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G = C24.97D4  order 128 = 27

52nd non-split extension by C24 of D4 acting via D4/C2=C22

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

Aliases: C24.97D4, C25.52C22, C23.522C24, C24.588C23, C22.2992+ 1+4, C24⋊3C4⋊20C2, C23.194(C2×D4), C23.8Q8⋊82C2, C23.240(C4○D4), C23.11D4⋊56C2, C23.23D4⋊68C2, C23.10D4⋊58C2, C2.24(C23⋊3D4), (C22×C4).132C23, (C23×C4).136C22, C22.347(C22×D4), C2.C42⋊30C22, (C22×D4).193C22, C2.37(C22.32C24), C22.37(C22.D4), (C2×C4⋊C4)⋊26C22, (C2×C22≀C2).12C2, (C22×C22⋊C4)⋊12C2, (C2×C22⋊C4)⋊24C22, C22.394(C2×C4○D4), C2.40(C2×C22.D4), SmallGroup(128,1354)

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

Derived series
C1 — C23 — C24.97D4
Chief series
C1 — C2 — C22 — C23 — C24 — C22×D4 — C23.23D4 — C24.97D4
Lower central
C1 — C23 — C24.97D4
Upper central
C1 — C23 — C24.97D4
Jennings
C1 — C23 — C24.97D4

Generators and relations for C24.97D4
 G = < a,b,c,d,e,f | a2=b2=c2=d2=e4=1, f2=d, ab=ba, faf-1=ac=ca, ad=da, eae-1=acd, ebe-1=bc=cb, bd=db, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=de-1 >

Subgroups: 836 in 351 conjugacy classes, 100 normal (14 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, 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×C22⋊C4, C2×C4⋊C4, C22≀C2, C23×C4, C22×D4, C22×D4, C25, C24⋊3C4, C23.8Q8, C23.23D4, C23.10D4, C23.11D4, C22×C22⋊C4, C2×C22≀C2, C24.97D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C22.D4, C22×D4, C2×C4○D4, 2+ 1+4, C2×C22.D4, C23⋊3D4, C22.32C24, C24.97D4

Smallest permutation representation of C24.97D4
►On 32 points
Generators in S32
(2 30)(4 32)(5 18)(6 22)(7 20)(8 24)(10 28)(12 26)(13 23)(14 17)(15 21)(16 19)

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

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

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

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

32 conjugacy classes

class 1 2A···2G2H2I2J2K2L2M2N2O2P4A···4H4I···4O
order12···22222222224···44···4
size11···12222444484···48···8

32 irreducible representations

dim11111111224
type++++++++++
imageC1C2C2C2C2C2C2C2D4C4○D42+ 1+4
kernelC24.97D4C24⋊3C4C23.8Q8C23.23D4C23.10D4C23.11D4C22×C22⋊C4C2×C22≀C2C24C23C22
# reps11224411484

Matrix representation of C24.97D4 ►in GL8(𝔽5)

10000000
01000000
00400000
00010000
00001000
00000100
00000140
00001004
,
40000000
04000000
00100000
00010000
00001000
00000400
00000410
00001004
,
10000000
01000000
00100000
00010000
00004000
00000400
00000040
00000004
,
10000000
01000000
00400000
00040000
00001000
00000100
00000010
00000001
,
01000000
40000000
00030000
00200000
00001003
00000130
00000040
00000004
,
40000000
01000000
00300000
00020000
00000130
00001003
00000004
00000040

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

C24.97D4 in GAP, Magma, Sage, TeX 

C_2^4._{97}D_4
 
% in TeX
 
G:=Group("C2^4.97D4");
 
// GroupNames label
 
G:=SmallGroup(128,1354);
 
// by ID
 
G=gap.SmallGroup(128,1354);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,253,232,758,723,185]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=e^4=1,f^2=d,a*b=b*a,f*a*f^-1=a*c=c*a,a*d=d*a,e*a*e^-1=a*c*d,e*b*e^-1=b*c=c*b,b*d=d*b,b*f=f*b,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=d*e^-1>;
 
// generators/relations
 

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