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G = C42:8D4  order 128 = 27

2nd semidirect product of C42 and D4 acting via D4/C2=C22

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

Aliases: C42:8D4, C4:Q8:19C4, C4.16(C4xD4), C4:1D4:15C4, C42:4C4:7C2, C4.4D4:17C4, C4.51(C4:1D4), C42.157(C2xC4), (C22xC4).299D4, C23.570(C2xD4), C4:M4(2):27C2, C4.91(C4.4D4), C22.17(C4:D4), (C2xC42).318C22, (C22xC4).1407C23, C2.43(C42:C22), (C2xM4(2)).207C22, C22.26C24.21C2, C2.15(C24.3C22), (C2xC4wrC2):19C2, (C2xC4).741(C2xD4), (C2xQ8).93(C2xC4), (C2xD4).108(C2xC4), (C2xC4).598(C4oD4), (C2xC4).421(C22xC4), (C2xC4oD4).38C22, (C2xC4).139(C22:C4), C22.285(C2xC22:C4), SmallGroup(128,695)

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

Derived series
C1 — C2xC4 — C42:8D4
Chief series
C1 — C2 — C4 — C2xC4 — C22xC4 — C2xC42 — C42:4C4 — C42:8D4
Lower central
C1 — C2 — C2xC4 — C42:8D4
Upper central
C1 — C2xC4 — C2xC42 — C42:8D4
Jennings
C1 — C2 — C2 — C22xC4 — C42:8D4

Generators and relations for C42:8D4
 G = < a,b,c,d | a4=b4=c4=d2=1, ab=ba, cac-1=ab2, dad=a-1b, bc=cb, bd=db, dcd=c-1 >

Subgroups: 340 in 160 conjugacy classes, 56 normal (18 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2xC4, C2xC4, C2xC4, D4, Q8, C23, C23, C42, C42, C42, C22:C4, C4:C4, C2xC8, M4(2), C22xC4, C22xC4, C22xC4, C2xD4, C2xD4, C2xQ8, C4oD4, C2.C42, C4wrC2, C4:C8, C2xC42, C2xC42, C4xD4, C4:D4, C4.4D4, C4:1D4, C4:Q8, C2xM4(2), C2xC4oD4, C42:4C4, C2xC4wrC2, C4:M4(2), C22.26C24, C42:8D4
Quotients: C1, C2, C4, C22, C2xC4, D4, C23, C22:C4, C22xC4, C2xD4, C4oD4, C2xC22:C4, C4xD4, C4:D4, C4.4D4, C4:1D4, C24.3C22, C42:C22, C42:8D4

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

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

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

32 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E4F4G···4R4S4T8A8B8C8D
order122222224444444···4448888
size111122881111224···4888888

32 irreducible representations

dim111111112224
type+++++++
imageC1C2C2C2C2C4C4C4D4D4C4oD4C42:C22
kernelC42:8D4C42:4C4C2xC4wrC2C4:M4(2)C22.26C24C4.4D4C4:1D4C4:Q8C42C22xC4C2xC4C2
# reps114114226244

Matrix representation of C42:8D4 ►in GL6(F17)

040000
1300000
000400
0013000
000001
0000160
,
1600000
0160000
004000
000400
000040
000004
,
0160000
100000
000100
001000
0000016
0000160
,
1140000
460000
0000130
000004
004000
0001300

G:=sub<GL(6,GF(17))| [0,13,0,0,0,0,4,0,0,0,0,0,0,0,0,13,0,0,0,0,4,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[0,1,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,16,0],[11,4,0,0,0,0,4,6,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,13,0,0,13,0,0,0,0,0,0,4,0,0] >;
 

C42:8D4 in GAP, Magma, Sage, TeX 

C_4^2\rtimes_8D_4
 
% in TeX
 
G:=Group("C4^2:8D4");
 
// GroupNames label
 
G:=SmallGroup(128,695);
 
// by ID
 
G=gap.SmallGroup(128,695);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,288,422,723,100,2019,248,124]);
 
// 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*b^2,d*a*d=a^-1*b,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
 
// generators/relations
 

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