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G = C2×D4⋊C4order 64 = 26

Direct product of C2 and D4⋊C4

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

Aliases: C2×D4⋊C4, C22.12D8, C23.54D4, C22.10SD16, D43(C2×C4), (C2×D4)⋊7C4, C2.1(C2×D8), C4⋊C47C22, (C22×C8)⋊3C2, C4.47(C2×D4), (C2×C4).69D4, (C2×C8)⋊11C22, C4.1(C22×C4), C2.1(C2×SD16), (C2×C4).59C23, (C22×D4).6C2, C22.41(C2×D4), C4.12(C22⋊C4), (C2×D4).46C22, C22.32(C22⋊C4), (C22×C4).107C22, (C2×C4⋊C4)⋊9C2, (C2×C4).43(C2×C4), C2.17(C2×C22⋊C4), SmallGroup(64,95)

Series: Derived Chief Lower central Upper central Jennings

C1C4 — C2×D4⋊C4
C1C2C22C2×C4C22×C4C22×D4 — C2×D4⋊C4
C1C2C4 — C2×D4⋊C4
C1C23C22×C4 — C2×D4⋊C4
C1C2C2C2×C4 — C2×D4⋊C4

Generators and relations for C2×D4⋊C4
 G = < a,b,c,d | a2=b4=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=dbd-1=b-1, dcd-1=bc >

Subgroups: 201 in 101 conjugacy classes, 49 normal (15 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×2], C4 [×2], C4 [×2], C22, C22 [×6], C22 [×16], C8 [×2], C2×C4 [×2], C2×C4 [×4], C2×C4 [×4], D4 [×4], D4 [×6], C23, C23 [×10], C4⋊C4 [×2], C4⋊C4, C2×C8 [×2], C2×C8 [×2], C22×C4, C22×C4, C2×D4 [×6], C2×D4 [×3], C24, D4⋊C4 [×4], C2×C4⋊C4, C22×C8, C22×D4, C2×D4⋊C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], D8 [×2], SD16 [×2], C22×C4, C2×D4 [×2], D4⋊C4 [×4], C2×C22⋊C4, C2×D8, C2×SD16, C2×D4⋊C4

Character table of C2×D4⋊C4

 class 12A2B2C2D2E2F2G2H2I2J2K4A4B4C4D4E4F4G4H8A8B8C8D8E8F8G8H
 size 1111111144442222444422222222
ρ11111111111111111111111111111    trivial
ρ211111111-1-1-1-11111-1-1-1-111111111    linear of order 2
ρ31-11-1-11-11-1-1111-1-111-11-1-11-11-11-11    linear of order 2
ρ41-11-1-11-1111-1-11-1-11-11-11-11-11-11-11    linear of order 2
ρ511111111-1-1-1-111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ61-11-1-11-11-1-1111-1-11-11-111-11-11-11-1    linear of order 2
ρ71111111111111111-1-1-1-1-1-1-1-1-1-1-1-1    linear of order 2
ρ81-11-1-11-1111-1-11-1-111-11-11-11-11-11-1    linear of order 2
ρ911-11-11-1-11-1-11-11-11-i-iiii-ii-i-ii-ii    linear of order 4
ρ101-1-1-1111-11-11-1-1-111i-i-ii-i-i-i-iiiii    linear of order 4
ρ111-1-1-1111-11-11-1-1-111-iii-iiiii-i-i-i-i    linear of order 4
ρ1211-11-11-1-11-1-11-11-11ii-i-i-ii-iii-ii-i    linear of order 4
ρ131-1-1-1111-1-11-11-1-111-iii-i-i-i-i-iiiii    linear of order 4
ρ1411-11-11-1-1-111-1-11-11ii-i-ii-ii-i-ii-ii    linear of order 4
ρ1511-11-11-1-1-111-1-11-11-i-iii-ii-iii-ii-i    linear of order 4
ρ161-1-1-1111-1-11-11-1-111i-i-iiiiii-i-i-i-i    linear of order 4
ρ1722-22-22-2-200002-22-2000000000000    orthogonal lifted from D4
ρ182-22-2-22-220000-222-2000000000000    orthogonal lifted from D4
ρ19222222220000-2-2-2-2000000000000    orthogonal lifted from D4
ρ202-2-2-2222-2000022-2-2000000000000    orthogonal lifted from D4
ρ212-2222-2-2-2000000000000-2-222-2-222    orthogonal lifted from D8
ρ22222-2-2-22-2000000000000-222-2-222-2    orthogonal lifted from D8
ρ23222-2-2-22-20000000000002-2-222-2-22    orthogonal lifted from D8
ρ242-2222-2-2-200000000000022-2-222-2-2    orthogonal lifted from D8
ρ2522-2-22-2-22000000000000-2-2--2--2--2--2-2-2    complex lifted from SD16
ρ262-2-22-2-222000000000000--2-2-2--2-2--2--2-2    complex lifted from SD16
ρ272-2-22-2-222000000000000-2--2--2-2--2-2-2--2    complex lifted from SD16
ρ2822-2-22-2-22000000000000--2--2-2-2-2-2--2--2    complex lifted from SD16

