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G = C42⋊16D4  order 128 = 27

10th semidirect product of C42 and D4 acting via D4/C2=C22

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

Aliases: C42⋊16D4, C23.297C24, C24.239C23, C42⋊4C4⋊18C2, C4.54(C4⋊1D4), C4○3(C23⋊2D4), C23⋊2D4⋊53C2, C23.17(C4○D4), (C23×C4).68C22, C4○3(C23.4Q8), C23.4Q8⋊73C2, (C2×C42).457C22, (C22×C4).783C23, C22.180(C22×D4), (C22×D4).496C22, C2.13(C22.19C24), C2.C42.534C22, (C2×C4×D4)⋊18C2, C2.6(C2×C4⋊1D4), (C2×C4).297(C2×D4), (C2×C4⋊C4).840C22, C22.177(C2×C4○D4), (C2×C22⋊C4).489C22, SmallGroup(128,1129)

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

Derived series
C1 — C23 — C42⋊16D4
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C23×C4 — C2×C4×D4 — C42⋊16D4
Lower central
C1 — C23 — C42⋊16D4
Upper central
C1 — C22×C4 — C42⋊16D4
Jennings
C1 — C23 — C42⋊16D4

Generators and relations for C42⋊16D4
 G = < a,b,c,d | a4=b4=c4=d2=1, ab=ba, cac-1=ab2, dad=a-1b2, bc=cb, bd=db, dcd=c-1 >

Subgroups: 804 in 420 conjugacy classes, 124 normal (6 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C2×C4, C2×C4, D4, C23, C23, C23, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C24, C2.C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C4×D4, C23×C4, C22×D4, C42⋊4C4, C23⋊2D4, C23.4Q8, C2×C4×D4, C42⋊16D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C4⋊1D4, C22×D4, C2×C4○D4, C22.19C24, C2×C4⋊1D4, C42⋊16D4

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

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

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

44 conjugacy classes

class 1 2A···2G2H···2O4A···4H4I···4AB
order12···22···24···44···4
size11···14···41···14···4

44 irreducible representations

dim1111122
type++++++
imageC1C2C2C2C2D4C4○D4
kernelC42⋊16D4C42⋊4C4C23⋊2D4C23.4Q8C2×C4×D4C42C23
# reps114461216

Matrix representation of C42⋊16D4 ►in GL6(𝔽5)

220000
030000
001000
000100
000013
000004
,
100000
010000
004000
000400
000030
000003
,
220000
030000
000400
001000
000020
000023
,
220000
130000
001000
000400
000021
000023

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

C42⋊16D4 in GAP, Magma, Sage, TeX 

C_4^2\rtimes_{16}D_4
 
% in TeX
 
G:=Group("C4^2:16D4");
 
// GroupNames label
 
G:=SmallGroup(128,1129);
 
// by ID
 
G=gap.SmallGroup(128,1129);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,112,253,758,723,192]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^4=b^4=c^4=d^2=1,a*b=b*a,c*a*c^-1=a*b^2,d*a*d=a^-1*b^2,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
 
// generators/relations
 

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