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G = C42.275C23  order 128 = 27

136th non-split extension by C42 of C23 acting via C23/C2=C22

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

Aliases: C42.275C23, D8⋊5(C2×C4), (C4×D8)⋊28C2, C8⋊C22⋊5C4, C4.77(C4×D4), D8⋊C4⋊2C2, C4⋊C4.398D4, (C4×C8)⋊21C22, SD16⋊3(C2×C4), C2.3(D4○D8), (C4×D4)⋊1C22, C8.2(C22×C4), (C4×Q8)⋊1C22, (C4×SD16)⋊13C2, M4(2)⋊9(C2×C4), C4.23(C23×C4), D4.6(C22×C4), C22.16(C4×D4), C2.D8⋊66C22, C4.Q8⋊46C22, C8⋊C4⋊37C22, SD16⋊C4⋊1C2, Q8.6(C22×C4), C4⋊C4.363C23, C8○2M4(2)⋊4C2, (C2×C4).203C24, (C2×C8).414C23, C22⋊C4.185D4, C2.4(D4○SD16), C23.435(C2×D4), D4⋊C4⋊99C22, C22.11C24⋊7C2, Q8⋊C4⋊92C22, (C2×D8).158C22, (C2×D4).372C23, (C2×Q8).345C23, M4(2)⋊C4⋊10C2, C23.24D4⋊38C2, (C22×C8).440C22, (C22×C4).924C23, C22.147(C22×D4), C23.33C23⋊4C2, C42⋊C2.80C22, (C2×SD16).109C22, (C22×D4).321C22, (C2×M4(2)).260C22, C2.63(C2×C4×D4), C4○D4⋊4(C2×C4), (C2×D4)⋊26(C2×C4), C4.11(C2×C4○D4), (C2×C4).910(C2×D4), (C2×C8⋊C22).9C2, (C2×D4⋊C4)⋊52C2, (C2×C4).70(C22×C4), (C2×C4).474(C4○D4), (C2×C4⋊C4).574C22, (C2×C4○D4).87C22, SmallGroup(128,1678)

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

Derived series
C1 — C4 — C42.275C23
Chief series
C1 — C2 — C22 — C2×C4 — C22×C4 — C42⋊C2 — C22.11C24 — C42.275C23
Lower central
C1 — C2 — C4 — C42.275C23
Upper central
C1 — C22 — C42⋊C2 — C42.275C23
Jennings
C1 — C2 — C2 — C2×C4 — C42.275C23

Generators and relations for C42.275C23
 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, ce=ec, de=ed >

Subgroups: 476 in 253 conjugacy classes, 140 normal (38 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C8, C8, C2×C4, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C2×C8, M4(2), D8, SD16, C22×C4, C22×C4, C2×D4, C2×D4, C2×D4, C2×Q8, C4○D4, C4○D4, C24, C4×C8, C8⋊C4, D4⋊C4, D4⋊C4, Q8⋊C4, C4.Q8, C2.D8, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C42⋊C2, C42⋊C2, C4×D4, C4×D4, C4×Q8, C22×C8, C2×M4(2), C2×D8, C2×SD16, C8⋊C22, C22×D4, C2×C4○D4, C8○2M4(2), C2×D4⋊C4, C23.24D4, M4(2)⋊C4, C4×D8, C4×SD16, SD16⋊C4, D8⋊C4, C22.11C24, C23.33C23, C2×C8⋊C22, C42.275C23
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, D4○SD16, C42.275C23

Smallest permutation representation of C42.275C23
►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 21 3 23)(2 22 4 24)(5 28 7 26)(6 25 8 27)(9 18 11 20)(10 19 12 17)(13 31 15 29)(14 32 16 30)

(5 19)(6 20)(7 17)(8 18)(9 22)(10 23)(11 24)(12 21)(25 30)(26 31)(27 32)(28 29)

(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), (1,21,3,23)(2,22,4,24)(5,28,7,26)(6,25,8,27)(9,18,11,20)(10,19,12,17)(13,31,15,29)(14,32,16,30), (5,19)(6,20)(7,17)(8,18)(9,22)(10,23)(11,24)(12,21)(25,30)(26,31)(27,32)(28,29), (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), (1,21,3,23)(2,22,4,24)(5,28,7,26)(6,25,8,27)(9,18,11,20)(10,19,12,17)(13,31,15,29)(14,32,16,30), (5,19)(6,20)(7,17)(8,18)(9,22)(10,23)(11,24)(12,21)(25,30)(26,31)(27,32)(28,29), (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)], [(1,21,3,23),(2,22,4,24),(5,28,7,26),(6,25,8,27),(9,18,11,20),(10,19,12,17),(13,31,15,29),(14,32,16,30)], [(5,19),(6,20),(7,17),(8,18),(9,22),(10,23),(11,24),(12,21),(25,30),(26,31),(27,32),(28,29)], [(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···2K4A···4L4M···4V8A8B8C8D8E···8J
order1222222···24···44···488888···8
size1111224···42···24···422224···4

44 irreducible representations

dim111111111111122244
type+++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2C2C4D4D4C4○D4D4○D8D4○SD16
kernelC42.275C23C8○2M4(2)C2×D4⋊C4C23.24D4M4(2)⋊C4C4×D8C4×SD16SD16⋊C4D8⋊C4C22.11C24C23.33C23C2×C8⋊C22C8⋊C22C22⋊C4C4⋊C4C2×C4C2C2
# reps1111122221111622422

Matrix representation of C42.275C23 ►in GL6(𝔽17)

400000
040000
000010
000001
0016000
0001600
,
1600000
0160000
000100
0016000
000001
0000160
,
1020000
970000
0012500
005500
0000125
000055
,
100000
7160000
001000
0001600
000010
0000016
,
1600000
0160000
0016000
0001600
000010
000001

G:=sub<GL(6,GF(17))| [4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,1,0,0,0,0,0,0,1,0,0],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0],[10,9,0,0,0,0,2,7,0,0,0,0,0,0,12,5,0,0,0,0,5,5,0,0,0,0,0,0,12,5,0,0,0,0,5,5],[1,7,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;
 

C42.275C23 in GAP, Magma, Sage, TeX 

C_4^2._{275}C_2^3
 
% in TeX
 
G:=Group("C4^2.275C2^3");
 
// GroupNames label
 
G:=SmallGroup(128,1678);
 
// by ID
 
G=gap.SmallGroup(128,1678);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,184,521,2804,1411,172]);
 
// 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,c*e=e*c,d*e=e*d>;
 
// generators/relations
 

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