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G = C2×C22.D4order 64 = 26

Direct product of C2 and C22.D4

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

Aliases: C2×C22.D4, C23.49D4, C22.18C24, C23.35C23, C24.31C22, (C23×C4)⋊4C2, C4⋊C411C22, C2.7(C22×D4), (C2×C4).12C23, C22.18(C2×D4), C22⋊C414C22, (C22×C4)⋊17C22, (C22×D4).10C2, (C2×D4).60C22, C22.31(C4○D4), (C2×C4⋊C4)⋊16C2, C2.7(C2×C4○D4), (C2×C22⋊C4)⋊10C2, SmallGroup(64,205)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C2×C22.D4
C1C2C22C23C22×C4C23×C4 — C2×C22.D4
C1C22 — C2×C22.D4
C1C23 — C2×C22.D4
C1C22 — C2×C22.D4

Generators and relations for C2×C22.D4
 G = < a,b,c,d,e | a2=b2=c2=d4=e2=1, ab=ba, ac=ca, ad=da, ae=ea, dbd-1=ebe=bc=cb, cd=dc, ce=ec, ede=cd-1 >

Subgroups: 265 in 171 conjugacy classes, 89 normal (11 characteristic)
C1, C2, C2 [×6], C2 [×6], C4 [×10], C22, C22 [×10], C22 [×22], C2×C4 [×10], C2×C4 [×18], D4 [×8], C23, C23 [×8], C23 [×10], C22⋊C4 [×12], C4⋊C4 [×8], C22×C4, C22×C4 [×8], C22×C4 [×4], C2×D4 [×4], C2×D4 [×4], C24 [×2], C2×C22⋊C4, C2×C22⋊C4 [×2], C2×C4⋊C4 [×2], C22.D4 [×8], C23×C4, C22×D4, C2×C22.D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×4], C24, C22.D4 [×4], C22×D4, C2×C4○D4 [×2], C2×C22.D4

Character table of C2×C22.D4

 class 12A2B2C2D2E2F2G2H2I2J2K2L2M4A4B4C4D4E4F4G4H4I4J4K4L4M4N
 size 1111111122224422222222444444
ρ11111111111111111111111111111    trivial
ρ211111111-1-1-1-11111-1-1-1-111-1-1-1-111    linear of order 2
ρ3111-11-1-1-11-11-11-1-1-1-11-11111-11-1-11    linear of order 2
ρ4111-11-1-1-1-11-111-1-1-11-11-111-11-11-11    linear of order 2
ρ511111111-1-1-1-111-1-11111-1-111-1-1-1-1    linear of order 2
ρ611111111111111-1-1-1-1-1-1-1-1-1-111-1-1    linear of order 2
ρ7111-11-1-1-1-11-111-111-11-11-1-11-1-111-1    linear of order 2
ρ8111-11-1-1-11-11-11-1111-11-1-1-1-111-11-1    linear of order 2
ρ9111-11-1-1-1-11-11-1111-11-11-1-1-111-1-11    linear of order 2
ρ10111-11-1-1-11-11-1-11111-11-1-1-11-1-11-11    linear of order 2
ρ1111111111-1-1-1-1-1-1-1-11111-1-1-1-11111    linear of order 2
ρ12111111111111-1-1-1-1-1-1-1-1-1-111-1-111    linear of order 2
ρ13111-11-1-1-11-11-1-11-1-1-11-1111-11-111-1    linear of order 2
ρ14111-11-1-1-1-11-11-11-1-11-11-1111-11-11-1    linear of order 2
ρ15111111111111-1-111111111-1-1-1-1-1-1    linear of order 2
ρ1611111111-1-1-1-1-1-111-1-1-1-1111111-1-1    linear of order 2
ρ1722-22-2-22-22-2-220000000000000000    orthogonal lifted from D4
ρ1822-2-2-22-2222-2-20000000000000000    orthogonal lifted from D4
ρ1922-22-2-22-2-222-20000000000000000    orthogonal lifted from D4
ρ2022-2-2-22-22-2-2220000000000000000    orthogonal lifted from D4
ρ212-2-222-2-220000002i-2i00002i-2i000000    complex lifted from C4○D4
ρ222-2-2-2222-20000002i-2i0000-2i2i000000    complex lifted from C4○D4
ρ232-22-2-2-22200000000-2i2i2i-2i00000000    complex lifted from C4○D4
ρ242-222-22-2-200000000-2i-2i2i2i00000000    complex lifted from C4○D4
ρ252-2-2-2222-2000000-2i2i00002i-2i000000    complex lifted from C4○D4
ρ262-22-2-2-222000000002i-2i-2i2i00000000    complex lifted from C4○D4
ρ272-222-22-2-2000000002i2i-2i-2i00000000    complex lifted from C4○D4
ρ282-2-222-2-22000000-2i2i0000-2i2i000000    complex lifted from C4○D4

