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G = C42⋊43D4  order 128 = 27

2nd semidirect product of C42 and D4 acting via D4/C22=C2

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

Aliases: C42⋊43D4, C23.752C24, C24.120C23, (C22×C42)⋊7C2, C42⋊5C4⋊39C2, C22⋊1(C42⋊2C2), C23.252(C4○D4), (C23×C4).652C22, C23.8Q8⋊147C2, C22.462(C22×D4), (C22×C4).1263C23, (C2×C42).1013C22, C23.23D4.79C2, (C22×D4).310C22, C24.C22⋊183C2, C2.95(C22.19C24), C23.63C23⋊203C2, C2.C42.449C22, C2.110(C23.36C23), (C2×C4).1206(C2×D4), (C2×C42⋊2C2)⋊29C2, (C2×C4).528(C4○D4), (C2×C4⋊C4).555C22, C2.25(C2×C42⋊2C2), C22.593(C2×C4○D4), (C2×C22⋊C4).362C22, SmallGroup(128,1584)

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

Derived series
C1 — C23 — C42⋊43D4
Chief series
C1 — C2 — C22 — C23 — C24 — C23×C4 — C22×C42 — C42⋊43D4
Lower central
C1 — C23 — C42⋊43D4
Upper central
C1 — C23 — C42⋊43D4
Jennings
C1 — C23 — C42⋊43D4

Generators and relations for C42⋊43D4
 G = < a,b,c,d | a4=b4=c4=d2=1, ab=ba, cac-1=dad=ab2, cbc-1=dbd=a2b-1, dcd=c-1 >

Subgroups: 516 in 291 conjugacy classes, 108 normal (13 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C2×C4, C2×C4, D4, C23, C23, C23, C42, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C24, C2.C42, C2×C42, C2×C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C42⋊2C2, C23×C4, C22×D4, C42⋊5C4, C23.8Q8, C23.23D4, C23.63C23, C24.C22, C22×C42, C2×C42⋊2C2, C42⋊43D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C42⋊2C2, C22×D4, C2×C4○D4, C22.19C24, C2×C42⋊2C2, C23.36C23, C42⋊43D4

Smallest permutation representation of C42⋊43D4
►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 46 55 52)(2 47 56 49)(3 48 53 50)(4 45 54 51)(5 43 62 21)(6 44 63 22)(7 41 64 23)(8 42 61 24)(9 37 15 35)(10 38 16 36)(11 39 13 33)(12 40 14 34)(17 26 59 32)(18 27 60 29)(19 28 57 30)(20 25 58 31)

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

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

44 conjugacy classes

class 1 2A···2G2H2I2J2K2L4A···4X4Y···4AE
order12···2222224···44···4
size11···1222282···28···8

44 irreducible representations

dim11111111222
type+++++++++
imageC1C2C2C2C2C2C2C2D4C4○D4C4○D4
kernelC42⋊43D4C42⋊5C4C23.8Q8C23.23D4C23.63C23C24.C22C22×C42C2×C42⋊2C2C42C2×C4C23
# reps1133331141212

Matrix representation of C42⋊43D4 ►in GL6(𝔽5)

300000
030000
000300
003000
000030
000003
,
010000
100000
002000
000200
000001
000010
,
040000
100000
000400
001000
000010
000004
,
100000
040000
001000
000400
000010
000004

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

C42⋊43D4 in GAP, Magma, Sage, TeX 

C_4^2\rtimes_{43}D_4
 
% in TeX
 
G:=Group("C4^2:43D4");
 
// GroupNames label
 
G:=SmallGroup(128,1584);
 
// by ID
 
G=gap.SmallGroup(128,1584);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,253,344,758,100,2019]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^4=b^4=c^4=d^2=1,a*b=b*a,c*a*c^-1=d*a*d=a*b^2,c*b*c^-1=d*b*d=a^2*b^-1,d*c*d=c^-1>;
 
// generators/relations
 

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