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G = C2xC33:4C8order 432 = 24·33

Direct product of C2 and C33:4C8

direct product, metabelian, soluble, monomial, A-group

Aliases: C2xC33:4C8, C62.12Dic3, C6:(C32:2C8), (C32xC6):4C8, C33:15(C2xC8), (C3xC62).3C4, C3:Dic3.42D6, C3:Dic3.7Dic3, C22.2(C33:C4), (C3xC6):4(C3:C8), C32:8(C2xC3:C8), C3:2(C2xC32:2C8), C6.12(C2xC32:C4), (C2xC6).4(C32:C4), C2.3(C2xC33:C4), (C6xC3:Dic3).14C2, (C3xC3:Dic3).10C4, (C2xC3:Dic3).10S3, (C32xC6).19(C2xC4), (C3xC6).26(C2xDic3), (C3xC3:Dic3).50C22, SmallGroup(432,639)

Series: Derived Chief Lower central Upper central

C1C33 — C2xC33:4C8
C1C3C33C32xC6C3xC3:Dic3C33:4C8 — C2xC33:4C8
C33 — C2xC33:4C8
C1C22

Generators and relations for C2xC33:4C8
 G = < a,b,c,d,e | a2=b3=c3=d3=e8=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, ebe-1=bc-1, cd=dc, ece-1=b-1c-1, ede-1=d-1 >

Subgroups: 392 in 96 conjugacy classes, 29 normal (19 characteristic)
C1, C2, C2, C3, C3, C4, C22, C6, C6, C6, C8, C2xC4, C32, C32, Dic3, C12, C2xC6, C2xC6, C2xC8, C3xC6, C3xC6, C3xC6, C3:C8, C2xDic3, C2xC12, C33, C3xDic3, C3:Dic3, C62, C62, C2xC3:C8, C32xC6, C32xC6, C32:2C8, C6xDic3, C2xC3:Dic3, C3xC3:Dic3, C3xC62, C2xC32:2C8, C33:4C8, C6xC3:Dic3, C2xC33:4C8
Quotients: C1, C2, C4, C22, S3, C8, C2xC4, Dic3, D6, C2xC8, C3:C8, C2xDic3, C32:C4, C2xC3:C8, C32:2C8, C2xC32:C4, C33:C4, C2xC32:2C8, C33:4C8, C2xC33:C4, C2xC33:4C8

Smallest permutation representation of C2xC33:4C8
On 48 points
Generators in S48
(1 25)(2 26)(3 27)(4 28)(5 29)(6 30)(7 31)(8 32)(9 46)(10 47)(11 48)(12 41)(13 42)(14 43)(15 44)(16 45)(17 33)(18 34)(19 35)(20 36)(21 37)(22 38)(23 39)(24 40)
(1 13 17)(2 18 14)(3 19 15)(4 16 20)(5 9 21)(6 22 10)(7 23 11)(8 12 24)(25 42 33)(26 34 43)(27 35 44)(28 45 36)(29 46 37)(30 38 47)(31 39 48)(32 41 40)
(1 17 13)(3 15 19)(5 21 9)(7 11 23)(25 33 42)(27 44 35)(29 37 46)(31 48 39)
(1 17 13)(2 14 18)(3 19 15)(4 16 20)(5 21 9)(6 10 22)(7 23 11)(8 12 24)(25 33 42)(26 43 34)(27 35 44)(28 45 36)(29 37 46)(30 47 38)(31 39 48)(32 41 40)
(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)

G:=sub<Sym(48)| (1,25)(2,26)(3,27)(4,28)(5,29)(6,30)(7,31)(8,32)(9,46)(10,47)(11,48)(12,41)(13,42)(14,43)(15,44)(16,45)(17,33)(18,34)(19,35)(20,36)(21,37)(22,38)(23,39)(24,40), (1,13,17)(2,18,14)(3,19,15)(4,16,20)(5,9,21)(6,22,10)(7,23,11)(8,12,24)(25,42,33)(26,34,43)(27,35,44)(28,45,36)(29,46,37)(30,38,47)(31,39,48)(32,41,40), (1,17,13)(3,15,19)(5,21,9)(7,11,23)(25,33,42)(27,44,35)(29,37,46)(31,48,39), (1,17,13)(2,14,18)(3,19,15)(4,16,20)(5,21,9)(6,10,22)(7,23,11)(8,12,24)(25,33,42)(26,43,34)(27,35,44)(28,45,36)(29,37,46)(30,47,38)(31,39,48)(32,41,40), (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)>;

