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G = C42:18D4  order 128 = 27

12nd semidirect product of C42 and D4 acting via D4/C2=C22

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

Aliases: C42:18D4, C24.320C23, C23.437C24, C22.2262+ 1+4, C22.1742- 1+4, C2.28D42, (C2xD4):34D4, C4:5(C4:D4), C23.47(C2xD4), C42:9C4:26C2, C2.49(D4:6D4), (C22xC4).94C23, C23.7Q8:65C2, C23.10D4:39C2, (C2xC42).543C22, (C23xC4).390C22, C22.288(C22xD4), (C22xD4).161C22, C2.C42.180C22, C2.12(C22.31C24), C2.24(C22.49C24), (C2xC4xD4):43C2, (C2xC4:D4):15C2, (C2xC4).351(C2xD4), C2.32(C2xC4:D4), (C2xC4).819(C4oD4), (C2xC4:C4).297C22, C22.314(C2xC4oD4), (C2xC22:C4).173C22, SmallGroup(128,1269)

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

Derived series
C1 — C23 — C42:18D4
Chief series
C1 — C2 — C22 — C23 — C22xC4 — C23xC4 — C2xC4xD4 — C42:18D4
Lower central
C1 — C23 — C42:18D4
Upper central
C1 — C23 — C42:18D4
Jennings
C1 — C23 — C42:18D4

Generators and relations for C42:18D4
 G = < a,b,c,d | a4=b4=c4=d2=1, ab=ba, cac-1=a-1, dad=ab2, cbc-1=dbd=b-1, dcd=c-1 >

Subgroups: 804 in 394 conjugacy classes, 124 normal (12 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C2xC4, C2xC4, D4, C23, C23, C23, C42, C22:C4, C4:C4, C22xC4, C22xC4, C22xC4, C2xD4, C2xD4, C24, C2.C42, C2xC42, C2xC22:C4, C2xC4:C4, C4xD4, C4:D4, C23xC4, C22xD4, C23.7Q8, C42:9C4, C23.10D4, C2xC4xD4, C2xC4:D4, C42:18D4
Quotients: C1, C2, C22, D4, C23, C2xD4, C4oD4, C24, C4:D4, C22xD4, C2xC4oD4, 2+ 1+4, 2- 1+4, C2xC4:D4, C22.31C24, D42, D4:6D4, C22.49C24, C42:18D4

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

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

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

38 conjugacy classes

class 1 2A···2G2H···2O4A···4H4I···4R4S4T4U4V
order12···22···24···44···44444
size11···14···42···24···48888

38 irreducible representations

dim11111122244
type+++++++++-
imageC1C2C2C2C2C2D4D4C4oD42+ 1+42- 1+4
kernelC42:18D4C23.7Q8C42:9C4C23.10D4C2xC4xD4C2xC4:D4C42C2xD4C2xC4C22C22
# reps14142448811

Matrix representation of C42:18D4 ►in GL6(F5)

100000
010000
000100
004000
000020
000023
,
100000
010000
004000
000400
000020
000023
,
420000
410000
002000
000300
000013
000014
,
420000
010000
000300
002000
000042
000001

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

C42:18D4 in GAP, Magma, Sage, TeX 

C_4^2\rtimes_{18}D_4
 
% in TeX
 
G:=Group("C4^2:18D4");
 
// GroupNames label
 
G:=SmallGroup(128,1269);
 
// by ID
 
G=gap.SmallGroup(128,1269);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,253,568,758,723,675,80]);
 
// 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,d*a*d=a*b^2,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
 
// generators/relations
 

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