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G = C43⋊2C2  order 128 = 27

2nd semidirect product of C43 and C2 acting faithfully

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

Aliases: C43⋊2C2, C24.186C23, C23.180C24, C42⋊4C4⋊2C2, C42⋊C2⋊23C4, C42.176(C2×C4), C22.71(C23×C4), C23.77(C22×C4), C4.79(C42⋊C2), (C23×C4).284C22, C4○3(C24.C22), (C2×C42).1006C22, (C22×C4).1240C23, C4○3(C23.63C23), C24.C22.87C2, C23.63C23⋊209C2, C2.C42.467C22, C2.5(C23.36C23), (C4×C4⋊C4)⋊18C2, C2.10(C4×C4○D4), C4⋊C4.200(C2×C4), (C4×C22⋊C4).18C2, C22⋊C4.54(C2×C4), C22.72(C2×C4○D4), (C2×C4).515(C4○D4), (C2×C4⋊C4).795C22, (C2×C4).213(C22×C4), (C22×C4).296(C2×C4), C2.18(C2×C42⋊C2), (C2×C42⋊C2).24C2, (C2×C22⋊C4).419C22, (C2×C4)○(C24.C22), (C2×C4)○(C23.63C23), SmallGroup(128,1030)

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

Derived series
C1 — C22 — C43⋊2C2
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C2×C42 — C43 — C43⋊2C2
Lower central
C1 — C22 — C43⋊2C2
Upper central
C1 — C22×C4 — C43⋊2C2
Jennings
C1 — C23 — C43⋊2C2

Generators and relations for C43⋊2C2
 G = < a,b,c,d | a4=b4=c4=d2=1, ab=ba, ac=ca, dad=ab2, bc=cb, dbd=bc2, cd=dc >

Subgroups: 396 in 262 conjugacy classes, 148 normal (14 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C2×C4, C2×C4, C23, C23, C23, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C24, C2.C42, C2×C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C23×C4, C43, C42⋊4C4, C4×C22⋊C4, C4×C4⋊C4, C23.63C23, C24.C22, C2×C42⋊C2, C43⋊2C2
Quotients: C1, C2, C4, C22, C2×C4, C23, C22×C4, C4○D4, C24, C42⋊C2, C23×C4, C2×C4○D4, C2×C42⋊C2, C4×C4○D4, C23.36C23, C43⋊2C2

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

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

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

56 conjugacy classes

class 1 2A···2G2H2I4A···4H4I···4AF4AG···4AT
order12···2224···44···44···4
size11···1441···12···24···4

56 irreducible representations

dim1111111112
type++++++++
imageC1C2C2C2C2C2C2C2C4C4○D4
kernelC43⋊2C2C43C42⋊4C4C4×C22⋊C4C4×C4⋊C4C23.63C23C24.C22C2×C42⋊C2C42⋊C2C2×C4
# reps111224411624

Matrix representation of C43⋊2C2 ►in GL5(𝔽5)

30000
02200
00300
00010
00034
,
10000
03300
00200
00030
00003
,
10000
03000
00300
00040
00004
,
40000
01000
03400
00011
00004

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

C43⋊2C2 in GAP, Magma, Sage, TeX 

C_4^3\rtimes_2C_2
 
% in TeX
 
G:=Group("C4^3:2C2");
 
// GroupNames label
 
G:=SmallGroup(128,1030);
 
// by ID
 
G=gap.SmallGroup(128,1030);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,456,758,100,136]);
 
// 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*b^2,b*c=c*b,d*b*d=b*c^2,c*d=d*c>;
 
// generators/relations
 

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