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G = C4.4D4⋊13C4  order 128 = 27

7th semidirect product of C4.4D4 and C4 acting via C4/C2=C2

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

Aliases: C4.89(C4×D4), C4⋊C4.318D4, (C2×Q8).71D4, C4.4D4⋊13C4, C4.8(C4⋊D4), (C22×C4).65D4, C42⋊6C4⋊24C2, C42.149(C2×C4), C23.563(C2×D4), C2.5(D4.8D4), C22.102C22≀C2, (C22×C4).686C23, (C2×C42).287C22, C23.37D4.4C2, (C22×D4).21C22, (C22×Q8).17C22, C23.32C23⋊1C2, C42⋊C2.23C22, C2.27(C23.23D4), (C2×M4(2)).183C22, C22.51(C22.D4), (C2×D4).77(C2×C4), (C2×Q8).68(C2×C4), (C2×C4).58(C4○D4), (C2×C4).1007(C2×D4), (C2×C4.4D4).6C2, (C2×C4.10D4)⋊16C2, (C2×C4).14(C22⋊C4), (C2×C4).188(C22×C4), C22.43(C2×C22⋊C4), SmallGroup(128,620)

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

Derived series
C1 — C2×C4 — C4.4D4⋊13C4
Chief series
C1 — C2 — C4 — C2×C4 — C22×C4 — C22×Q8 — C23.32C23 — C4.4D4⋊13C4
Lower central
C1 — C2 — C2×C4 — C4.4D4⋊13C4
Upper central
C1 — C22 — C22×C4 — C4.4D4⋊13C4
Jennings
C1 — C2 — C2 — C22×C4 — C4.4D4⋊13C4

Generators and relations for C4.4D4⋊13C4
 G = < a,b,c,d | a4=b4=d4=1, c2=a2, ab=ba, cac-1=dad-1=a-1, cbc-1=a2b-1, dbd-1=a-1b, cd=dc >

Subgroups: 372 in 172 conjugacy classes, 56 normal (16 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, Q8, 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, C2×Q8, C24, C4.10D4, D4⋊C4, C2×C42, C2×C22⋊C4, C42⋊C2, C42⋊C2, C4×Q8, C4.4D4, C4.4D4, C2×M4(2), C22×D4, C22×Q8, C42⋊6C4, C2×C4.10D4, C23.37D4, C23.32C23, C2×C4.4D4, C4.4D4⋊13C4
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, D4.8D4, C4.4D4⋊13C4

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

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

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

32 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E···4T8A8B8C8D
order1222222244444···48888
size1111228822224···48888

32 irreducible representations

dim111111122224
type+++++++++
imageC1C2C2C2C2C2C4D4D4D4C4○D4D4.8D4
kernelC4.4D4⋊13C4C42⋊6C4C2×C4.10D4C23.37D4C23.32C23C2×C4.4D4C4.4D4C4⋊C4C22×C4C2×Q8C2×C4C2
# reps121211842244

Matrix representation of C4.4D4⋊13C4 ►in GL6(𝔽17)

1600000
0160000
0001600
001000
000001
0000160
,
1600000
010000
0001300
004000
000040
000004
,
100000
010000
0001300
0013000
0000013
0000130
,
0130000
1300000
000010
000001
0016000
0001600

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

C4.4D4⋊13C4 in GAP, Magma, Sage, TeX 

C_4._4D_4\rtimes_{13}C_4
 
% in TeX
 
G:=Group("C4.4D4:13C4");
 
// GroupNames label
 
G:=SmallGroup(128,620);
 
// by ID
 
G=gap.SmallGroup(128,620);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,736,422,352,1018,521,248,2804,1411,2028,1027,124]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^4=b^4=d^4=1,c^2=a^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1,c*b*c^-1=a^2*b^-1,d*b*d^-1=a^-1*b,c*d=d*c>;
 
// generators/relations
 

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