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G = C42.310C23  order 128 = 27

171st non-split extension by C42 of C23 acting via C23/C2=C22

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

Aliases: C42.310C23, C4.1832+ 1+4, C4.1302- 1+4, C8⋊9D4⋊51C2, C8⋊6D4⋊49C2, C8⋊4Q8⋊49C2, C4⋊D4.33C4, C22⋊Q8.33C4, C4⋊C8.242C22, C42.240(C2×C4), (C2×C8).450C23, (C4×C8).349C22, (C2×C4).692C24, C42⋊2C2.7C4, C4.4D4.26C4, (C4×D4).66C22, C42.C2.26C4, (C4×Q8).63C22, C4⋊M4(2)⋊40C2, C42.6C4⋊57C2, C23.49(C22×C4), C8⋊C4.109C22, C2.39(Q8○M4(2)), C22⋊C8.151C22, C22.214(C23×C4), (C22×C4).952C23, (C2×C42).799C22, (C22×C8).456C22, C22.D4.14C4, C42.7C22⋊32C2, C42⋊C2.94C22, C42.6C22⋊35C2, (C2×M4(2)).255C22, C23.36C23.19C2, C2.50(C23.33C23), C4⋊C4.125(C2×C4), (C2×D4).147(C2×C4), C22⋊C4.27(C2×C4), (C2×C4).90(C22×C4), (C2×Q8).130(C2×C4), (C22×C4).370(C2×C4), SmallGroup(128,1727)

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

Derived series
C1 — C22 — C42.310C23
Chief series
C1 — C2 — C4 — C2×C4 — C42 — C2×C42 — C23.36C23 — C42.310C23
Lower central
C1 — C22 — C42.310C23
Upper central
C1 — C2×C4 — C42.310C23
Jennings
C1 — C2 — C2 — C2×C4 — C42.310C23

Generators and relations for C42.310C23
 G = < a,b,c,d,e | a4=b4=d2=e2=1, c2=b, ab=ba, ac=ca, dad=a-1, eae=ab2, bc=cb, bd=db, be=eb, dcd=b2c, ece=a2b2c, ede=a2d >

Subgroups: 252 in 173 conjugacy classes, 124 normal (40 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4×C8, C8⋊C4, C22⋊C8, C4⋊C8, C4⋊C8, C2×C42, C42⋊C2, C4×D4, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C42.C2, C42⋊2C2, C22×C8, C2×M4(2), C2×M4(2), C4⋊M4(2), C42.6C22, C42.6C4, C42.7C22, C8⋊9D4, C8⋊6D4, C8⋊4Q8, C23.36C23, C42.310C23
Quotients: C1, C2, C4, C22, C2×C4, C23, C22×C4, C24, C23×C4, 2+ 1+4, 2- 1+4, C23.33C23, Q8○M4(2), C42.310C23

Smallest permutation representation of C42.310C23
►On 64 points
Generators in S64
(1 39 51 46)(2 40 52 47)(3 33 53 48)(4 34 54 41)(5 35 55 42)(6 36 56 43)(7 37 49 44)(8 38 50 45)(9 30 58 20)(10 31 59 21)(11 32 60 22)(12 25 61 23)(13 26 62 24)(14 27 63 17)(15 28 64 18)(16 29 57 19)

(1 3 5 7)(2 4 6 8)(9 11 13 15)(10 12 14 16)(17 19 21 23)(18 20 22 24)(25 27 29 31)(26 28 30 32)(33 35 37 39)(34 36 38 40)(41 43 45 47)(42 44 46 48)(49 51 53 55)(50 52 54 56)(57 59 61 63)(58 60 62 64)

(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)(33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56)(57 58 59 60 61 62 63 64)

(2 6)(4 8)(9 62)(10 59)(11 64)(12 61)(13 58)(14 63)(15 60)(16 57)(18 22)(20 24)(26 30)(28 32)(33 48)(34 45)(35 42)(36 47)(37 44)(38 41)(39 46)(40 43)(50 54)(52 56)

