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G = C23⋊2C42  order 128 = 27

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

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

Aliases: C23⋊2C42, 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(C4⋊1D4), C23.241(C22×C4), (C23×C4).627C22, C2.3(C23.23D4), C22.46(C4.4D4), C22.22(C42⋊2C2), 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

Derived series
C1 — C22 — C23⋊2C42
Chief series
C1 — C2 — C22 — C23 — C24 — C23×C4 — C22×C42 — C23⋊2C42
Lower central
C1 — C22 — C23⋊2C42
Upper central
C1 — C24 — C23⋊2C42
Jennings
C1 — C24 — C23⋊2C42

Generators and relations for C23⋊2C42
 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, C2, C4, C22, C22, C22, C2×C4, C2×C4, C23, C23, C42, C22⋊C4, C22×C4, C22×C4, C24, C24, C24, C2.C42, C2×C42, C2×C22⋊C4, C2×C22⋊C4, C23×C4, C25, C2×C2.C42, C22×C42, C22×C22⋊C4, C23⋊2C42
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C42, C22⋊C4, C22×C4, C2×D4, C4○D4, C2×C42, C2×C22⋊C4, C42⋊C2, C4×D4, C22≀C2, C4⋊D4, C22.D4, C4.4D4, C42⋊2C2, C4⋊1D4, C4×C22⋊C4, C23.23D4, C24.C22, C24.3C22, C23⋊2C42

Smallest permutation representation of C23⋊2C42
►On 64 points
Generators in S64
(1 3)(2 42)(4 44)(5 48)(6 37)(7 46)(8 39)(9 11)(10 16)(12 14)(13 15)(17 47)(18 40)(19 45)(20 38)(21 23)(22 28)(24 26)(25 27)(29 59)(30 64)(31 57)(32 62)(33 61)(34 58)(35 63)(36 60)(41 43)(49 51)(50 56)(52 54)(53 55)

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

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

56 conjugacy classes

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

56 irreducible representations

dim1111122
type+++++
imageC1C2C2C2C4D4C4○D4
kernelC23⋊2C42C2×C2.C42C22×C42C22×C22⋊C4C2×C22⋊C4C22×C4C23
# reps1313241212

Matrix representation of C23⋊2C42 ►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] >;
 

C23⋊2C42 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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