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G = C24⋊13D4  order 128 = 27

1st semidirect product of C24 and D4 acting via D4/C4=C2

p-group, metabelian, nilpotent (class 2), monomial

Aliases: C24⋊13D4, C25.89C22, C23.747C24, C24.118C23, C4⋊3C22≀C2, (C24×C4)⋊7C2, (C22×C4)⋊54D4, C22⋊3(C4⋊D4), C23.629(C2×D4), (C22×D4)⋊16C22, C23.247(C4○D4), (C23×C4).681C22, (C22×C4).257C23, C23.7Q8⋊115C2, C22.457(C22×D4), C23.23D4⋊111C2, C2.C42⋊45C22, C2.90(C22.19C24), (C2×C4⋊D4)⋊41C2, (C2×C4⋊C4)⋊40C22, C2.46(C2×C4⋊D4), (C2×C22≀C2)⋊18C2, C2.30(C2×C22≀C2), (C2×C4).1202(C2×D4), (C2×C22⋊C4)⋊34C22, C22.588(C2×C4○D4), SmallGroup(128,1579)

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

Derived series
C1 — C23 — C24⋊13D4
Chief series
C1 — C2 — C22 — C23 — C24 — C25 — C24×C4 — C24⋊13D4
Lower central
C1 — C23 — C24⋊13D4
Upper central
C1 — C23 — C24⋊13D4
Jennings
C1 — C23 — C24⋊13D4

Subgroups: 1188 in 612 conjugacy classes, 144 normal (9 characteristic)
C1, C2, C2 [×6], C2 [×14], C4 [×4], C4 [×12], C22, C22 [×18], C22 [×82], C2×C4 [×12], C2×C4 [×72], D4 [×28], C23, C23 [×18], C23 [×82], C22⋊C4 [×24], C4⋊C4 [×6], C22×C4, C22×C4 [×21], C22×C4 [×62], C2×D4 [×42], C24 [×9], C24 [×12], C2.C42 [×6], C2×C22⋊C4 [×12], C2×C4⋊C4 [×3], C22≀C2 [×8], C4⋊D4 [×12], C23×C4 [×6], C23×C4 [×12], C22×D4, C22×D4 [×6], C25, C23.7Q8 [×3], C23.23D4 [×6], C2×C22≀C2 [×2], C2×C4⋊D4 [×3], C24×C4, C24⋊13D4

Quotients:
C1, C2 [×15], C22 [×35], D4 [×16], C23 [×15], C2×D4 [×24], C4○D4 [×6], C24, C22≀C2 [×4], C4⋊D4 [×12], C22×D4 [×4], C2×C4○D4 [×3], C2×C22≀C2, C2×C4⋊D4 [×3], C22.19C24 [×3], C24⋊13D4

Generators and relations
 G = < a,b,c,d,e,f | a2=b2=c2=d2=e4=f2=1, ab=ba, faf=ac=ca, ad=da, ae=ea, bc=cb, fbf=bd=db, be=eb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef=e-1 >

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

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

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

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

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

Matrix representation ►G ⊆ GL6(𝔽5)

100000
040000
004000
000400
000040
000001
,
400000
010000
001000
000100
000040
000004
,
400000
040000
001000
000100
000040
000004
,
400000
040000
001000
000100
000010
000001
,
100000
010000
003000
000200
000010
000001
,
010000
100000
000200
003000
000001
000010

G:=sub<GL(6,GF(5))| [1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,1],[4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,3,0,0,0,0,0,0,2,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,3,0,0,0,0,2,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;
 

44 conjugacy classes

class 1 2A···2G2H···2S2T2U4A···4P4Q···4V
order12···22···2224···44···4
size11···12···2882···28···8

44 irreducible representations

dim111111222
type++++++++
imageC1C2C2C2C2C2D4D4C4○D4
kernelC24⋊13D4C23.7Q8C23.23D4C2×C22≀C2C2×C4⋊D4C24×C4C22×C4C24C23
# reps13623112412

In GAP, Magma, Sage, TeX 

C_2^4\rtimes_{13}D_4
 
% in TeX
 
G:=Group("C2^4:13D4");
 
// GroupNames label
 
G:=SmallGroup(128,1579);
 
// by ID
 
G=gap.SmallGroup(128,1579);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,253,120,758,2019]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=e^4=f^2=1,a*b=b*a,f*a*f=a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,f*b*f=b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f=e^-1>;
 
// generators/relations
 

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