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G = C43⋊5C2  order 128 = 27

5th semidirect product of C43 and C2 acting faithfully

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

Aliases: C43⋊5C2, C24.127C23, C23.766C24, (C22×C4).1268C23, (C2×C42).1020C22, C24.C22.86C2, C23.63C23⋊207C2, C2.C42.461C22, C2.119(C23.36C23), (C2×C4).535(C4○D4), (C2×C4⋊C4).569C22, C22.607(C2×C4○D4), (C2×C22⋊C4).372C22, SmallGroup(128,1598)

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

Derived series
C1 — C23 — C43⋊5C2
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C2×C42 — C43 — C43⋊5C2
Lower central
C1 — C23 — C43⋊5C2
Upper central
C1 — C23 — C43⋊5C2
Jennings
C1 — C23 — C43⋊5C2

Generators and relations for C43⋊5C2
 G = < a,b,c,d | a4=b4=c4=d2=1, ab=ba, ac=ca, dad=ac2, bc=cb, dbd=b-1c2, dcd=a2b2c >

Subgroups: 356 in 209 conjugacy classes, 96 normal (5 characteristic)
C1, C2, C2, C4, C22, C22, C2×C4, C2×C4, C23, C23, C42, C22⋊C4, C4⋊C4, C22×C4, C24, C2.C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C43, C23.63C23, C24.C22, C43⋊5C2
Quotients: C1, C2, C22, C23, C4○D4, C24, C2×C4○D4, C23.36C23, C43⋊5C2

Smallest permutation representation of C43⋊5C2
►On 64 points
Generators in S64
(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)

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

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

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

44 conjugacy classes

class 1 2A···2G2H4A···4AB4AC···4AI
order12···224···44···4
size11···182···28···8

44 irreducible representations

dim11112
type++++
imageC1C2C2C2C4○D4
kernelC43⋊5C2C43C23.63C23C24.C22C2×C4
# reps117728

Matrix representation of C43⋊5C2 ►in GL6(𝔽5)

100000
010000
003000
000300
000012
000044
,
030000
300000
004000
000400
000020
000002
,
010000
100000
001300
000400
000030
000003
,
100000
040000
001000
001400
000010
000044

G:=sub<GL(6,GF(5))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,0,1,4,0,0,0,0,2,4],[0,3,0,0,0,0,3,0,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,2,0,0,0,0,0,0,2],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,3,4,0,0,0,0,0,0,3,0,0,0,0,0,0,3],[1,0,0,0,0,0,0,4,0,0,0,0,0,0,1,1,0,0,0,0,0,4,0,0,0,0,0,0,1,4,0,0,0,0,0,4] >;
 

C43⋊5C2 in GAP, Magma, Sage, TeX 

C_4^3\rtimes_5C_2
 
% in TeX
 
G:=Group("C4^3:5C2");
 
// GroupNames label
 
G:=SmallGroup(128,1598);
 
// by ID
 
G=gap.SmallGroup(128,1598);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,253,792,758,268,2019,80]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^4=b^4=c^4=d^2=1,a*b=b*a,a*c=c*a,d*a*d=a*c^2,b*c=c*b,d*b*d=b^-1*c^2,d*c*d=a^2*b^2*c>;
 
// generators/relations
 

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