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G = C232C42order 128 = 27

1st semidirect product of C23 and C42 acting via C42/C22=C22

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

Aliases: C232C42, C25.2C22, C24.627C23, C24.40(C2×C4), (C22×C42)⋊1C2, C22.76(C4×D4), C23.717(C2×D4), (C22×C4).647D4, C22.64C22≀C2, C22.39(C2×C42), C23.340(C4○D4), C22.96(C4⋊D4), C22.22(C41D4), C23.241(C22×C4), (C23×C4).627C22, C2.3(C23.23D4), C22.46(C4.4D4), C22.22(C422C2), C22.44(C42⋊C2), C2.2(C24.3C22), C2.3(C24.C22), C22.69(C22.D4), C2.7(C4×C22⋊C4), (C2×C22⋊C4)⋊11C4, (C2×C4)⋊8(C22⋊C4), (C22×C4).97(C2×C4), (C2×C2.C42)⋊2C2, (C22×C22⋊C4).2C2, C22.87(C2×C22⋊C4), SmallGroup(128,169)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C232C42
C1C2C22C23C24C23×C4C22×C42 — C232C42
C1C22 — C232C42
C1C24 — C232C42
C1C24 — C232C42

Generators and relations for C232C42
 G = < a,b,c,d,e | a2=b2=c2=d4=e4=1, dad-1=ab=ba, eae-1=ac=ca, bc=cb, bd=db, be=eb, cd=dc, ce=ec, de=ed >

Subgroups: 812 in 414 conjugacy classes, 140 normal (8 characteristic)
C1, C2 [×15], C2 [×4], C4 [×18], C22 [×3], C22 [×32], C22 [×36], C2×C4 [×12], C2×C4 [×66], C23 [×19], C23 [×52], C42 [×8], C22⋊C4 [×24], C22×C4 [×24], C22×C4 [×30], C24, C24 [×6], C24 [×12], C2.C42 [×6], C2×C42 [×6], C2×C22⋊C4 [×12], C2×C22⋊C4 [×12], C23×C4 [×6], C25, C2×C2.C42 [×3], C22×C42, C22×C22⋊C4 [×3], C232C42
Quotients: C1, C2 [×7], C4 [×12], C22 [×7], C2×C4 [×18], D4 [×12], C23, C42 [×4], C22⋊C4 [×12], C22×C4 [×3], C2×D4 [×6], C4○D4 [×6], C2×C42, C2×C22⋊C4 [×3], C42⋊C2 [×3], C4×D4 [×12], C22≀C2, C4⋊D4 [×6], C22.D4 [×3], C4.4D4 [×3], C422C2 [×2], C41D4, C4×C22⋊C4 [×3], C23.23D4 [×3], C24.C22 [×6], C24.3C22 [×3], C232C42

Smallest permutation representation of C232C42
On 64 points
Generators in S64
(1 3)(2 44)(4 42)(5 18)(6 37)(7 20)(8 39)(9 11)(10 16)(12 14)(13 15)(17 47)(19 45)(21 23)(22 28)(24 26)(25 27)(29 59)(30 64)(31 57)(32 62)(33 61)(34 58)(35 63)(36 60)(38 46)(40 48)(41 43)(49 51)(50 56)(52 54)(53 55)
(1 41)(2 42)(3 43)(4 44)(5 48)(6 45)(7 46)(8 47)(9 13)(10 14)(11 15)(12 16)(17 39)(18 40)(19 37)(20 38)(21 25)(22 26)(23 27)(24 28)(29 35)(30 36)(31 33)(32 34)(49 53)(50 54)(51 55)(52 56)(57 61)(58 62)(59 63)(60 64)
(1 9)(2 10)(3 11)(4 12)(5 38)(6 39)(7 40)(8 37)(13 41)(14 42)(15 43)(16 44)(17 45)(18 46)(19 47)(20 48)(21 49)(22 50)(23 51)(24 52)(25 53)(26 54)(27 55)(28 56)(29 57)(30 58)(31 59)(32 60)(33 63)(34 64)(35 61)(36 62)
(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 31 23 45)(2 32 24 46)(3 29 21 47)(4 30 22 48)(5 44 36 26)(6 41 33 27)(7 42 34 28)(8 43 35 25)(9 59 51 17)(10 60 52 18)(11 57 49 19)(12 58 50 20)(13 63 55 39)(14 64 56 40)(15 61 53 37)(16 62 54 38)

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

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

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

56 conjugacy classes

class 1 2A···2O2P2Q2R2S4A···4X4Y···4AJ
order12···222224···44···4
size11···144442···24···4

56 irreducible representations

dim1111122
type+++++
imageC1C2C2C2C4D4C4○D4
kernelC232C42C2×C2.C42C22×C42C22×C22⋊C4C2×C22⋊C4C22×C4C23
# reps1313241212

Matrix representation of C232C42 in GL6(𝔽5)

100000
040000
001000
000400
000040
000001
,
100000
010000
004000
000400
000010
000001
,
100000
010000
004000
000400
000040
000004
,
200000
010000
000200
003000
000030
000003
,
300000
020000
000100
004000
000001
000040

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

C232C42 in GAP, Magma, Sage, TeX

C_2^3\rtimes_2C_4^2
% in TeX

G:=Group("C2^3:2C4^2");
// GroupNames label

G:=SmallGroup(128,169);
// by ID

G=gap.SmallGroup(128,169);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,2,224,141,456,422,268]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^4=e^4=1,d*a*d^-1=a*b=b*a,e*a*e^-1=a*c=c*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,d*e=e*d>;
// generators/relations

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