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G = C42⋊22D4  order 128 = 27

16th semidirect product of C42 and D4 acting via D4/C2=C22

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

Aliases: C42⋊22D4, C24.349C23, C23.498C24, C22.2802+ 1+4, C42⋊5C4⋊22C2, C23.60(C4○D4), C23⋊2D4.12C2, C23.4Q8⋊25C2, C23.8Q8⋊76C2, C23.23D4⋊61C2, C23.10D4⋊47C2, (C23×C4).130C22, (C22×C4).120C23, (C2×C42).585C22, C22.328(C22×D4), C24.C22⋊97C2, (C22×D4).183C22, C23.83C23⋊54C2, C2.71(C22.19C24), C2.28(C22.29C24), C2.65(C22.45C24), C2.C42.228C22, C2.75(C22.47C24), (C2×C4×D4)⋊48C2, (C2×C4).367(C2×D4), (C2×C42⋊C2)⋊32C2, (C2×C4).407(C4○D4), (C2×C4⋊C4).883C22, C22.374(C2×C4○D4), (C2×C22⋊C4).200C22, SmallGroup(128,1330)

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

Derived series
C1 — C23 — C42⋊22D4
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C23×C4 — C2×C42⋊C2 — C42⋊22D4
Lower central
C1 — C23 — C42⋊22D4
Upper central
C1 — C23 — C42⋊22D4
Jennings
C1 — C23 — C42⋊22D4

Generators and relations for C42⋊22D4
 G = < a,b,c,d | a4=b4=c4=d2=1, ab=ba, cac-1=a-1b2, dad=a-1, cbc-1=dbd=a2b, dcd=c-1 >

Subgroups: 612 in 302 conjugacy classes, 100 normal (34 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C2×C4, C2×C4, D4, C23, C23, C23, C42, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C24, C24, C2.C42, C2.C42, C2×C42, C2×C22⋊C4, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C42⋊C2, C4×D4, C23×C4, C23×C4, C22×D4, C22×D4, C42⋊5C4, C23.8Q8, C23.23D4, C24.C22, C23⋊2D4, C23.10D4, C23.4Q8, C23.83C23, C2×C42⋊C2, C2×C4×D4, C42⋊22D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C22×D4, C2×C4○D4, 2+ 1+4, C22.19C24, C22.29C24, C22.45C24, C22.47C24, C42⋊22D4

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

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

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

38 conjugacy classes

class 1 2A···2G2H···2M4A···4H4I···4T4U4V4W4X
order12···22···24···44···44444
size11···14···42···24···48888

38 irreducible representations

dim111111111112224
type+++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2D4C4○D4C4○D42+ 1+4
kernelC42⋊22D4C42⋊5C4C23.8Q8C23.23D4C24.C22C23⋊2D4C23.10D4C23.4Q8C23.83C23C2×C42⋊C2C2×C4×D4C42C2×C4C23C22
# reps1124211111144122

Matrix representation of C42⋊22D4 ►in GL6(𝔽5)

040000
400000
001000
000100
000030
000002
,
300000
030000
001000
000100
000010
000004
,
010000
400000
004200
004100
000001
000040
,
010000
100000
004200
000100
000001
000010

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

C42⋊22D4 in GAP, Magma, Sage, TeX 

C_4^2\rtimes_{22}D_4
 
% in TeX
 
G:=Group("C4^2:22D4");
 
// GroupNames label
 
G:=SmallGroup(128,1330);
 
// by ID
 
G=gap.SmallGroup(128,1330);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,253,120,758,723,675,248]);
 
// 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^-1*b^2,d*a*d=a^-1,c*b*c^-1=d*b*d=a^2*b,d*c*d=c^-1>;
 
// generators/relations
 

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