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G = C43⋊4C2  order 128 = 27

4th semidirect product of C43 and C2 acting faithfully

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

Aliases: C43⋊4C2, C24.126C23, C23.765C24, (C2×C42).1019C22, (C22×C4).1267C23, C24.C22.85C2, C23.63C23⋊206C2, C2.C42.460C22, C2.118(C23.36C23), (C2×C4).534(C4○D4), (C2×C4⋊C4).568C22, C22.606(C2×C4○D4), (C2×C22⋊C4).371C22, SmallGroup(128,1597)

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

Derived series
C1 — C23 — C43⋊4C2
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C2×C42 — C43 — C43⋊4C2
Lower central
C1 — C23 — C43⋊4C2
Upper central
C1 — C23 — C43⋊4C2
Jennings
C1 — C23 — C43⋊4C2

Generators and relations for C43⋊4C2
 G = < a,b,c,d | a4=b4=c4=d2=1, ab=ba, ac=ca, dad=ac2, bc=cb, dbd=b-1c2, dcd=a2b2c-1 >

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⋊4C2
Quotients: C1, C2, C22, C23, C4○D4, C24, C2×C4○D4, C23.36C23, C43⋊4C2

Smallest permutation representation of C43⋊4C2
►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 45 37 59)(2 46 38 60)(3 47 39 57)(4 48 40 58)(5 31 23 13)(6 32 24 14)(7 29 21 15)(8 30 22 16)(9 35 27 17)(10 36 28 18)(11 33 25 19)(12 34 26 20)(41 63 55 49)(42 64 56 50)(43 61 53 51)(44 62 54 52)

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

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

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⋊4C2C43C23.63C23C24.C22C2×C4
# reps117728

Matrix representation of C43⋊4C2 ►in GL6(𝔽5)

200000
020000
002000
000200
000001
000040
,
100000
010000
003300
000200
000020
000002
,
010000
100000
004000
000400
000001
000040
,
100000
040000
001000
003400
000010
000004

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

C43⋊4C2 in GAP, Magma, Sage, TeX 

C_4^3\rtimes_4C_2
 
% in TeX
 
G:=Group("C4^3:4C2");
 
// GroupNames label
 
G:=SmallGroup(128,1597);
 
// by ID
 
G=gap.SmallGroup(128,1597);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,253,680,758,268,2019,80]);
 
// 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^-1>;
 
// generators/relations
 

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