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G = C24.68D4  order 128 = 27

23rd non-split extension by C24 of D4 acting via D4/C2=C22

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

Aliases: C24.68D4, C25.8C22, C24.168C23, (C23×C4)⋊4C4, C24⋊3C4.3C2, C24.117(C2×C4), C23.549(C2×D4), C23.9D4⋊2C2, C23.113(C4○D4), C23.74(C22⋊C4), C22.29(C23⋊C4), C23.188(C22×C4), C2.7(C23.34D4), C22.28(C42⋊C2), C22.42(C22.D4), (C2×C22⋊C4)⋊15C4, C2.26(C2×C23⋊C4), (C22×C4).52(C2×C4), (C22×C22⋊C4).7C2, (C2×C22⋊C4).6C22, C22.253(C2×C22⋊C4), SmallGroup(128,551)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series
C1 — C23 — C24.68D4
Chief series
C1 — C2 — C22 — C23 — C24 — C2×C22⋊C4 — C22×C22⋊C4 — C24.68D4
Lower central
C1 — C2 — C23 — C24.68D4
Upper central
C1 — C22 — C25 — C24.68D4
Jennings
C1 — C2 — C24 — C24.68D4

Generators and relations for C24.68D4
 G = < a,b,c,d,e,f | a2=b2=c2=d2=e4=1, f2=c, eae-1=faf-1=ab=ba, ac=ca, ad=da, bc=cb, bd=db, be=eb, bf=fb, ece-1=cd=dc, cf=fc, de=ed, df=fd, fef-1=bce-1 >

Subgroups: 660 in 252 conjugacy classes, 56 normal (10 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C2×C4, C23, C23, C23, C22⋊C4, C22×C4, C22×C4, C24, C24, C24, C2×C22⋊C4, C2×C22⋊C4, C23×C4, C25, C23.9D4, C24⋊3C4, C22×C22⋊C4, C24.68D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C4○D4, C23⋊C4, C2×C22⋊C4, C42⋊C2, C22.D4, C23.34D4, C2×C23⋊C4, C24.68D4

Permutation representations of C24.68D4
►On 16 points - transitive group 16T274
Generators in S16
(2 4)(5 12)(7 10)(13 15)

(1 3)(2 4)(5 12)(6 9)(7 10)(8 11)(13 15)(14 16)

(2 15)(4 13)(6 11)(8 9)

(1 14)(2 15)(3 16)(4 13)(5 10)(6 11)(7 12)(8 9)

(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)

(1 10)(2 11 15 6)(3 7)(4 8 13 9)(5 14)(12 16)
 
G:=sub<Sym(16)| (2,4)(5,12)(7,10)(13,15), (1,3)(2,4)(5,12)(6,9)(7,10)(8,11)(13,15)(14,16), (2,15)(4,13)(6,11)(8,9), (1,14)(2,15)(3,16)(4,13)(5,10)(6,11)(7,12)(8,9), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,10)(2,11,15,6)(3,7)(4,8,13,9)(5,14)(12,16)>;
 
G:=Group( (2,4)(5,12)(7,10)(13,15), (1,3)(2,4)(5,12)(6,9)(7,10)(8,11)(13,15)(14,16), (2,15)(4,13)(6,11)(8,9), (1,14)(2,15)(3,16)(4,13)(5,10)(6,11)(7,12)(8,9), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,10)(2,11,15,6)(3,7)(4,8,13,9)(5,14)(12,16) );
 
G=PermutationGroup([[(2,4),(5,12),(7,10),(13,15)], [(1,3),(2,4),(5,12),(6,9),(7,10),(8,11),(13,15),(14,16)], [(2,15),(4,13),(6,11),(8,9)], [(1,14),(2,15),(3,16),(4,13),(5,10),(6,11),(7,12),(8,9)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16)], [(1,10),(2,11,15,6),(3,7),(4,8,13,9),(5,14),(12,16)]])
 
G:=TransitiveGroup(16,274);
 

32 conjugacy classes

class 1 2A2B2C2D···2M2N2O4A···4H4I···4P
order12222···2224···44···4
size11112···2444···48···8

32 irreducible representations

dim111111224
type++++++
imageC1C2C2C2C4C4D4C4○D4C23⋊C4
kernelC24.68D4C23.9D4C24⋊3C4C22×C22⋊C4C2×C22⋊C4C23×C4C24C23C22
# reps142144484

Matrix representation of C24.68D4 ►in GL6(𝔽5)

100000
040000
001000
000100
000040
000004
,
400000
040000
004000
000400
000040
000004
,
100000
010000
004000
000100
000040
000001
,
100000
010000
004000
000400
000040
000004
,
040000
100000
000001
000040
000100
001000
,
020000
300000
000010
000001
004000
000100

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

C24.68D4 in GAP, Magma, Sage, TeX 

C_2^4._{68}D_4
 
% in TeX
 
G:=Group("C2^4.68D4");
 
// GroupNames label
 
G:=SmallGroup(128,551);
 
// by ID
 
G=gap.SmallGroup(128,551);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,422,58,2804,1027]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=e^4=1,f^2=c,e*a*e^-1=f*a*f^-1=a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,e*c*e^-1=c*d=d*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=b*c*e^-1>;
 
// generators/relations
 

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