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G = C4315C2order 128 = 27

15th semidirect product of C43 and C2 acting faithfully

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

Aliases: C4315C2, C4241D4, C23.767C24, C24.128C23, C41(C41D4), C22.473(C22×D4), (C2×C42).1098C22, (C22×C4).1487C23, (C22×D4).317C22, (C2×C41D4)⋊13C2, (C2×C4).837(C2×D4), C2.18(C2×C41D4), SmallGroup(128,1599)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C4315C2
C1C2C22C23C22×C4C2×C42C43 — C4315C2
C1C23 — C4315C2
C1C23 — C4315C2
C1C23 — C4315C2

Generators and relations for C4315C2
 G = < a,b,c,d | a4=b4=c4=d2=1, ab=ba, ac=ca, dad=a-1, bc=cb, dbd=b-1, dcd=c-1 >

Subgroups: 1476 in 636 conjugacy classes, 180 normal (4 characteristic)
C1, C2 [×7], C2 [×8], C4 [×28], C22 [×7], C22 [×56], C2×C4 [×42], D4 [×112], C23, C23 [×56], C42 [×28], C22×C4 [×7], C2×D4 [×168], C24 [×8], C2×C42 [×7], C41D4 [×56], C22×D4 [×28], C43, C2×C41D4 [×14], C4315C2
Quotients: C1, C2 [×15], C22 [×35], D4 [×28], C23 [×15], C2×D4 [×42], C24, C41D4 [×28], C22×D4 [×7], C2×C41D4 [×7], C4315C2

Smallest permutation representation of C4315C2
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 49 41 61)(2 50 42 62)(3 51 43 63)(4 52 44 64)(5 60 20 12)(6 57 17 9)(7 58 18 10)(8 59 19 11)(13 34 25 21)(14 35 26 22)(15 36 27 23)(16 33 28 24)(29 53 45 38)(30 54 46 39)(31 55 47 40)(32 56 48 37)
(1 47 11 25)(2 48 12 26)(3 45 9 27)(4 46 10 28)(5 22 50 37)(6 23 51 38)(7 24 52 39)(8 21 49 40)(13 41 31 59)(14 42 32 60)(15 43 29 57)(16 44 30 58)(17 36 63 53)(18 33 64 54)(19 34 61 55)(20 35 62 56)
(1 34)(2 33)(3 36)(4 35)(5 30)(6 29)(7 32)(8 31)(9 53)(10 56)(11 55)(12 54)(13 49)(14 52)(15 51)(16 50)(17 45)(18 48)(19 47)(20 46)(21 41)(22 44)(23 43)(24 42)(25 61)(26 64)(27 63)(28 62)(37 58)(38 57)(39 60)(40 59)

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

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

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

44 conjugacy classes

class 1 2A···2G2H···2O4A···4AB
order12···22···24···4
size11···18···82···2

44 irreducible representations

dim1112
type++++
imageC1C2C2D4
kernelC4315C2C43C2×C41D4C42
# reps111428

Matrix representation of C4315C2 in GL6(ℤ)

010000
-100000
00-1200
00-1100
0000-10
00000-1
,
-100000
0-10000
001-200
001-100
00000-1
000010
,
010000
-100000
001-200
001-100
000001
0000-10
,
0-10000
-100000
001000
001-100
0000-10
000001

G:=sub<GL(6,Integers())| [0,-1,0,0,0,0,1,0,0,0,0,0,0,0,-1,-1,0,0,0,0,2,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1],[-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,1,0,0,0,0,-2,-1,0,0,0,0,0,0,0,1,0,0,0,0,-1,0],[0,-1,0,0,0,0,1,0,0,0,0,0,0,0,1,1,0,0,0,0,-2,-1,0,0,0,0,0,0,0,-1,0,0,0,0,1,0],[0,-1,0,0,0,0,-1,0,0,0,0,0,0,0,1,1,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1] >;

C4315C2 in GAP, Magma, Sage, TeX

C_4^3\rtimes_{15}C_2
% in TeX

G:=Group("C4^3:15C2");
// GroupNames label

G:=SmallGroup(128,1599);
// by ID

G=gap.SmallGroup(128,1599);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,253,120,758,184,2019,248]);
// 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^-1,b*c=c*b,d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations

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