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G = C42.277C23  order 128 = 27

138th non-split extension by C42 of C23 acting via C23/C2=C22

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

Aliases: C42.277C23, C22⋊C4○D8, (C4×D8)⋊38C2, D8⋊15(C2×C4), (C2×D8)⋊18C4, D8⋊C4⋊3C2, C4⋊C4.400D4, (C4×C8)⋊36C22, C2.4(D4○D8), C4.101(C4×D4), (C4×D4)⋊2C22, C8.20(C22×C4), C4.25(C23×C4), D4.8(C22×C4), C22.68(C4×D4), C4.Q8⋊47C22, C8⋊C4⋊38C22, C2.D8⋊75C22, C4⋊C4.365C23, C8○2M4(2)⋊6C2, (C2×C4).205C24, (C2×C8).416C23, C22⋊C4.187D4, (C22×D8).16C2, C23.437(C2×D4), D4⋊C4⋊89C22, C22.11C24⋊8C2, (C2×D8).159C22, (C2×D4).374C23, C23.25D4⋊24C2, C23.37D4⋊32C2, (C22×C8).248C22, (C22×C4).926C23, C22.149(C22×D4), C42⋊C2.82C22, (C22×D4).322C22, (C2×M4(2)).352C22, C2.65(C2×C4×D4), (C2×C8)⋊14(C2×C4), (C2×D4)⋊27(C2×C4), C4.13(C2×C4○D4), (C2×C4).912(C2×D4), (C2×C4).264(C4○D4), (C2×C4).264(C22×C4), SmallGroup(128,1680)

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

Derived series
C1 — C4 — C42.277C23
Chief series
C1 — C2 — C22 — C2×C4 — C22×C4 — C42⋊C2 — C22.11C24 — C42.277C23
Lower central
C1 — C2 — C4 — C42.277C23
Upper central
C1 — C22 — C42⋊C2 — C42.277C23
Jennings
C1 — C2 — C2 — C2×C4 — C42.277C23

Generators and relations for C42.277C23
 G = < a,b,c,d,e | a4=b4=d2=e2=1, c2=a2, ab=ba, ac=ca, ad=da, eae=ab2, cbc-1=dbd=b-1, be=eb, dcd=bc, ece=b2c, de=ed >

Subgroups: 548 in 264 conjugacy classes, 140 normal (18 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C8, C8, C2×C4, C2×C4, C2×C4, D4, D4, C23, C23, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), D8, C22×C4, C22×C4, C2×D4, C2×D4, C24, C4×C8, C8⋊C4, D4⋊C4, C4.Q8, C2.D8, C2×C22⋊C4, C42⋊C2, C42⋊C2, C4×D4, C4×D4, C22×C8, C2×M4(2), C2×D8, C22×D4, C8○2M4(2), C23.37D4, C23.25D4, C4×D8, D8⋊C4, C22.11C24, C22×D8, C42.277C23
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22×C4, C2×D4, C4○D4, C24, C4×D4, C23×C4, C22×D4, C2×C4○D4, C2×C4×D4, D4○D8, C42.277C23

Smallest permutation representation of C42.277C23
►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 19 15 5)(2 20 16 6)(3 17 13 7)(4 18 14 8)(9 22 25 30)(10 23 26 31)(11 24 27 32)(12 21 28 29)

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

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

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

44 conjugacy classes

class 1 2A2B2C2D2E2F···2M4A···4L4M···4T8A8B8C8D8E···8J
order1222222···24···44···488888···8
size1111224···42···24···422224···4

44 irreducible representations

dim1111111112224
type+++++++++++
imageC1C2C2C2C2C2C2C2C4D4D4C4○D4D4○D8
kernelC42.277C23C8○2M4(2)C23.37D4C23.25D4C4×D8D8⋊C4C22.11C24C22×D8C2×D8C22⋊C4C4⋊C4C2×C4C2
# reps11214421162244

Matrix representation of C42.277C23 ►in GL6(𝔽17)

1300000
0130000
0001150
00414015
00117016
00117133
,
1600000
0160000
000100
0016000
0061001
001011160
,
400000
9130000
0001620
00414015
00107016
0011343
,
16160000
010000
00141400
0014300
001371414
0014143
,
1600000
0160000
0016000
0001600
0001610
0013301

G:=sub<GL(6,GF(17))| [13,0,0,0,0,0,0,13,0,0,0,0,0,0,0,4,11,11,0,0,1,14,7,7,0,0,15,0,0,13,0,0,0,15,16,3],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,16,6,10,0,0,1,0,10,11,0,0,0,0,0,16,0,0,0,0,1,0],[4,9,0,0,0,0,0,13,0,0,0,0,0,0,0,4,10,11,0,0,16,14,7,3,0,0,2,0,0,4,0,0,0,15,16,3],[16,0,0,0,0,0,16,1,0,0,0,0,0,0,14,14,13,1,0,0,14,3,7,4,0,0,0,0,14,14,0,0,0,0,14,3],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,13,0,0,0,16,16,3,0,0,0,0,1,0,0,0,0,0,0,1] >;
 

C42.277C23 in GAP, Magma, Sage, TeX 

C_4^2._{277}C_2^3
 
% in TeX
 
G:=Group("C4^2.277C2^3");
 
// GroupNames label
 
G:=SmallGroup(128,1680);
 
// by ID
 
G=gap.SmallGroup(128,1680);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,184,2019,521,2804,1411,172]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^4=b^4=d^2=e^2=1,c^2=a^2,a*b=b*a,a*c=c*a,a*d=d*a,e*a*e=a*b^2,c*b*c^-1=d*b*d=b^-1,b*e=e*b,d*c*d=b*c,e*c*e=b^2*c,d*e=e*d>;
 
// generators/relations
 

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