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G = C42⋊7D4  order 128 = 27

1st semidirect product of C42 and D4 acting via D4/C2=C22

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

Aliases: C42⋊7D4, C24.77D4, C4⋊D4⋊15C4, C4.117(C4×D4), C22⋊Q8⋊15C4, C42⋊6C4⋊26C2, C23.567(C2×D4), (C22×C4).292D4, C4.189(C4⋊D4), C24.4C4⋊28C2, C22.28C22≀C2, C23.83(C22⋊C4), C22.19C24.8C2, (C23×C4).261C22, (C2×C42).288C22, (C22×C4).1373C23, C42⋊C2.26C22, C2.42(C42⋊C22), C2.36(C23.23D4), (C2×M4(2)).186C22, C22.26(C22.D4), (C2×C4≀C2)⋊16C2, C4⋊C4.75(C2×C4), (C2×D4).83(C2×C4), (C2×Q8).71(C2×C4), (C2×C42⋊C2)⋊2C2, (C2×C4).1335(C2×D4), (C2×C4).570(C4○D4), (C2×C4).391(C22×C4), (C22×C4).282(C2×C4), (C2×C4○D4).22C22, (C2×C4).134(C22⋊C4), C22.272(C2×C22⋊C4), SmallGroup(128,629)

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

Derived series
C1 — C2×C4 — C42⋊7D4
Chief series
C1 — C2 — C4 — C2×C4 — C22×C4 — C23×C4 — C2×C42⋊C2 — C42⋊7D4
Lower central
C1 — C2 — C2×C4 — C42⋊7D4
Upper central
C1 — C2×C4 — C23×C4 — C42⋊7D4
Jennings
C1 — C2 — C2 — C22×C4 — C42⋊7D4

Generators and relations for C42⋊7D4
 G = < a,b,c,d | a4=b4=c4=d2=1, ab=ba, cac-1=a-1b, dad=a-1b-1, bc=cb, bd=db, dcd=c-1 >

Subgroups: 372 in 179 conjugacy classes, 56 normal (20 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C23, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C24, C22⋊C8, C4≀C2, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C42⋊C2, C4×D4, C22≀C2, C4⋊D4, C22⋊Q8, C22.D4, C2×M4(2), C23×C4, C2×C4○D4, C42⋊6C4, C24.4C4, C2×C4≀C2, C2×C42⋊C2, C22.19C24, C42⋊7D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C4○D4, C2×C22⋊C4, C4×D4, C22≀C2, C4⋊D4, C22.D4, C23.23D4, C42⋊C22, C42⋊7D4

Smallest permutation representation of C42⋊7D4
►On 32 points
Generators in S32
(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 3 9 12)(2 4 10 11)(5 8 13 15)(6 7 14 16)(17 31 19 29)(18 32 20 30)(21 28 23 26)(22 25 24 27)

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

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

32 conjugacy classes

class 1 2A2B2C2D2E2F2G2H4A4B4C4D4E4F4G···4P4Q4R4S8A8B8C8D
order1222222224444444···44448888
size1111224481111224···48888888

32 irreducible representations

dim1111111122224
type+++++++++
imageC1C2C2C2C2C2C4C4D4D4D4C4○D4C42⋊C22
kernelC42⋊7D4C42⋊6C4C24.4C4C2×C4≀C2C2×C42⋊C2C22.19C24C4⋊D4C22⋊Q8C42C22×C4C24C2×C4C2
# reps1212114443144

Matrix representation of C42⋊7D4 ►in GL6(𝔽17)

400000
13130000
0001600
001000
0000013
000040
,
100000
010000
004000
000400
000040
000004
,
16150000
110000
000010
0000016
0016000
000100
,
120000
0160000
0000016
000010
000100
0016000

G:=sub<GL(6,GF(17))| [4,13,0,0,0,0,0,13,0,0,0,0,0,0,0,1,0,0,0,0,16,0,0,0,0,0,0,0,0,4,0,0,0,0,13,0],[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],[16,1,0,0,0,0,15,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,16,0,0],[1,0,0,0,0,0,2,16,0,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,1,0,0,0,0,16,0,0,0] >;
 

C42⋊7D4 in GAP, Magma, Sage, TeX 

C_4^2\rtimes_7D_4
 
% in TeX
 
G:=Group("C4^2:7D4");
 
// GroupNames label
 
G:=SmallGroup(128,629);
 
// by ID
 
G=gap.SmallGroup(128,629);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,288,422,723,2019,248,124]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^4=b^4=c^4=d^2=1,a*b=b*a,c*a*c^-1=a^-1*b,d*a*d=a^-1*b^-1,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
 
// generators/relations
 

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