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G = C24.157D4order 128 = 27

12nd non-split extension by C24 of D4 acting via D4/C22=C2

p-group, metabelian, nilpotent (class 3), monomial

Aliases: C24.157D4, C23.18Q16, C23.42SD16, C22.4Q166C2, C23.748(C2×D4), (C22×C4).278D4, C22.26(C2×Q16), (C22×C8).17C22, C22.50(C2×SD16), C4.19(C42⋊C2), C22.63(C8⋊C22), (C23×C4).244C22, C23.7Q8.10C2, C23.200(C22⋊C4), (C22×C4).1335C23, C22.24(Q8⋊C4), C2.1(C23.48D4), C2.1(C23.46D4), C2.12(C23.34D4), C4.104(C22.D4), C2.19(C23.37D4), C22.79(C22.D4), (C2×C4⋊C4)⋊29C4, C4⋊C4.195(C2×C4), (C2×C4).1325(C2×D4), (C22×C4⋊C4).14C2, (C2×C4⋊C4).42C22, (C2×C22⋊C8).18C2, C2.19(C2×Q8⋊C4), (C2×C4).741(C4○D4), (C2×C4).368(C22×C4), (C22×C4).267(C2×C4), (C2×C4).127(C22⋊C4), C22.258(C2×C22⋊C4), SmallGroup(128,556)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C24.157D4
C1C2C22C2×C4C22×C4C2×C4⋊C4C22×C4⋊C4 — C24.157D4
C1C2C2×C4 — C24.157D4
C1C23C23×C4 — C24.157D4
C1C2C2C22×C4 — C24.157D4

Generators and relations for C24.157D4
 G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e4=d, f2=c, eae-1=faf-1=ab=ba, ac=ca, ad=da, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=bce3 >

Subgroups: 348 in 172 conjugacy classes, 68 normal (18 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×4], C4 [×8], C22 [×3], C22 [×8], C22 [×12], C8 [×2], C2×C4 [×2], C2×C4 [×6], C2×C4 [×32], C23, C23 [×6], C23 [×4], C22⋊C4 [×2], C4⋊C4 [×4], C4⋊C4 [×10], C2×C8 [×6], C22×C4 [×2], C22×C4 [×4], C22×C4 [×16], C24, C2.C42, C22⋊C8 [×2], C2×C22⋊C4, C2×C4⋊C4 [×8], C2×C4⋊C4 [×3], C22×C8 [×2], C23×C4, C23×C4, C22.4Q16 [×4], C23.7Q8, C2×C22⋊C8, C22×C4⋊C4, C24.157D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], SD16 [×2], Q16 [×2], C22×C4, C2×D4 [×2], C4○D4 [×4], Q8⋊C4 [×4], C2×C22⋊C4, C42⋊C2 [×2], C22.D4 [×4], C2×SD16, C2×Q16, C8⋊C22 [×2], C23.34D4, C2×Q8⋊C4, C23.37D4, C23.46D4 [×2], C23.48D4 [×2], C24.157D4

Smallest permutation representation of C24.157D4
On 64 points
Generators in S64
(1 63)(2 27)(3 57)(4 29)(5 59)(6 31)(7 61)(8 25)(9 28)(10 58)(11 30)(12 60)(13 32)(14 62)(15 26)(16 64)(17 47)(18 50)(19 41)(20 52)(21 43)(22 54)(23 45)(24 56)(33 51)(34 42)(35 53)(36 44)(37 55)(38 46)(39 49)(40 48)
(1 15)(2 16)(3 9)(4 10)(5 11)(6 12)(7 13)(8 14)(17 39)(18 40)(19 33)(20 34)(21 35)(22 36)(23 37)(24 38)(25 62)(26 63)(27 64)(28 57)(29 58)(30 59)(31 60)(32 61)(41 51)(42 52)(43 53)(44 54)(45 55)(46 56)(47 49)(48 50)
(1 63)(2 64)(3 57)(4 58)(5 59)(6 60)(7 61)(8 62)(9 28)(10 29)(11 30)(12 31)(13 32)(14 25)(15 26)(16 27)(17 47)(18 48)(19 41)(20 42)(21 43)(22 44)(23 45)(24 46)(33 51)(34 52)(35 53)(36 54)(37 55)(38 56)(39 49)(40 50)
(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)(17 21)(18 22)(19 23)(20 24)(25 29)(26 30)(27 31)(28 32)(33 37)(34 38)(35 39)(36 40)(41 45)(42 46)(43 47)(44 48)(49 53)(50 54)(51 55)(52 56)(57 61)(58 62)(59 63)(60 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 34 63 52)(2 45 64 23)(3 40 57 50)(4 43 58 21)(5 38 59 56)(6 41 60 19)(7 36 61 54)(8 47 62 17)(9 18 28 48)(10 53 29 35)(11 24 30 46)(12 51 31 33)(13 22 32 44)(14 49 25 39)(15 20 26 42)(16 55 27 37)

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

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

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

38 conjugacy classes

class 1 2A···2G2H2I2J2K4A4B4C4D4E···4N4O4P4Q4R8A···8H
order12···2222244444···444448···8
size11···1222222224···488884···4

38 irreducible representations

dim111111222224
type+++++++-+
imageC1C2C2C2C2C4D4D4C4○D4SD16Q16C8⋊C22
kernelC24.157D4C22.4Q16C23.7Q8C2×C22⋊C8C22×C4⋊C4C2×C4⋊C4C22×C4C24C2×C4C23C23C22
# reps141118318442

Matrix representation of C24.157D4 in GL5(𝔽17)

10000
01000
001600
000160
000016
,
10000
016000
001600
00010
00001
,
160000
01000
00100
000160
000016
,
10000
01000
00100
000160
000016
,
40000
001300
013000
000314
00033
,
130000
00100
01000
0001610
000101

G:=sub<GL(5,GF(17))| [1,0,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,16],[1,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,1,0,0,0,0,0,1],[16,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,16],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,16],[4,0,0,0,0,0,0,13,0,0,0,13,0,0,0,0,0,0,3,3,0,0,0,14,3],[13,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,16,10,0,0,0,10,1] >;

C24.157D4 in GAP, Magma, Sage, TeX

C_2^4._{157}D_4
% in TeX

G:=Group("C2^4.157D4");
// GroupNames label

G:=SmallGroup(128,556);
// by ID

G=gap.SmallGroup(128,556);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,422,394,2804,718,172]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^4=d,f^2=c,e*a*e^-1=f*a*f^-1=a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=b*c*e^3>;
// generators/relations

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