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G = C24.155D4order 128 = 27

10th non-split extension by C24 of D4 acting via D4/C22=C2

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

Aliases: C24.155D4, C23.17Q16, C23.41SD16, C4.53C22≀C2, Q83(C22⋊C4), (C2×Q8).206D4, (C22×Q8)⋊14C4, (Q8×C23).3C2, C2.1(Q8⋊D4), (C22×C4).263D4, C23.738(C2×D4), (C22×C8).6C22, C22.23(C2×Q16), C22.73C22≀C2, C223(Q8⋊C4), C2.1(C22⋊Q16), C22.45(C2×SD16), C2.12(C243C4), C23.7Q8.5C2, (C23×C4).232C22, C23.195(C22⋊C4), (C22×C4).1321C23, C22.46(C8.C22), (C22×Q8).380C22, C2.17(C23.38D4), C4.2(C2×C22⋊C4), (C2×Q8⋊C4)⋊1C2, (C2×C4).1311(C2×D4), (C2×C22⋊C8).12C2, (C2×C4⋊C4).29C22, (C2×Q8).183(C2×C4), C2.17(C2×Q8⋊C4), (C22×C4).261(C2×C4), (C2×C4).359(C22×C4), (C2×C4).120(C22⋊C4), C22.240(C2×C22⋊C4), SmallGroup(128,519)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C24.155D4
C1C2C22C23C22×C4C23×C4Q8×C23 — C24.155D4
C1C2C2×C4 — C24.155D4
C1C23C23×C4 — C24.155D4
C1C2C2C22×C4 — C24.155D4

Generators and relations for C24.155D4
 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=ce3 >

Subgroups: 540 in 296 conjugacy classes, 84 normal (18 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×4], C4 [×12], C22 [×3], C22 [×8], C22 [×12], C8 [×2], C2×C4 [×2], C2×C4 [×6], C2×C4 [×44], Q8 [×8], Q8 [×28], C23, C23 [×6], C23 [×4], C22⋊C4 [×2], C4⋊C4 [×4], C2×C8 [×6], C22×C4 [×2], C22×C4 [×4], C22×C4 [×20], C2×Q8 [×12], C2×Q8 [×50], C24, C2.C42, C22⋊C8 [×2], Q8⋊C4 [×8], C2×C22⋊C4, C2×C4⋊C4 [×2], C22×C8 [×2], C23×C4, C23×C4, C22×Q8 [×6], C22×Q8 [×11], C23.7Q8, C2×C22⋊C8, C2×Q8⋊C4 [×4], Q8×C23, C24.155D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×12], C23, C22⋊C4 [×12], SD16 [×2], Q16 [×2], C22×C4, C2×D4 [×6], Q8⋊C4 [×4], C2×C22⋊C4 [×3], C22≀C2 [×4], C2×SD16, C2×Q16, C8.C22 [×2], C243C4, C2×Q8⋊C4, C23.38D4, Q8⋊D4 [×2], C22⋊Q16 [×2], C24.155D4

Smallest permutation representation of C24.155D4
On 64 points
Generators in S64
(1 23)(2 9)(3 17)(4 11)(5 19)(6 13)(7 21)(8 15)(10 61)(12 63)(14 57)(16 59)(18 62)(20 64)(22 58)(24 60)(25 46)(26 34)(27 48)(28 36)(29 42)(30 38)(31 44)(32 40)(33 53)(35 55)(37 49)(39 51)(41 56)(43 50)(45 52)(47 54)
(1 59)(2 60)(3 61)(4 62)(5 63)(6 64)(7 57)(8 58)(9 24)(10 17)(11 18)(12 19)(13 20)(14 21)(15 22)(16 23)(25 53)(26 54)(27 55)(28 56)(29 49)(30 50)(31 51)(32 52)(33 46)(34 47)(35 48)(36 41)(37 42)(38 43)(39 44)(40 45)
(1 23)(2 24)(3 17)(4 18)(5 19)(6 20)(7 21)(8 22)(9 60)(10 61)(11 62)(12 63)(13 64)(14 57)(15 58)(16 59)(25 33)(26 34)(27 35)(28 36)(29 37)(30 38)(31 39)(32 40)(41 56)(42 49)(43 50)(44 51)(45 52)(46 53)(47 54)(48 55)
(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 39 23 31)(2 26 24 34)(3 37 17 29)(4 32 18 40)(5 35 19 27)(6 30 20 38)(7 33 21 25)(8 28 22 36)(9 47 60 54)(10 49 61 42)(11 45 62 52)(12 55 63 48)(13 43 64 50)(14 53 57 46)(15 41 58 56)(16 51 59 44)

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

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

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

38 conjugacy classes

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

38 irreducible representations

dim111111222224
type++++++++--
imageC1C2C2C2C2C4D4D4D4SD16Q16C8.C22
kernelC24.155D4C23.7Q8C2×C22⋊C8C2×Q8⋊C4Q8×C23C22×Q8C22×C4C2×Q8C24C23C23C22
# reps111418381442

Matrix representation of C24.155D4 in GL6(𝔽17)

1600000
0160000
001000
000100
0000160
000071
,
100000
010000
001000
000100
0000160
0000016
,
1600000
0160000
001000
000100
0000160
0000016
,
1600000
0160000
0016000
0001600
000010
000001
,
3140000
330000
0012500
00121200
00001015
000077
,
400000
0130000
000100
001000
00001015
000087

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,7,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[3,3,0,0,0,0,14,3,0,0,0,0,0,0,12,12,0,0,0,0,5,12,0,0,0,0,0,0,10,7,0,0,0,0,15,7],[4,0,0,0,0,0,0,13,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,10,8,0,0,0,0,15,7] >;

C24.155D4 in GAP, Magma, Sage, TeX

C_2^4._{155}D_4
% in TeX

G:=Group("C2^4.155D4");
// GroupNames label

G:=SmallGroup(128,519);
// by ID

G=gap.SmallGroup(128,519);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,456,422,2019,1018,248]);
// 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=c*e^3>;
// generators/relations

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