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G = C432C2order 128 = 27

2nd semidirect product of C43 and C2 acting faithfully

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

Aliases: C432C2, C24.186C23, C23.180C24, C424C42C2, C42⋊C223C4, C42.176(C2×C4), C22.71(C23×C4), C23.77(C22×C4), C4.79(C42⋊C2), (C23×C4).284C22, C43(C24.C22), (C2×C42).1006C22, (C22×C4).1240C23, C43(C23.63C23), C24.C22.87C2, C23.63C23209C2, 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

C1C22 — C432C2
C1C2C22C23C22×C4C2×C42C43 — C432C2
C1C22 — C432C2
C1C22×C4 — C432C2
C1C23 — C432C2

Generators and relations for C432C2
 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 [×6], C2 [×2], C4 [×4], C4 [×22], C22, C22 [×6], C22 [×10], C2×C4 [×24], C2×C4 [×38], C23, C23 [×2], C23 [×6], C42 [×8], C42 [×16], C22⋊C4 [×8], C22⋊C4 [×8], C4⋊C4 [×8], C4⋊C4 [×8], C22×C4 [×2], C22×C4 [×16], C22×C4 [×4], C24, C2.C42 [×12], C2×C42 [×2], C2×C42 [×8], C2×C22⋊C4 [×6], C2×C4⋊C4 [×6], C42⋊C2 [×8], C23×C4, C43, C424C4, C4×C22⋊C4 [×2], C4×C4⋊C4 [×2], C23.63C23 [×4], C24.C22 [×4], C2×C42⋊C2, C432C2
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], C22×C4 [×14], C4○D4 [×12], C24, C42⋊C2 [×4], C23×C4, C2×C4○D4 [×6], C2×C42⋊C2, C4×C4○D4 [×2], C23.36C23 [×4], C432C2

Smallest permutation representation of C432C2
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 19 11 47)(2 20 12 48)(3 17 9 45)(4 18 10 46)(5 14 38 42)(6 15 39 43)(7 16 40 44)(8 13 37 41)(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 43)(2 56 52 44)(3 53 49 41)(4 54 50 42)(5 18 36 30)(6 19 33 31)(7 20 34 32)(8 17 35 29)(9 25 21 13)(10 26 22 14)(11 27 23 15)(12 28 24 16)(37 45 61 57)(38 46 62 58)(39 47 63 59)(40 48 64 60)
(2 12)(4 10)(5 62)(6 33)(7 64)(8 35)(14 42)(16 44)(17 29)(18 58)(19 31)(20 60)(22 50)(24 52)(26 54)(28 56)(30 46)(32 48)(34 40)(36 38)(37 61)(39 63)(45 57)(47 59)

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

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

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

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
kernelC432C2C43C424C4C4×C22⋊C4C4×C4⋊C4C23.63C23C24.C22C2×C42⋊C2C42⋊C2C2×C4
# reps111224411624

Matrix representation of C432C2 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] >;

C432C2 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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