(1 10)(2 64)(3 12)(4 58)(5 14)(6 60)(7 16)(8 62)(9 54)(11 56)(13 50)(15 52)(17 46)(18 36)(19 48)(20 38)(21 42)(22 40)(23 44)(24 34)(25 37)(26 41)(27 39)(28 43)(29 33)(30 45)(31 35)(32 47)(49 57)(51 59)(53 61)(55 63)
 
G:=sub<Sym(64)| (1,39,51,46)(2,40,52,47)(3,33,53,48)(4,34,54,41)(5,35,55,42)(6,36,56,43)(7,37,49,44)(8,38,50,45)(9,30,58,20)(10,31,59,21)(11,32,60,22)(12,25,61,23)(13,26,62,24)(14,27,63,17)(15,28,64,18)(16,29,57,19), (1,3,5,7)(2,4,6,8)(9,11,13,15)(10,12,14,16)(17,19,21,23)(18,20,22,24)(25,27,29,31)(26,28,30,32)(33,35,37,39)(34,36,38,40)(41,43,45,47)(42,44,46,48)(49,51,53,55)(50,52,54,56)(57,59,61,63)(58,60,62,64), (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)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64), (2,6)(4,8)(9,62)(10,59)(11,64)(12,61)(13,58)(14,63)(15,60)(16,57)(18,22)(20,24)(26,30)(28,32)(33,48)(34,45)(35,42)(36,47)(37,44)(38,41)(39,46)(40,43)(50,54)(52,56), (1,10)(2,64)(3,12)(4,58)(5,14)(6,60)(7,16)(8,62)(9,54)(11,56)(13,50)(15,52)(17,46)(18,36)(19,48)(20,38)(21,42)(22,40)(23,44)(24,34)(25,37)(26,41)(27,39)(28,43)(29,33)(30,45)(31,35)(32,47)(49,57)(51,59)(53,61)(55,63)>;
 
G:=Group( (1,39,51,46)(2,40,52,47)(3,33,53,48)(4,34,54,41)(5,35,55,42)(6,36,56,43)(7,37,49,44)(8,38,50,45)(9,30,58,20)(10,31,59,21)(11,32,60,22)(12,25,61,23)(13,26,62,24)(14,27,63,17)(15,28,64,18)(16,29,57,19), (1,3,5,7)(2,4,6,8)(9,11,13,15)(10,12,14,16)(17,19,21,23)(18,20,22,24)(25,27,29,31)(26,28,30,32)(33,35,37,39)(34,36,38,40)(41,43,45,47)(42,44,46,48)(49,51,53,55)(50,52,54,56)(57,59,61,63)(58,60,62,64), (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)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64), (2,6)(4,8)(9,62)(10,59)(11,64)(12,61)(13,58)(14,63)(15,60)(16,57)(18,22)(20,24)(26,30)(28,32)(33,48)(34,45)(35,42)(36,47)(37,44)(38,41)(39,46)(40,43)(50,54)(52,56), (1,10)(2,64)(3,12)(4,58)(5,14)(6,60)(7,16)(8,62)(9,54)(11,56)(13,50)(15,52)(17,46)(18,36)(19,48)(20,38)(21,42)(22,40)(23,44)(24,34)(25,37)(26,41)(27,39)(28,43)(29,33)(30,45)(31,35)(32,47)(49,57)(51,59)(53,61)(55,63) );
 
