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G = C2×C23.37D4  order 128 = 27

Direct product of C2 and C23.37D4

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

Aliases: C2×C23.37D4, C24.168D4, C4⋊C4⋊12C23, (C2×C8)⋊10C23, C4.4(C23×C4), (C22×D4)⋊27C4, (C2×C4).174C24, (C22×C8)⋊47C22, D4.18(C22×C4), (D4×C23).16C2, C4.139(C22×D4), (C22×C4).778D4, C23.637(C2×D4), D4⋊C4⋊83C22, (C2×D4).359C23, C42⋊C2⋊74C22, (C22×M4(2))⋊18C2, (C2×M4(2))⋊68C22, (C22×C4).898C23, (C23×C4).512C22, C22.124(C22×D4), C22.104(C8⋊C22), C23.207(C22⋊C4), (C22×D4).552C22, (C2×D4)⋊47(C2×C4), C2.1(C2×C8⋊C22), C4.73(C2×C22⋊C4), (C2×D4⋊C4)⋊48C2, (C2×C4⋊C4)⋊113C22, (C2×C4).1404(C2×D4), (C2×C42⋊C2)⋊41C2, (C2×C4).459(C22×C4), (C22×C4).322(C2×C4), C22.80(C2×C22⋊C4), C2.36(C22×C22⋊C4), (C2×C4).156(C22⋊C4), SmallGroup(128,1625)

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

Derived series
C1 — C4 — C2×C23.37D4
Chief series
C1 — C2 — C22 — C2×C4 — C22×C4 — C23×C4 — D4×C23 — C2×C23.37D4
Lower central
C1 — C2 — C4 — C2×C23.37D4
Upper central
C1 — C23 — C23×C4 — C2×C23.37D4
Jennings
C1 — C2 — C2 — C2×C4 — C2×C23.37D4

Generators and relations for C2×C23.37D4
 G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e4=d, f2=c, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, ebe-1=fbf-1=bd=db, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=cde3 >

Subgroups: 1100 in 504 conjugacy classes, 180 normal (16 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, D4, C23, C23, C23, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C24, C24, D4⋊C4, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C42⋊C2, C22×C8, C2×M4(2), C2×M4(2), C23×C4, C22×D4, C22×D4, C25, C2×D4⋊C4, C23.37D4, C2×C42⋊C2, C22×M4(2), D4×C23, C2×C23.37D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C24, C2×C22⋊C4, C8⋊C22, C23×C4, C22×D4, C23.37D4, C22×C22⋊C4, C2×C8⋊C22, C2×C23.37D4

Smallest permutation representation of C2×C23.37D4
►On 32 points
Generators in S32
(1 19)(2 20)(3 21)(4 22)(5 23)(6 24)(7 17)(8 18)(9 27)(10 28)(11 29)(12 30)(13 31)(14 32)(15 25)(16 26)

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

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

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

44 conjugacy classes

class 1 2A···2G2H2I2J2K2L···2S4A···4H4I···4P8A···8H
order12···222222···24···44···48···8
size11···122224···42···24···44···4

44 irreducible representations

dim1111111224
type+++++++++
imageC1C2C2C2C2C2C4D4D4C8⋊C22
kernelC2×C23.37D4C2×D4⋊C4C23.37D4C2×C42⋊C2C22×M4(2)D4×C23C22×D4C22×C4C24C22
# reps14811116714

Matrix representation of C2×C23.37D4 ►in GL8(𝔽17)

160000000
016000000
00100000
00010000
000016000
000001600
000000160
000000016
,
160000000
016000000
00100000
00010000
00001000
00000100
000000160
000000016
,
10000000
01000000
001600000
000160000
00001000
00000100
00000010
00000001
,
10000000
01000000
00100000
00010000
000016000
000001600
000000160
000000016
,
19000000
1316000000
00420000
001130000
00000010
000000016
000001600
000016000
,
168000000
01000000
00420000
000130000
00000010
00000001
00001000
00000100

G:=sub<GL(8,GF(17))| [16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16],[16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16],[1,13,0,0,0,0,0,0,9,16,0,0,0,0,0,0,0,0,4,1,0,0,0,0,0,0,2,13,0,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0],[16,0,0,0,0,0,0,0,8,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,2,13,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0] >;
 

C2×C23.37D4 in GAP, Magma, Sage, TeX 

C_2\times C_2^3._{37}D_4
 
% in TeX
 
G:=Group("C2xC2^3.37D4");
 
// GroupNames label
 
G:=SmallGroup(128,1625);
 
// by ID
 
G=gap.SmallGroup(128,1625);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,1430,2804,1411,172]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^4=d,f^2=c,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,e*b*e^-1=f*b*f^-1=b*d=d*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=c*d*e^3>;
 
// generators/relations
 

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