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G = C24.73D4order 128 = 27

28th non-split extension by C24 of D4 acting via D4/C2=C22

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

Aliases: C24.73D4, C4.60(C4×D4), C4⋊C4.309D4, C22⋊Q811C4, (C22×C4).287D4, C23.765(C2×D4), C4.126(C4⋊D4), C22.4Q1611C2, C22.90C22≀C2, C2.5(D4.7D4), C22.53(C4○D8), (C22×C8).32C22, C23.79(C22⋊C4), (C23×C4).255C22, (C22×Q8).12C22, (C22×C4).1360C23, C2.3(C23.20D4), C22.63(C8.C22), C2.20(C23.38D4), C2.24(C23.24D4), C2.12(C23.23D4), C22.85(C22.D4), C4⋊C4.69(C2×C4), (C2×Q8⋊C4)⋊4C2, (C2×C4).993(C2×D4), (C2×Q8).62(C2×C4), (C2×C22⋊Q8).7C2, (C2×C22⋊C8).23C2, (C2×C4).756(C4○D4), (C2×C4⋊C4).762C22, (C22×C4).277(C2×C4), (C2×C4).378(C22×C4), (C2×C4).191(C22⋊C4), (C2×C42⋊C2).20C2, C22.264(C2×C22⋊C4), SmallGroup(128,605)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C24.73D4
C1C2C4C2×C4C22×C4C23×C4C2×C42⋊C2 — C24.73D4
C1C2C2×C4 — C24.73D4
C1C23C23×C4 — C24.73D4
C1C2C2C22×C4 — C24.73D4

Generators and relations for C24.73D4
 G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e4=f2=d, ab=ba, eae-1=ac=ca, ad=da, faf-1=acd, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=be3 >

Subgroups: 348 in 180 conjugacy classes, 64 normal (20 characteristic)
C1, C2 [×3], C2 [×4], C2 [×2], C4 [×4], C4 [×10], C22 [×3], C22 [×4], C22 [×10], C8 [×2], C2×C4 [×2], C2×C4 [×6], C2×C4 [×26], Q8 [×6], C23, C23 [×2], C23 [×6], C42 [×4], C22⋊C4 [×8], C4⋊C4 [×6], C4⋊C4 [×7], C2×C8 [×6], C22×C4 [×2], C22×C4 [×4], C22×C4 [×8], C2×Q8 [×2], C2×Q8 [×5], C24, C22⋊C8 [×2], Q8⋊C4 [×4], C2×C42, C2×C22⋊C4 [×2], C2×C4⋊C4, C2×C4⋊C4 [×2], C2×C4⋊C4, C42⋊C2 [×4], C22⋊Q8 [×4], C22⋊Q8 [×2], C22×C8 [×2], C23×C4, C22×Q8, C22.4Q16 [×2], C2×C22⋊C8, C2×Q8⋊C4 [×2], C2×C42⋊C2, C2×C22⋊Q8, C24.73D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×8], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×4], C4○D4 [×2], C2×C22⋊C4, C4×D4 [×2], C22≀C2, C4⋊D4 [×2], C22.D4, C4○D8 [×2], C8.C22 [×2], C23.23D4, C23.24D4, C23.38D4, D4.7D4 [×2], C23.20D4 [×2], C24.73D4

Smallest permutation representation of C24.73D4
On 64 points
Generators in S64
(1 17)(2 26)(3 19)(4 28)(5 21)(6 30)(7 23)(8 32)(9 64)(10 41)(11 58)(12 43)(13 60)(14 45)(15 62)(16 47)(18 56)(20 50)(22 52)(24 54)(25 55)(27 49)(29 51)(31 53)(33 59)(34 44)(35 61)(36 46)(37 63)(38 48)(39 57)(40 42)
(1 21)(2 22)(3 23)(4 24)(5 17)(6 18)(7 19)(8 20)(9 48)(10 41)(11 42)(12 43)(13 44)(14 45)(15 46)(16 47)(25 51)(26 52)(27 53)(28 54)(29 55)(30 56)(31 49)(32 50)(33 59)(34 60)(35 61)(36 62)(37 63)(38 64)(39 57)(40 58)
(1 55)(2 56)(3 49)(4 50)(5 51)(6 52)(7 53)(8 54)(9 38)(10 39)(11 40)(12 33)(13 34)(14 35)(15 36)(16 37)(17 25)(18 26)(19 27)(20 28)(21 29)(22 30)(23 31)(24 32)(41 57)(42 58)(43 59)(44 60)(45 61)(46 62)(47 63)(48 64)
(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 46 5 42)(2 10 6 14)(3 44 7 48)(4 16 8 12)(9 23 13 19)(11 21 15 17)(18 45 22 41)(20 43 24 47)(25 40 29 36)(26 61 30 57)(27 38 31 34)(28 59 32 63)(33 50 37 54)(35 56 39 52)(49 60 53 64)(51 58 55 62)

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

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

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

38 conjugacy classes

class 1 2A···2G2H2I4A···4H4I···4P4Q4R4S4T8A···8H
order12···2224···44···444448···8
size11···1442···24···488884···4

38 irreducible representations

dim1111111222224
type+++++++++-
imageC1C2C2C2C2C2C4D4D4D4C4○D4C4○D8C8.C22
kernelC24.73D4C22.4Q16C2×C22⋊C8C2×Q8⋊C4C2×C42⋊C2C2×C22⋊Q8C22⋊Q8C4⋊C4C22×C4C24C2×C4C22C22
# reps1212118431482

Matrix representation of C24.73D4 in GL5(𝔽17)

10000
01000
001600
000160
00011
,
160000
016000
001600
000160
000016
,
10000
01000
00100
000160
000016
,
10000
016000
001600
00010
00001
,
130000
015000
00800
0001615
00011
,
160000
00800
02000
000139
00044

G:=sub<GL(5,GF(17))| [1,0,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,16,1,0,0,0,0,1],[16,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,16],[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],[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],[13,0,0,0,0,0,15,0,0,0,0,0,8,0,0,0,0,0,16,1,0,0,0,15,1],[16,0,0,0,0,0,0,2,0,0,0,8,0,0,0,0,0,0,13,4,0,0,0,9,4] >;

C24.73D4 in GAP, Magma, Sage, TeX

C_2^4._{73}D_4
% in TeX

G:=Group("C2^4.73D4");
// GroupNames label

G:=SmallGroup(128,605);
// by ID

G=gap.SmallGroup(128,605);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,288,422,352,1018,248,2028,124]);
// Polycyclic

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

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