G=PermutationGroup([[(1,39,51,46),(2,40,52,47),(3,33,53,48),(4,34,54,41),(5,35,55,42),(6,36,56,43),(7,37,49,44),(8,38,50,45),(9,30,58,20),(10,31,59,21),(11,32,60,22),(12,25,61,23),(13,26,62,24),(14,27,63,17),(15,28,64,18),(16,29,57,19)], [(1,3,5,7),(2,4,6,8),(9,11,13,15),(10,12,14,16),(17,19,21,23),(18,20,22,24),(25,27,29,31),(26,28,30,32),(33,35,37,39),(34,36,38,40),(41,43,45,47),(42,44,46,48),(49,51,53,55),(50,52,54,56),(57,59,61,63),(58,60,62,64)], [(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),(33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56),(57,58,59,60,61,62,63,64)], [(2,6),(4,8),(9,62),(10,59),(11,64),(12,61),(13,58),(14,63),(15,60),(16,57),(18,22),(20,24),(26,30),(28,32),(33,48),(34,45),(35,42),(36,47),(37,44),(38,41),(39,46),(40,43),(50,54),(52,56)], [(1,10),(2,64),(3,12),(4,58),(5,14),(6,60),(7,16),(8,62),(9,54),(11,56),(13,50),(15,52),(17,46),(18,36),(19,48),(20,38),(21,42),(22,40),(23,44),(24,34),(25,37),(26,41),(27,39),(28,43),(29,33),(30,45),(31,35),(32,47),(49,57),(51,59),(53,61),(55,63)]])
 

38 conjugacy classes

class 1 2A2B2C2D2E2F4A4B4C4D4E···4O8A···8P
order122222244444···48···8
size111144411114···44···4

38 irreducible representations

dim111111111111111444
type++++++++++-
imageC1C2C2C2C2C2C2C2C2C4C4C4C4C4C42+ 1+42- 1+4Q8○M4(2)
kernelC42.310C23C4⋊M4(2)C42.6C22C42.6C4C42.7C22C8⋊9D4C8⋊6D4C8⋊4Q8C23.36C23C4⋊D4C22⋊Q8C22.D4C4.4D4C42.C2C42⋊2C2C4C4C2
# reps112124221224224114

Matrix representation of C42.310C23 ►in GL8(𝔽17)

100150000
00140000
1316040000
100160000
00001000
000041600
00001810
0000150016
,
130000000
013000000
001300000
000130000
00004000
00000400
00000040
00000004
,
304100000
3014160000
2129140000
32250000
00006410
0000101328
000013932
0000741612
,
10000000
01000000
1301600000
100160000
00001000
00000100
000059160
000093016
,
1301500000
160410000
160400000
011600000
000041500
0000161300
0000104016
0000215160

G:=sub<GL(8,GF(17))| [1,0,13,1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,15,4,4,16,0,0,0,0,0,0,0,0,1,4,1,15,0,0,0,0,0,16,8,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16],[13,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[3,3,2,3,0,0,0,0,0,0,12,2,0,0,0,0,4,14,9,2,0,0,0,0,10,16,14,5,0,0,0,0,0,0,0,0,6,10,13,7,0,0,0,0,4,13,9,4,0,0,0,0,1,2,3,16,0,0,0,0,0,8,2,12],[1,0,13,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,5,9,0,0,0,0,0,1,9,3,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16],[13,16,16,0,0,0,0,0,0,0,0,1,0,0,0,0,15,4,4,16,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,4,16,10,2,0,0,0,0,15,13,4,15,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0] >;
 

C42.310C23 in GAP, Magma, Sage, TeX 

C_4^2._{310}C_2^3
 
% in TeX
 
G:=Group("C4^2.310C2^3");
 
// GroupNames label
 
G:=SmallGroup(128,1727);
 
// by ID
 
G=gap.SmallGroup(128,1727);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,1430,891,675,1018,80,124]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^4=b^4=d^2=e^2=1,c^2=b,a*b=b*a,a*c=c*a,d*a*d=a^-1,e*a*e=a*b^2,b*c=c*b,b*d=d*b,b*e=e*b,d*c*d=b^2*c,e*c*e=a^2*b^2*c,e*d*e=a^2*d>;
 
// generators/relations
 

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