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G = C2×C4⋊D4order 64 = 26

Direct product of C2 and C4⋊D4

direct product, p-group, metabelian, nilpotent (class 2), monomial

Aliases: C2×C4⋊D4, C234D4, C23.5C23, C24.12C22, C22.16C24, C43(C2×D4), (C2×C4)⋊10D4, C4⋊C49C22, (C23×C4)⋊5C2, C221(C2×D4), (C22×D4)⋊4C2, C2.5(C22×D4), (C2×D4)⋊10C22, (C2×C4).52C23, C22⋊C413C22, (C22×C4)⋊18C22, C22.29(C4○D4), (C2×C4⋊C4)⋊14C2, C2.5(C2×C4○D4), (C2×C22⋊C4)⋊9C2, SmallGroup(64,203)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C2×C4⋊D4
C1C2C22C23C24C23×C4 — C2×C4⋊D4
C1C22 — C2×C4⋊D4
C1C23 — C2×C4⋊D4
C1C22 — C2×C4⋊D4

Generators and relations for C2×C4⋊D4
 G = < a,b,c,d | a2=b4=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >

Subgroups: 353 in 213 conjugacy classes, 97 normal (13 characteristic)
C1, C2 [×3], C2 [×4], C2 [×8], C4 [×4], C4 [×6], C22, C22 [×10], C22 [×32], C2×C4 [×12], C2×C4 [×14], D4 [×24], C23, C23 [×10], C23 [×16], C22⋊C4 [×8], C4⋊C4 [×4], C22×C4 [×2], C22×C4 [×6], C22×C4 [×4], C2×D4 [×12], C2×D4 [×12], C24, C24 [×2], C2×C22⋊C4 [×2], C2×C4⋊C4, C4⋊D4 [×8], C23×C4, C22×D4, C22×D4 [×2], C2×C4⋊D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×8], C23 [×15], C2×D4 [×12], C4○D4 [×2], C24, C4⋊D4 [×4], C22×D4 [×2], C2×C4○D4, C2×C4⋊D4

Character table of C2×C4⋊D4

 class 12A2B2C2D2E2F2G2H2I2J2K2L2M2N2O4A4B4C4D4E4F4G4H4I4J4K4L
 size 1111111122224444222222224444
ρ11111111111111111111111111111    trivial
ρ2111111111111-1-1-1-111111111-1-1-1-1    linear of order 2
ρ311111111-1-1-1-111-1-11-111-1-11-111-1-1    linear of order 2
ρ411111111-1-1-1-1-1-1111-111-1-11-1-1-111    linear of order 2
ρ511-111-1-1-1-11-11-111-1-111-1-111-1-111-1    linear of order 2
ρ611-111-1-1-11-11-11-11-1-1-11-11-1111-11-1    linear of order 2
ρ711-111-1-1-1-11-111-1-11-111-1-111-11-1-11    linear of order 2
ρ811-111-1-1-11-11-1-11-11-1-11-11-111-11-11    linear of order 2
ρ911-111-1-1-11-11-11-1-1111-11-11-1-1-111-1    linear of order 2
ρ1011-111-1-1-1-11-11-11-111-1-111-1-111-11-1    linear of order 2
ρ1111-111-1-1-11-11-1-111-111-11-11-1-11-1-11    linear of order 2
ρ1211-111-1-1-1-11-111-11-11-1-111-1-11-11-11    linear of order 2
ρ1311111111-1-1-1-11111-11-1-111-11-1-1-1-1    linear of order 2
ρ14111111111111-1-111-1-1-1-1-1-1-1-111-1-1    linear of order 2
ρ1511111111-1-1-1-1-1-1-1-1-11-1-111-111111    linear of order 2
ρ1611111111111111-1-1-1-1-1-1-1-1-1-1-1-111    linear of order 2
ρ17222-2-22-2-222-2-20000000000000000    orthogonal lifted from D4
ρ1822-2-2-2-222-222-20000000000000000    orthogonal lifted from D4
ρ192-222-2-22-2000000000200-2-2020000    orthogonal lifted from D4
ρ202-222-2-22-2000000000-200220-20000    orthogonal lifted from D4
ρ212-2-22-22-220000000002002-20-20000    orthogonal lifted from D4
ρ222-2-22-22-22000000000-200-22020000    orthogonal lifted from D4
ρ2322-2-2-2-2222-2-220000000000000000    orthogonal lifted from D4
ρ24222-2-22-2-2-2-2220000000000000000    orthogonal lifted from D4
ρ252-22-22-2-22000000002i0-2i-2i002i00000    complex lifted from C4○D4
ρ262-22-22-2-2200000000-2i02i2i00-2i00000    complex lifted from C4○D4
ρ272-2-2-2222-200000000-2i0-2i2i002i00000    complex lifted from C4○D4
ρ282-2-2-2222-2000000002i02i-2i00-2i00000    complex lifted from C4○D4

