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G = C24.78D4order 128 = 27

33rd non-split extension by C24 of D4 acting via D4/C2=C22

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

Aliases: C24.78D4, C25.10C22, C24.173C23, C241(C2×C4), C22⋊C430D4, (C22×D4)⋊7C4, C243C43C2, C22.49(C4×D4), C221(C23⋊C4), C23.568(C2×D4), C23.9D48C2, C22.29C22≀C2, C23.120(C4○D4), C22.47(C4⋊D4), C23.192(C22×C4), (C22×D4).26C22, C23.124(C22⋊C4), C2.37(C23.23D4), C22.27(C22.D4), (C2×C22⋊C4)⋊4C4, (C2×C23⋊C4)⋊5C2, (C22×C4)⋊1(C2×C4), C2.27(C2×C23⋊C4), (C2×C22≀C2).2C2, (C22×C22⋊C4)⋊2C2, (C2×C22⋊C4).12C22, C22.273(C2×C22⋊C4), SmallGroup(128,630)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C24.78D4
C1C2C22C23C24C25C22×C22⋊C4 — C24.78D4
C1C2C23 — C24.78D4
C1C22C25 — C24.78D4
C1C2C24 — C24.78D4

Generators and relations for C24.78D4
 G = < a,b,c,d,e,f | a2=b2=c2=d2=e4=1, f2=d, faf-1=ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, bf=fb, ece-1=fcf-1=cd=dc, de=ed, df=fd, fef-1=cde-1 >

Subgroups: 788 in 299 conjugacy classes, 60 normal (20 characteristic)
C1, C2, C2 [×2], C2 [×13], C4 [×9], C22 [×3], C22 [×8], C22 [×53], C2×C4 [×29], D4 [×12], C23 [×3], C23 [×8], C23 [×53], C22⋊C4 [×4], C22⋊C4 [×20], C22×C4, C22×C4 [×2], C22×C4 [×12], C2×D4 [×14], C24 [×4], C24 [×4], C24 [×8], C23⋊C4 [×4], C2×C22⋊C4, C2×C22⋊C4 [×6], C2×C22⋊C4 [×7], C22≀C2 [×4], C23×C4, C22×D4, C22×D4 [×2], C25, C23.9D4 [×2], C243C4, C2×C23⋊C4 [×2], C22×C22⋊C4, C2×C22≀C2, C24.78D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×8], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×4], C4○D4 [×2], C23⋊C4 [×4], C2×C22⋊C4, C4×D4 [×2], C22≀C2, C4⋊D4 [×2], C22.D4, C23.23D4, C2×C23⋊C4 [×2], C24.78D4

Permutation representations of C24.78D4
On 16 points - transitive group 16T230
Generators in S16
(9 14)(10 15)(11 16)(12 13)
(1 3)(2 4)(5 7)(6 8)(9 14)(10 15)(11 16)(12 13)
(1 7)(2 8)(3 5)(4 6)(9 12)(10 11)(13 14)(15 16)
(1 2)(3 4)(5 6)(7 8)(9 11)(10 12)(13 15)(14 16)
(5 6)(7 8)(9 10 11 12)(13 14 15 16)
(1 14 2 16)(3 9 4 11)(5 10 6 12)(7 15 8 13)

G:=sub<Sym(16)| (9,14)(10,15)(11,16)(12,13), (1,3)(2,4)(5,7)(6,8)(9,14)(10,15)(11,16)(12,13), (1,7)(2,8)(3,5)(4,6)(9,12)(10,11)(13,14)(15,16), (1,2)(3,4)(5,6)(7,8)(9,11)(10,12)(13,15)(14,16), (5,6)(7,8)(9,10,11,12)(13,14,15,16), (1,14,2,16)(3,9,4,11)(5,10,6,12)(7,15,8,13)>;

G:=Group( (9,14)(10,15)(11,16)(12,13), (1,3)(2,4)(5,7)(6,8)(9,14)(10,15)(11,16)(12,13), (1,7)(2,8)(3,5)(4,6)(9,12)(10,11)(13,14)(15,16), (1,2)(3,4)(5,6)(7,8)(9,11)(10,12)(13,15)(14,16), (5,6)(7,8)(9,10,11,12)(13,14,15,16), (1,14,2,16)(3,9,4,11)(5,10,6,12)(7,15,8,13) );

G=PermutationGroup([(9,14),(10,15),(11,16),(12,13)], [(1,3),(2,4),(5,7),(6,8),(9,14),(10,15),(11,16),(12,13)], [(1,7),(2,8),(3,5),(4,6),(9,12),(10,11),(13,14),(15,16)], [(1,2),(3,4),(5,6),(7,8),(9,11),(10,12),(13,15),(14,16)], [(5,6),(7,8),(9,10,11,12),(13,14,15,16)], [(1,14,2,16),(3,9,4,11),(5,10,6,12),(7,15,8,13)])

G:=TransitiveGroup(16,230);