Smallest permutation representation of C2×D4⋊C4
On 32 points
Generators in S32
(1 20)(2 17)(3 18)(4 19)(5 24)(6 21)(7 22)(8 23)(9 16)(10 13)(11 14)(12 15)(25 31)(26 32)(27 29)(28 30)
(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 4)(2 3)(5 7)(9 12)(10 11)(13 14)(15 16)(17 18)(19 20)(22 24)(26 28)(30 32)
(1 7 11 32)(2 6 12 31)(3 5 9 30)(4 8 10 29)(13 27 19 23)(14 26 20 22)(15 25 17 21)(16 28 18 24)

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

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

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

C2×D4⋊C4 is a maximal subgroup of
D4⋊C42  C23.35D8  C24.65D4  C42.98D4  C42.100D4  C2.(C4×D8)  D4⋊(C4⋊C4)  C23.38D8  C24.74D4  (C2×SD16)⋊14C4  (C2×C4)⋊9D8  (C2×SD16)⋊15C4  M4(2).48D4  D4⋊C4⋊C4  C4.67(C4×D4)  C4.D43C4  C42.433D4  C42.118D4  C42.119D4  C233SD16  (C2×C8).41D4  (C2×D4)⋊Q8  C4⋊C4.84D4  C4⋊C47D4  C4⋊C4.94D4  M4(2).10D4  (C2×C4)⋊5SD16  M4(2).12D4  (C2×C4).24D8  C428C4⋊C2  (C2×C4).27D8  2+ 1+45C4  C2×C4×D8  C2×C4×SD16  C42.275C23  (C2×D4)⋊21D4  C42.18C23  M4(2)⋊16D4  C42.20C23  (C2×D4).301D4  C42.366C23  D44D8  D47SD16  C42.461C23  C42.462C23  C42.41C23  C42.45C23  C42.49C23  C42.53C23
 C2.(C8pD4): C23.23D8  C24.76D4  C2.(C87D4)  C2.(C82D4)  C42.432D4  C42.110D4  C42.112D4  (C2×C4)⋊9SD16 ...
C2×D4⋊C4 is a maximal quotient of
C42.45D4  D4⋊M4(2)  C42.315D4  C42.403D4  C42.61D4  C24.60D4  C42.409D4  C42.413D4  C42.414D4  C42.78D4  C23.35D8  C42.98D4  C23.36D8  C23.38D8  C23.23D8  C42.432D4  C42.118D4  C42.122D4  C42.436D4  C23.24D8  C23.39D8  C23.40D8  C23.41D8  C23.20SD16  C23.13D8  C23.21SD16

Matrix representation of C2×D4⋊C4 in GL4(𝔽17) generated by

16000
0100
0010
0001
,
1000
0100
0001
00160
,
16000
01600
0001
0010
,
4000
01300
00125
0055
G:=sub<GL(4,GF(17))| [16,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,0,16,0,0,1,0],[16,0,0,0,0,16,0,0,0,0,0,1,0,0,1,0],[4,0,0,0,0,13,0,0,0,0,12,5,0,0,5,5] >;

C2×D4⋊C4 in GAP, Magma, Sage, TeX

C_2\times D_4\rtimes C_4
% in TeX

G:=Group("C2xD4:C4");
// GroupNames label

G:=SmallGroup(64,95);
// by ID

G=gap.SmallGroup(64,95);
# by ID

G:=PCGroup([6,-2,2,2,-2,2,-2,96,121,963,489,117]);
// Polycyclic

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

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Character table of C2×D4⋊C4 in TeX

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