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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(C4×D4), C4⋊1D4⋊15C4, C42⋊4C4⋊7C2, C4.4D4⋊17C4, C4.51(C4⋊1D4), C42.157(C2×C4), (C22×C4).299D4, C23.570(C2×D4), C4⋊M4(2)⋊27C2, C4.91(C4.4D4), C22.17(C4⋊D4), (C2×C42).318C22, (C22×C4).1407C23, C2.43(C42⋊C22), (C2×M4(2)).207C22, C22.26C24.21C2, C2.15(C24.3C22), (C2×C4≀C2)⋊19C2, (C2×C4).741(C2×D4), (C2×Q8).93(C2×C4), (C2×D4).108(C2×C4), (C2×C4).598(C4○D4), (C2×C4).421(C22×C4), (C2×C4○D4).38C22, (C2×C4).139(C22⋊C4), C22.285(C2×C22⋊C4), SmallGroup(128,695)

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

Derived series
C1 — C2×C4 — C42⋊8D4
Chief series
C1 — C2 — C4 — C2×C4 — C22×C4 — C2×C42 — C42⋊4C4 — C42⋊8D4
Lower central
C1 — C2 — C2×C4 — C42⋊8D4
Upper central
C1 — C2×C4 — C2×C42 — C42⋊8D4
Jennings
C1 — C2 — C2 — C22×C4 — 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, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C42, C42, C22⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C2.C42, C4≀C2, C4⋊C8, C2×C42, C2×C42, C4×D4, C4⋊D4, C4.4D4, C4⋊1D4, C4⋊Q8, C2×M4(2), C2×C4○D4, C42⋊4C4, C2×C4≀C2, C4⋊M4(2), C22.26C24, C42⋊8D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C4○D4, C2×C22⋊C4, C4×D4, 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+++++++
imageC1C2C2C2C2C4C4C4D4D4C4○D4C42⋊C22
kernelC42⋊8D4C42⋊4C4C2×C4≀C2C4⋊M4(2)C22.26C24C4.4D4C4⋊1D4C4⋊Q8C42C22×C4C2×C4C2
# 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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