Smallest permutation representation of C2×C22.D4
On 32 points
Generators in S32
(1 7)(2 8)(3 5)(4 6)(9 32)(10 29)(11 30)(12 31)(13 20)(14 17)(15 18)(16 19)(21 27)(22 28)(23 25)(24 26)
(1 28)(2 11)(3 26)(4 9)(5 24)(6 32)(7 22)(8 30)(10 13)(12 15)(14 25)(16 27)(17 23)(18 31)(19 21)(20 29)
(1 13)(2 14)(3 15)(4 16)(5 18)(6 19)(7 20)(8 17)(9 27)(10 28)(11 25)(12 26)(21 32)(22 29)(23 30)(24 31)
(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 5)(2 17)(3 7)(4 19)(6 16)(8 14)(9 32)(10 24)(11 30)(12 22)(13 18)(15 20)(21 27)(23 25)(26 29)(28 31)

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

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

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

C2×C22.D4 is a maximal subgroup of
C23⋊C8⋊C2  C24.26D4  C24.174C23  C24.31D4  C24.195C23  C24.204C23  C23.241C24  C24.223C23  C24.225C23  C24.94D4  C24.243C23  C23.311C24  C24.95D4  C23.313C24  C23.318C24  C24.563C23  C23.322C24  C24.258C23  C24.262C23  C24.269C23  C23.345C24  C24.276C23  C23.356C24  C24.278C23  C24.279C23  C24.282C23  C23.364C24  C24.289C23  C24.290C23  C24.299C23  C23.388C24  C23.398C24  C23.401C24  C23.434C24  C23.439C24  C24.326C23  C24.327C23  C23.457C24  C23.458C24  C23.502C24  C249D4  C23.514C24  C24.587C23  C24.589C23  C23.530C24  C24.377C23  C24.378C23  C23.571C24  C23.572C24  C23.573C24  C23.574C24  C23.578C24  C23.580C24  C23.581C24  C24.389C23  C24.394C23  C23.593C24  C24.401C23  C23.595C24  C24.403C23  C23.597C24  C24.407C23  C23.603C24  C23.605C24  C23.606C24  C23.607C24  C23.608C24  C24.411C23  C24.412C23  C23.617C24  C23.618C24  C23.624C24  C24.459C23  C23.714C24  C24.166D4  C23.753C24  C24.598C23  C22.74C25  C22.80C25  C22.102C25  C22.122C25  C22.123C25  C22.124C25
C2×C22.D4 is a maximal quotient of
C23.295C24  C24.94D4  C24.95D4  C23.318C24  C24.563C23  C24.254C23  C23.321C24  C23.322C24  C23.323C24  C24.269C23  C23.344C24  C23.345C24  C23.346C24  C24.271C23  C23.348C24  C23.382C24  C24.96D4  C24.576C23  C23.385C24  C24.299C23  C24.300C23  C23.398C24  C24.308C23  C23.400C24  C23.401C24  C23.402C24  C24.579C23  C23.404C24  C24.97D4  C24.589C23  C23.524C24  C23.525C24  C24.166D4  C23.753C24  C24.598C23  C24.599C23  C24.115D4  C24.183D4  C24.116D4  C24.117D4  C24.118D4  (C2×D4).301D4  (C2×D4).302D4  (C2×D4).303D4  (C2×D4).304D4

Matrix representation of C2×C22.D4 in GL6(𝔽5)

400000
040000
001000
000100
000010
000001
,
310000
220000
000200
003000
000010
000001
,
400000
040000
004000
000400
000010
000001
,
120000
040000
000400
004000
000004
000010
,
400000
110000
001000
000400
000040
000001

G:=sub<GL(6,GF(5))| [4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[3,2,0,0,0,0,1,2,0,0,0,0,0,0,0,3,0,0,0,0,2,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,2,4,0,0,0,0,0,0,0,4,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,4,0],[4,1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,1] >;

C2×C22.D4 in GAP, Magma, Sage, TeX

C_2\times C_2^2.D_4
% in TeX

G:=Group("C2xC2^2.D4");
// GroupNames label

G:=SmallGroup(64,205);
// by ID

G=gap.SmallGroup(64,205);
# by ID

G:=PCGroup([6,-2,2,2,2,-2,2,217,650,86]);
// Polycyclic

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

Export

Character table of C2×C22.D4 in TeX

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