2-2--2--2    complex lifted from SD16
ρ1822-2-2-2200-20200000000-2--2--2-2    complex lifted from SD16
ρ194-44-4002-2000000000000000    orthogonal lifted from C2≀C22
ρ204-44-400-22000000000000000    orthogonal lifted from C2≀C22
ρ2144-4-44-400000000000000000    symplectic lifted from C8.C22, Schur index 2
ρ224-4-4400000000-2i000002i0000    complex lifted from D4.8D4
ρ234-4-44000000002i00000-2i0000    complex lifted from D4.8D4

Smallest permutation representation of (C2×C4)⋊SD16
On 32 points
Generators in S32
(1 31)(3 25)(5 27)(7 29)(9 17)(11 19)(13 21)(15 23)
(1 19 31 11)(2 20 32 12)(3 13 25 21)(4 14 26 22)(5 23 27 15)(6 24 28 16)(7 9 29 17)(8 10 30 18)
(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 15)(2 10)(3 13)(4 16)(5 11)(6 14)(7 9)(8 12)(17 29)(18 32)(19 27)(20 30)(21 25)(22 28)(23 31)(24 26)

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

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

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

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

100000
010000
0016000
0001600
000010
000001
,
1600000
0160000
001200
00161600
000012
00001616
,
8100000
020000
00001615
000001
0013900
004400
,
100000
13160000
00161500
000100
000048
00001313

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],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,16,0,0,0,0,2,16,0,0,0,0,0,0,1,16,0,0,0,0,2,16],[8,0,0,0,0,0,10,2,0,0,0,0,0,0,0,0,13,4,0,0,0,0,9,4,0,0,16,0,0,0,0,0,15,1,0,0],[1,13,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,15,1,0,0,0,0,0,0,4,13,0,0,0,0,8,13] >;

(C2×C4)⋊SD16 in GAP, Magma, Sage, TeX

(C_2\times C_4)\rtimes {\rm SD}_{16}
% in TeX

G:=Group("(C2xC4):SD16");
// GroupNames label

G:=SmallGroup(128,331);
// by ID

G=gap.SmallGroup(128,331);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,141,232,422,352,1123,570,521,136,1411]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^4=c^8=d^2=1,d*b*d=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=c^3>;
// generators/relations

Export

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

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