On 16 points - transitive group 16T237
Generators in S16
(5 15)(6 16)(7 13)(8 14)
(1 3)(2 4)(5 15)(6 16)(7 13)(8 14)(9 11)(10 12)
(1 3)(2 10)(4 12)(5 7)(6 16)(8 14)(9 11)(13 15)
(1 11)(2 12)(3 9)(4 10)(5 13)(6 14)(7 15)(8 16)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(1 13 11 5)(2 6 12 14)(3 7 9 15)(4 16 10 8)

G:=sub<Sym(16)| (5,15)(6,16)(7,13)(8,14), (1,3)(2,4)(5,15)(6,16)(7,13)(8,14)(9,11)(10,12), (1,3)(2,10)(4,12)(5,7)(6,16)(8,14)(9,11)(13,15), (1,11)(2,12)(3,9)(4,10)(5,13)(6,14)(7,15)(8,16), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,13,11,5)(2,6,12,14)(3,7,9,15)(4,16,10,8)>;

G:=Group( (5,15)(6,16)(7,13)(8,14), (1,3)(2,4)(5,15)(6,16)(7,13)(8,14)(9,11)(10,12), (1,3)(2,10)(4,12)(5,7)(6,16)(8,14)(9,11)(13,15), (1,11)(2,12)(3,9)(4,10)(5,13)(6,14)(7,15)(8,16), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,13,11,5)(2,6,12,14)(3,7,9,15)(4,16,10,8) );

G=PermutationGroup([(5,15),(6,16),(7,13),(8,14)], [(1,3),(2,4),(5,15),(6,16),(7,13),(8,14),(9,11),(10,12)], [(1,3),(2,10),(4,12),(5,7),(6,16),(8,14),(9,11),(13,15)], [(1,11),(2,12),(3,9),(4,10),(5,13),(6,14),(7,15),(8,16)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16)], [(1,13,11,5),(2,6,12,14),(3,7,9,15),(4,16,10,8)])

G:=TransitiveGroup(16,237);

On 16 points - transitive group 16T249
Generators in S16
(1 4)(2 3)(5 8)(6 7)
(1 4)(2 3)(5 8)(6 7)(9 14)(10 15)(11 16)(12 13)
(2 6)(3 7)(9 11)(14 16)
(1 5)(2 6)(3 7)(4 8)(9 11)(10 12)(13 15)(14 16)
(1 2)(3 4)(5 6)(7 8)(9 10 11 12)(13 14 15 16)
(1 14 5 16)(2 13 6 15)(3 12 7 10)(4 9 8 11)

G:=sub<Sym(16)| (1,4)(2,3)(5,8)(6,7), (1,4)(2,3)(5,8)(6,7)(9,14)(10,15)(11,16)(12,13), (2,6)(3,7)(9,11)(14,16), (1,5)(2,6)(3,7)(4,8)(9,11)(10,12)(13,15)(14,16), (1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14,15,16), (1,14,5,16)(2,13,6,15)(3,12,7,10)(4,9,8,11)>;

G:=Group( (1,4)(2,3)(5,8)(6,7), (1,4)(2,3)(5,8)(6,7)(9,14)(10,15)(11,16)(12,13), (2,6)(3,7)(9,11)(14,16), (1,5)(2,6)(3,7)(4,8)(9,11)(10,12)(13,15)(14,16), (1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14,15,16), (1,14,5,16)(2,13,6,15)(3,12,7,10)(4,9,8,11) );

G=PermutationGroup([(1,4),(2,3),(5,8),(6,7)], [(1,4),(2,3),(5,8),(6,7),(9,14),(10,15),(11,16),(12,13)], [(2,6),(3,7),(9,11),(14,16)], [(1,5),(2,6),(3,7),(4,8),(9,11),(10,12),(13,15),(14,16)], [(1,2),(3,4),(5,6),(7,8),(9,10,11,12),(13,14,15,16)], [(1,14,5,16),(2,13,6,15),(3,12,7,10),(4,9,8,11)])

G:=TransitiveGroup(16,249);

32 conjugacy classes

class 1 2A2B2C2D···2M2N2O2P4A···4H4I···4O
order12222···22224···44···4
size11112···24484···48···8

32 irreducible representations

dim111111112224
type+++++++++
imageC1C2C2C2C2C2C4C4D4D4C4○D4C23⋊C4
kernelC24.78D4C23.9D4C243C4C2×C23⋊C4C22×C22⋊C4C2×C22≀C2C2×C22⋊C4C22×D4C22⋊C4C24C23C22
# reps121211444444

Matrix representation of C24.78D4 in GL6(𝔽5)

400000
010000
004000
000400
000040
000004
,
400000
040000
001000
000100
000010
000001
,
400000
040000
000100
001000
000001
000010
,
100000
010000
004000
000400
000040
000004
,
200000
020000
000400
001000
000040
000001
,
010000
100000
000040
000001
001000
000400

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

C24.78D4 in GAP, Magma, Sage, TeX

C_2^4._{78}D_4
% in TeX

G:=Group("C2^4.78D4");
// GroupNames label

G:=SmallGroup(128,630);
// by ID

G=gap.SmallGroup(128,630);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,288,422,2019,1018,2028]);
// Polycyclic

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

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