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G = C4312C2order 128 = 27

12nd semidirect product of C43 and C2 acting faithfully

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

Aliases: C4312C2, C4240D4, C24.123C23, C23.758C24, C41(C4.4D4), C4.9(C41D4), C22.468(C22×D4), (C22×C4).1265C23, (C2×C42).1093C22, (C22×D4).314C22, (C22×Q8).250C22, C24.3C22102C2, C2.58(C22.26C24), (C2×C4⋊Q8)⋊27C2, (C2×C4).835(C2×D4), C2.17(C2×C41D4), (C2×C41D4).20C2, (C2×C4.4D4)⋊35C2, C2.34(C2×C4.4D4), (C2×C4).674(C4○D4), (C2×C4⋊C4).561C22, C22.599(C2×C4○D4), (C2×C22⋊C4).368C22, SmallGroup(128,1590)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C4312C2
C1C2C22C23C22×C4C2×C42C43 — C4312C2
C1C23 — C4312C2
C1C23 — C4312C2
C1C23 — C4312C2

Generators and relations for C4312C2
 G = < a,b,c,d | a4=b4=c4=d2=1, ab=ba, ac=ca, dad=ac2, bc=cb, dbd=b-1, dcd=a2c >

Subgroups: 772 in 376 conjugacy classes, 132 normal (10 characteristic)
C1, C2, C2 [×6], C2 [×4], C4 [×12], C4 [×12], C22, C22 [×6], C22 [×28], C2×C4 [×26], C2×C4 [×20], D4 [×24], Q8 [×8], C23, C23 [×28], C42 [×12], C42 [×8], C22⋊C4 [×32], C4⋊C4 [×8], C22×C4, C22×C4 [×10], C2×D4 [×36], C2×Q8 [×12], C24 [×4], C2×C42, C2×C42 [×6], C2×C22⋊C4 [×16], C2×C4⋊C4 [×4], C4.4D4 [×16], C41D4 [×4], C4⋊Q8 [×4], C22×D4 [×6], C22×Q8 [×2], C43, C24.3C22 [×8], C2×C4.4D4 [×4], C2×C41D4, C2×C4⋊Q8, C4312C2
Quotients: C1, C2 [×15], C22 [×35], D4 [×12], C23 [×15], C2×D4 [×18], C4○D4 [×8], C24, C4.4D4 [×8], C41D4 [×4], C22×D4 [×3], C2×C4○D4 [×4], C2×C4.4D4 [×2], C2×C41D4, C22.26C24 [×4], C4312C2

Smallest permutation representation of C4312C2
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 59 51 45)(2 60 52 46)(3 57 49 47)(4 58 50 48)(5 44 62 54)(6 41 63 55)(7 42 64 56)(8 43 61 53)(9 31 23 17)(10 32 24 18)(11 29 21 19)(12 30 22 20)(13 33 27 39)(14 34 28 40)(15 35 25 37)(16 36 26 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)
(1 44)(2 13)(3 42)(4 15)(5 59)(6 32)(7 57)(8 30)(9 16)(10 41)(11 14)(12 43)(17 36)(18 63)(19 34)(20 61)(21 28)(22 53)(23 26)(24 55)(25 50)(27 52)(29 40)(31 38)(33 46)(35 48)(37 58)(39 60)(45 62)(47 64)(49 56)(51 54)

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

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

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

44 conjugacy classes

class 1 2A···2G2H2I2J2K4A···4AB4AC4AD4AE4AF
order12···222224···44444
size11···188882···28888

44 irreducible representations

dim11111122
type+++++++
imageC1C2C2C2C2C2D4C4○D4
kernelC4312C2C43C24.3C22C2×C4.4D4C2×C41D4C2×C4⋊Q8C42C2×C4
# reps1184111216

Matrix representation of C4312C2 in GL6(𝔽5)

400000
040000
003000
000300
000004
000010
,
010000
400000
001400
002400
000004
000010
,
100000
010000
002300
004300
000001
000040
,
400000
010000
001400
000400
000040
000001

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

C4312C2 in GAP, Magma, Sage, TeX

C_4^3\rtimes_{12}C_2
% in TeX

G:=Group("C4^3:12C2");
// GroupNames label

G:=SmallGroup(128,1590);
// by ID

G=gap.SmallGroup(128,1590);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,253,456,758,184,2019,80]);
// 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,d*c*d=a^2*c>;
// generators/relations

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