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

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

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

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

C2×C4⋊D4 is a maximal subgroup of
C24.(C2×C4)  C24.54D4  C24.56D4  C24.60D4  C23.38D8  C24.74D4  C23.23D8  C24.76D4  M4(2)⋊20D4  C24.175C23  C232D8  C233SD16  C24.83D4  C24.84D4  C4213D4  C24.198C23  C23.215C24  C24.215C23  C24.217C23  C24.218C23  C23.259C24  C24.244C23  C23.308C24  C248D4  C24.249C23  C23.316C24  C24.254C23  C23.322C24  C23.324C24  C24.258C23  C24.259C23  C23.327C24  C23.328C24  C24.262C23  C24.263C23  C23.333C24  C24.565C23  C24.269C23  C23.344C24  C23.356C24  C23.364C24  C24.293C23  C23.390C24  C23.391C24  C23.400C24  C23.401C24  C23.404C24  C4217D4  C4218D4  C23.439C24  C4219D4  C4220D4  C23.443C24  C23.455C24  C24.331C23  C23.491C24  C24.360C23  C2410D4  C24.587C23  C4227D4  C4228D4  C23.524C24  C23.535C24  C24.592C23  C23.556C24  C24.377C23  C23.568C24  C23.569C24  C23.571C24  C23.572C24  C23.573C24  C24.384C23  C23.576C24  C23.578C24  C23.581C24  C24.389C23  C24.393C23  C24.395C23  C23.591C24  C24.406C23  C24.407C23  C23.603C24  C23.608C24  C23.611C24  C24.413C23  C24.459C23  C23.715C24  C23.716C24  C2413D4  C4246D4  C24.598C23  C4247D4  C24.105D4  M4(2)⋊14D4  C24.117D4  C233D8  C234SD16  C24.121D4  C24.125D4  C24.126D4  C24.127D4  C2×D42  C22.77C25  C22.83C25  C4⋊2+ 1+4  C22.94C25  C22.108C25  C22.123C25  C22.125C25  C22.126C25  C22.131C25
C2×C4⋊D4 is a maximal quotient of
C4215D4  C23.308C24  C248D4  C23.311C24  C24.95D4  C23.313C24  C24.249C23  C23.315C24  C23.316C24  C24.252C23  C23.324C24  C24.258C23  C24.259C23  C23.327C24  C23.328C24  C23.329C24  C24.299C23  C23.434C24  C4217D4  C42.165D4  C4218D4  C42.166D4  C4220D4  C23.443C24  C4221D4  C42.168D4  C42.169D4  C42.170D4  C42.171D4  C2410D4  C24.587C23  C4227D4  C4228D4  C42.186D4  C2413D4  C4246D4  C24.598C23  C4247D4  C42.440D4  C42.443D4  C42.211D4  C42.212D4  C42.444D4  C42.445D4  C42.446D4  C42.14C23  C42.15C23  C42.16C23  C42.17C23  C42.18C23  C42.19C23  C24.144D4  C24.110D4  M4(2)⋊14D4  M4(2)⋊15D4  (C2×C8)⋊11D4  (C2×C8)⋊12D4  C8.D4⋊C2  (C2×C8)⋊13D4  (C2×C8)⋊14D4  M4(2)⋊16D4  M4(2)⋊17D4  M4(2).10C23  M4(2).37D4  M4(2).38D4

Matrix representation of C2×C4⋊D4 in GL5(𝔽5)

40000
01000
00100
00040
00004
,
40000
03100
00200
00013
00014
,
10000
01200
04400
00040
00041
,
10000
01000
04400
00010
00014

G:=sub<GL(5,GF(5))| [4,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,4,0,0,0,0,0,4],[4,0,0,0,0,0,3,0,0,0,0,1,2,0,0,0,0,0,1,1,0,0,0,3,4],[1,0,0,0,0,0,1,4,0,0,0,2,4,0,0,0,0,0,4,4,0,0,0,0,1],[1,0,0,0,0,0,1,4,0,0,0,0,4,0,0,0,0,0,1,1,0,0,0,0,4] >;

C2×C4⋊D4 in GAP, Magma, Sage, TeX

C_2\times C_4\rtimes D_4
% in TeX

G:=Group("C2xC4:D4");
// GroupNames label

G:=SmallGroup(64,203);
// by ID

G=gap.SmallGroup(64,203);
# by ID

G:=PCGroup([6,-2,2,2,2,-2,2,217,103,650]);
// Polycyclic

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

Export

Character table of C2×C4⋊D4 in TeX

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