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G = C42:3Dic7order 448 = 26·7

3rd semidirect product of C42 and Dic7 acting via Dic7/C7=C4

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C42:3Dic7, (C4xC28):3C4, (D4xC14):2C4, (C2xD4):2Dic7, C4:1D4.2D7, C7:2(C42:C4), (C2xD4).10D14, C23:Dic7:8C2, (C22xC14).17D4, C23.8(C7:D4), C14.25(C23:C4), (D4xC14).173C22, C2.10(C23:Dic7), C22.16(C23.D7), (C2xC28).10(C2xC4), (C7xC4:1D4).7C2, (C2xC4).3(C2xDic7), (C2xC14).103(C22:C4), SmallGroup(448,102)

Series: Derived Chief Lower central Upper central

C1C2xC28 — C42:3Dic7
C1C7C14C2xC14C22xC14D4xC14C23:Dic7 — C42:3Dic7
C7C14C2xC14C2xC28 — C42:3Dic7
C1C2C22C2xD4C4:1D4

Generators and relations for C42:3Dic7
 G = < a,b,c,d | a4=b4=c14=1, d2=c7, ab=ba, cac-1=a-1, dad-1=a-1b, cbc-1=b-1, dbd-1=a2b, dcd-1=c-1 >

Subgroups: 428 in 86 conjugacy classes, 23 normal (17 characteristic)
C1, C2, C2, C4, C22, C22, C7, C2xC4, C2xC4, D4, C23, C23, C14, C14, C42, C22:C4, C2xD4, C2xD4, Dic7, C28, C2xC14, C2xC14, C23:C4, C4:1D4, C2xDic7, C2xC28, C2xC28, C7xD4, C22xC14, C22xC14, C42:C4, C23.D7, C4xC28, D4xC14, D4xC14, C23:Dic7, C7xC4:1D4, C42:3Dic7
Quotients: C1, C2, C4, C22, C2xC4, D4, D7, C22:C4, Dic7, D14, C23:C4, C2xDic7, C7:D4, C42:C4, C23.D7, C23:Dic7, C42:3Dic7

Smallest permutation representation of C42:3Dic7
On 56 points
Generators in S56
(1 54)(2 55)(3 56)(4 43)(5 44)(6 45)(7 46)(8 47)(9 48)(10 49)(11 50)(12 51)(13 52)(14 53)(15 31 38 22)(16 23 39 32)(17 33 40 24)(18 25 41 34)(19 35 42 26)(20 27 29 36)(21 37 30 28)
(1 8 54 47)(2 48 55 9)(3 10 56 49)(4 50 43 11)(5 12 44 51)(6 52 45 13)(7 14 46 53)(15 22 38 31)(16 32 39 23)(17 24 40 33)(18 34 41 25)(19 26 42 35)(20 36 29 27)(21 28 30 37)
(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)
(1 30 8 37)(2 29 9 36)(3 42 10 35)(4 41 11 34)(5 40 12 33)(6 39 13 32)(7 38 14 31)(15 53 22 46)(16 52 23 45)(17 51 24 44)(18 50 25 43)(19 49 26 56)(20 48 27 55)(21 47 28 54)

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

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

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

55 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G7A7B7C14A···14I14J···14U28A···28R
order122222444444477714···1414···1428···28
size112448444565656562222···28···84···4

55 irreducible representations

dim111112222224444
type+++++--+++
imageC1C2C2C4C4D4D7Dic7Dic7D14C7:D4C23:C4C42:C4C23:Dic7C42:3Dic7
kernelC42:3Dic7C23:Dic7C7xC4:1D4C4xC28D4xC14C22xC14C4:1D4C42C2xD4C2xD4C23C14C7C2C1
# reps12122233331212612

Matrix representation of C42:3Dic7 in GL4(F29) generated by

28000
02800
00203
00219
,
20300
21900
00926
00820
,
131500
21600
00418
001425
,
0010
0001
92600
172000
G:=sub<GL(4,GF(29))| [28,0,0,0,0,28,0,0,0,0,20,21,0,0,3,9],[20,21,0,0,3,9,0,0,0,0,9,8,0,0,26,20],[13,2,0,0,15,16,0,0,0,0,4,14,0,0,18,25],[0,0,9,17,0,0,26,20,1,0,0,0,0,1,0,0] >;

C42:3Dic7 in GAP, Magma, Sage, TeX

C_4^2\rtimes_3{\rm Dic}_7
% in TeX

G:=Group("C4^2:3Dic7");
// GroupNames label

G:=SmallGroup(448,102);
// by ID

G=gap.SmallGroup(448,102);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,28,141,219,1571,570,297,136,1684,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^4=c^14=1,d^2=c^7,a*b=b*a,c*a*c^-1=a^-1,d*a*d^-1=a^-1*b,c*b*c^-1=b^-1,d*b*d^-1=a^2*b,d*c*d^-1=c^-1>;
// generators/relations

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