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G = C4314C2order 128 = 27

14th semidirect product of C43 and C2 acting faithfully

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

Aliases: C4314C2, C42.346D4, C24.125C23, C23.761C24, C4.16(C4.4D4), (C22×C4).266C23, C22.471(C22×D4), (C2×C42).1017C22, (C22×D4).316C22, (C22×Q8).251C22, C24.C22187C2, C24.3C22.82C2, C23.67C23111C2, C2.C42.456C22, C2.61(C22.26C24), C2.115(C23.36C23), (C2×C4).690(C2×D4), (C2×C42.C2)⋊30C2, C2.35(C2×C4.4D4), (C2×C4).531(C4○D4), (C2×C4⋊C4).564C22, (C2×C4.4D4).36C2, C22.602(C2×C4○D4), (C2×C22⋊C4).370C22, SmallGroup(128,1593)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C4314C2
C1C2C22C23C22×C4C2×C42C43 — C4314C2
C1C23 — C4314C2
C1C23 — C4314C2
C1C23 — C4314C2

Generators and relations for C4314C2
 G = < a,b,c,d | a4=b4=c4=d2=1, ab=ba, ac=ca, dad=ac2, bc=cb, dbd=b-1c2, dcd=a2c >

Subgroups: 484 in 262 conjugacy classes, 108 normal (12 characteristic)
C1, C2, C2 [×6], C2 [×2], C4 [×4], C4 [×18], C22, C22 [×6], C22 [×14], C2×C4 [×18], C2×C4 [×30], D4 [×4], Q8 [×4], C23, C23 [×14], C42 [×4], C42 [×12], C22⋊C4 [×24], C4⋊C4 [×12], C22×C4, C22×C4 [×12], C2×D4 [×6], C2×Q8 [×6], C24 [×2], C2.C42 [×8], C2×C42, C2×C42 [×6], C2×C22⋊C4 [×12], C2×C4⋊C4 [×6], C4.4D4 [×4], C42.C2 [×4], C22×D4, C22×Q8, C43, C24.C22 [×8], C24.3C22 [×2], C23.67C23 [×2], C2×C4.4D4, C2×C42.C2, C4314C2
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×12], C24, C4.4D4 [×4], C22×D4, C2×C4○D4 [×6], C2×C4.4D4, C23.36C23 [×4], C22.26C24 [×2], C4314C2

Smallest permutation representation of C4314C2
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 31 23 45)(2 32 24 46)(3 29 21 47)(4 30 22 48)(5 44 36 26)(6 41 33 27)(7 42 34 28)(8 43 35 25)(9 59 51 17)(10 60 52 18)(11 57 49 19)(12 58 50 20)(13 63 55 39)(14 64 56 40)(15 61 53 37)(16 62 54 38)
(1 13 9 41)(2 14 10 42)(3 15 11 43)(4 16 12 44)(5 48 38 20)(6 45 39 17)(7 46 40 18)(8 47 37 19)(21 53 49 25)(22 54 50 26)(23 55 51 27)(24 56 52 28)(29 61 57 35)(30 62 58 36)(31 63 59 33)(32 64 60 34)
(2 10)(4 12)(5 34)(6 61)(7 36)(8 63)(13 15)(14 44)(16 42)(17 31)(18 60)(19 29)(20 58)(22 50)(24 52)(25 27)(26 56)(28 54)(30 48)(32 46)(33 37)(35 39)(38 64)(40 62)(41 43)(45 59)(47 57)(53 55)

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

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

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

44 conjugacy classes

class 1 2A···2G2H2I4A···4AB4AC···4AH
order12···2224···44···4
size11···1882···28···8

44 irreducible representations

dim111111122
type++++++++
imageC1C2C2C2C2C2C2D4C4○D4
kernelC4314C2C43C24.C22C24.3C22C23.67C23C2×C4.4D4C2×C42.C2C42C2×C4
# reps1182211424

Matrix representation of C4314C2 in GL6(𝔽5)

010000
400000
000400
001000
000030
000003
,
030000
200000
003000
000300
000010
000001
,
010000
400000
000100
004000
000001
000010
,
100000
040000
001000
000400
000010
000004

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

C4314C2 in GAP, Magma, Sage, TeX

C_4^3\rtimes_{14}C_2
% in TeX

G:=Group("C4^3:14C2");
// GroupNames label

G:=SmallGroup(128,1593);
// by ID

G=gap.SmallGroup(128,1593);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,253,456,758,268,2019,80]);
// Polycyclic

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

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