G:=Group( (1,25)(2,26)(3,27)(4,28)(5,29)(6,30)(7,31)(8,32)(9,46)(10,47)(11,48)(12,41)(13,42)(14,43)(15,44)(16,45)(17,33)(18,34)(19,35)(20,36)(21,37)(22,38)(23,39)(24,40), (1,13,17)(2,18,14)(3,19,15)(4,16,20)(5,9,21)(6,22,10)(7,23,11)(8,12,24)(25,42,33)(26,34,43)(27,35,44)(28,45,36)(29,46,37)(30,38,47)(31,39,48)(32,41,40), (1,17,13)(3,15,19)(5,21,9)(7,11,23)(25,33,42)(27,44,35)(29,37,46)(31,48,39), (1,17,13)(2,14,18)(3,19,15)(4,16,20)(5,21,9)(6,10,22)(7,23,11)(8,12,24)(25,33,42)(26,43,34)(27,35,44)(28,45,36)(29,37,46)(30,47,38)(31,39,48)(32,41,40), (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) );

G=PermutationGroup([[(1,25),(2,26),(3,27),(4,28),(5,29),(6,30),(7,31),(8,32),(9,46),(10,47),(11,48),(12,41),(13,42),(14,43),(15,44),(16,45),(17,33),(18,34),(19,35),(20,36),(21,37),(22,38),(23,39),(24,40)], [(1,13,17),(2,18,14),(3,19,15),(4,16,20),(5,9,21),(6,22,10),(7,23,11),(8,12,24),(25,42,33),(26,34,43),(27,35,44),(28,45,36),(29,46,37),(30,38,47),(31,39,48),(32,41,40)], [(1,17,13),(3,15,19),(5,21,9),(7,11,23),(25,33,42),(27,44,35),(29,37,46),(31,48,39)], [(1,17,13),(2,14,18),(3,19,15),(4,16,20),(5,21,9),(6,10,22),(7,23,11),(8,12,24),(25,33,42),(26,43,34),(27,35,44),(28,45,36),(29,37,46),(30,47,38),(31,39,48),(32,41,40)], [(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)]])

48 conjugacy classes

class 1 2A2B2C3A3B···3G4A4B4C4D6A6B6C6D···6U8A···8H12A12B12C12D
order122233···344446666···68···812121212
size111124···499992224···427···2718181818

48 irreducible representations

dim11111122222444444
type++++-+-+-+
imageC1C2C2C4C4C8S3Dic3D6Dic3C3:C8C32:C4C32:2C8C2xC32:C4C33:C4C33:4C8C2xC33:C4
kernelC2xC33:4C8C33:4C8C6xC3:Dic3C3xC3:Dic3C3xC62C32xC6C2xC3:Dic3C3:Dic3C3:Dic3C62C3xC6C2xC6C6C6C22C2C2
# reps12122811114242484

Matrix representation of C2xC33:4C8 in GL6(F73)

7200000
0720000
001000
000100
000010
000001
,
100000
010000
00640019
0008190
0000640
000008
,
100000
010000
00804848
00064667
000010
000001
,
0720000
1720000
00805454
00081954
0000640
0000064
,
0220000
2200000
0067600
00676700
00063667
0010066

G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,64,0,0,0,0,0,0,8,0,0,0,0,0,19,64,0,0,0,19,0,0,8],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,8,0,0,0,0,0,0,64,0,0,0,0,48,6,1,0,0,0,48,67,0,1],[0,1,0,0,0,0,72,72,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,0,54,19,64,0,0,0,54,54,0,64],[0,22,0,0,0,0,22,0,0,0,0,0,0,0,67,67,0,10,0,0,6,67,63,0,0,0,0,0,6,6,0,0,0,0,67,6] >;

C2xC33:4C8 in GAP, Magma, Sage, TeX

C_2\times C_3^3\rtimes_4C_8
% in TeX

G:=Group("C2xC3^3:4C8");
// GroupNames label

G:=SmallGroup(432,639);
// by ID

G=gap.SmallGroup(432,639);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,3,-3,28,58,2804,298,2693,1027,14118]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^3=c^3=d^3=e^8=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,e*b*e^-1=b*c^-1,c*d=d*c,e*c*e^-1=b^-1*c^-1,e*d*e^-1=d^-1>;
// generators/relations

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