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G = C4.D4⋊3C4  order 128 = 27

2nd semidirect product of C4.D4 and C4 acting via C4/C2=C2

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

Aliases: C4.72(C4×D4), C4⋊C4.216D4, (C2×C8).330D4, C4.D4⋊3C4, C4.C42⋊7C2, M4(2).8(C2×C4), C8○2M4(2)⋊25C2, M4(2)⋊C4⋊4C2, C2.5(D4.3D4), C2.3(D4.4D4), C4.29(C4.4D4), C22.C42⋊19C2, C23.266(C4○D4), (C22×C8).396C22, (C22×C4).694C23, C23.37D4.5C2, (C22×D4).38C22, C22.129(C4⋊D4), C22.7(C42⋊2C2), C4.91(C22.D4), C42⋊C2.274C22, C22.10(C42⋊C2), (C2×M4(2)).199C22, C2.16(C24.C22), (C2×D4).97(C2×C4), (C2×C4).1339(C2×D4), (C2×C4⋊C4).69C22, (C2×C4).16(C22×C4), (C2×C4.D4).2C2, (C2×D4⋊C4).33C2, (C2×C4).335(C4○D4), SmallGroup(128,663)

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

Derived series
C1 — C2×C4 — C4.D4⋊3C4
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C2×M4(2) — C8○2M4(2) — C4.D4⋊3C4
Lower central
C1 — C2 — C2×C4 — C4.D4⋊3C4
Upper central
C1 — C22 — C22×C4 — C4.D4⋊3C4
Jennings
C1 — C2 — C2 — C22×C4 — C4.D4⋊3C4

Generators and relations for C4.D4⋊3C4
 G = < a,b,c,d | a4=d4=1, b4=a2, c2=a, bab-1=dad-1=a-1, ac=ca, cbc-1=a-1b3, dbd-1=b3, dcd-1=ab2c >

Subgroups: 276 in 116 conjugacy classes, 48 normal (38 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, D4, C23, C23, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), M4(2), C22×C4, C22×C4, C2×D4, C2×D4, C24, C4×C8, C8⋊C4, C4.D4, D4⋊C4, C4.Q8, C2.D8, C2×C4⋊C4, C42⋊C2, C22×C8, C2×M4(2), C22×D4, C4.C42, C22.C42, C8○2M4(2), C2×C4.D4, C2×D4⋊C4, C23.37D4, M4(2)⋊C4, C4.D4⋊3C4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22×C4, C2×D4, C4○D4, C42⋊C2, C4×D4, C4⋊D4, C22.D4, C4.4D4, C42⋊2C2, C24.C22, D4.3D4, D4.4D4, C4.D4⋊3C4

Smallest permutation representation of C4.D4⋊3C4
►On 32 points
Generators in S32
(1 17 5 21)(2 22 6 18)(3 19 7 23)(4 24 8 20)(9 25 13 29)(10 30 14 26)(11 27 15 31)(12 32 16 28)

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

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

32 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E4F4G4H4I4J8A8B8C8D8E···8J8K8L8M8N
order12222222444444444488888···88888
size11112288222244448822224···48888

32 irreducible representations

dim111111111222244
type+++++++++++
imageC1C2C2C2C2C2C2C2C4D4D4C4○D4C4○D4D4.3D4D4.4D4
kernelC4.D4⋊3C4C4.C42C22.C42C8○2M4(2)C2×C4.D4C2×D4⋊C4C23.37D4M4(2)⋊C4C4.D4C4⋊C4C2×C8C2×C4C23C2C2
# reps111111118226222

Matrix representation of C4.D4⋊3C4 ►in GL6(𝔽17)

1600000
0160000
000100
0016000
000001
0000160
,
1300000
1340000
000010
0000016
000100
001000
,
1300000
0130000
000010
000001
000100
0016000
,
1380000
040000
0014300
003300
0000143
000033

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0],[13,13,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,16,0,0],[13,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0],[13,0,0,0,0,0,8,4,0,0,0,0,0,0,14,3,0,0,0,0,3,3,0,0,0,0,0,0,14,3,0,0,0,0,3,3] >;
 

C4.D4⋊3C4 in GAP, Magma, Sage, TeX 

C_4.D_4\rtimes_3C_4
 
% in TeX
 
G:=Group("C4.D4:3C4");
 
// GroupNames label
 
G:=SmallGroup(128,663);
 
// by ID
 
G=gap.SmallGroup(128,663);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,736,422,58,2019,248,718]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^4=d^4=1,b^4=a^2,c^2=a,b*a*b^-1=d*a*d^-1=a^-1,a*c=c*a,c*b*c^-1=a^-1*b^3,d*b*d^-1=b^3,d*c*d^-1=a*b^2*c>;
 
// generators/relations
 

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