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G = C24⋊10D4  order 128 = 27

5th semidirect product of C24 and D4 acting via D4/C2=C22

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

Aliases: C24⋊10D4, C25.51C22, C23.517C24, C24.586C23, C22.2952+ 1+4, (C22×C4)⋊33D4, C23⋊2D4⋊24C2, C23.190(C2×D4), (C22×D4)⋊9C22, C23.7Q8⋊76C2, C23.238(C4○D4), C23.23D4⋊66C2, C23.10D4⋊54C2, C23.11D4⋊55C2, C22.29(C4⋊D4), C2.22(C23⋊3D4), (C23×C4).420C22, (C22×C4).127C23, C22.342(C22×D4), C2.C42⋊29C22, C2.33(C22.29C24), C2.36(C22.32C24), (C2×C4⋊D4)⋊22C2, (C2×C4⋊C4)⋊25C22, (C2×C4).377(C2×D4), C2.41(C2×C4⋊D4), (C2×C22≀C2)⋊10C2, (C22×C22⋊C4)⋊25C2, (C2×C22⋊C4)⋊23C22, C22.389(C2×C4○D4), SmallGroup(128,1349)

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

Derived series
C1 — C23 — C24⋊10D4
Chief series
C1 — C2 — C22 — C23 — C24 — C22×D4 — C2×C22≀C2 — C24⋊10D4
Lower central
C1 — C23 — C24⋊10D4
Upper central
C1 — C23 — C24⋊10D4
Jennings
C1 — C23 — C24⋊10D4

Generators and relations for C24⋊10D4
 G = < a,b,c,d,e,f | a2=b2=c2=d2=e4=f2=1, ab=ba, eae-1=ac=ca, ad=da, faf=acd, fbf=bc=cb, bd=db, be=eb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef=e-1 >

Subgroups: 996 in 412 conjugacy classes, 108 normal (20 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C2×C4, C2×C4, D4, C23, C23, C23, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C24, C24, C24, C2.C42, C2×C22⋊C4, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C22≀C2, C4⋊D4, C23×C4, C22×D4, C22×D4, C25, C23.7Q8, C23.23D4, C23⋊2D4, C23.10D4, C23.11D4, C22×C22⋊C4, C2×C22≀C2, C2×C4⋊D4, C24⋊10D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C4⋊D4, C22×D4, C2×C4○D4, 2+ 1+4, C2×C4⋊D4, C23⋊3D4, C22.29C24, C22.32C24, C24⋊10D4

Smallest permutation representation of C24⋊10D4
►On 32 points
Generators in S32
(1 25)(2 14)(3 27)(4 16)(5 7)(6 12)(8 10)(9 11)(13 23)(15 21)(17 32)(18 20)(19 30)(22 28)(24 26)(29 31)

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

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

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

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

32 conjugacy classes

class 1 2A···2G2H2I2J2K2L2M2N2O2P2Q4A···4H4I···4N
order12···222222222224···44···4
size11···122224444884···48···8

32 irreducible representations

dim1111111112224
type++++++++++++
imageC1C2C2C2C2C2C2C2C2D4D4C4○D42+ 1+4
kernelC24⋊10D4C23.7Q8C23.23D4C23⋊2D4C23.10D4C23.11D4C22×C22⋊C4C2×C22≀C2C2×C4⋊D4C22×C4C24C23C22
# reps1122421214444

Matrix representation of C24⋊10D4 ►in GL8(𝔽5)

10000000
04000000
00400000
00040000
00001000
00000400
00000040
00000001
,
10000000
01000000
00400000
00040000
00004000
00000100
00000040
00000001
,
10000000
01000000
00100000
00010000
00004000
00000400
00000040
00000004
,
40000000
04000000
00100000
00010000
00004000
00000400
00000040
00000004
,
10000000
01000000
00300000
00020000
00000010
00000001
00004000
00000400
,
01000000
10000000
00020000
00300000
00000001
00000010
00000100
00001000

G:=sub<GL(8,GF(5))| [1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0] >;
 

C24⋊10D4 in GAP, Magma, Sage, TeX 

C_2^4\rtimes_{10}D_4
 
% in TeX
 
G:=Group("C2^4:10D4");
 
// GroupNames label
 
G:=SmallGroup(128,1349);
 
// by ID
 
G=gap.SmallGroup(128,1349);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,224,253,758,723,185]);
 
// 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,e*a*e^-1=a*c=c*a,a*d=d*a,f*a*f=a*c*d,f*b*f=b*c=c*b,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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