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G = C24.71D4order 128 = 27

26th non-split extension by C24 of D4 acting via D4/C2=C22

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

Aliases: C24.71D4, C22⋊C88C4, C4⋊C4.299D4, C4.138(C4×D4), C4.3(C22⋊Q8), C2.2(D4⋊D4), (C22×C4).50Q8, C23.29(C4⋊C4), (C22×C4).681D4, C23.759(C2×D4), C22.4Q1634C2, C2.2(D4.7D4), C22.79C22≀C2, C22.48(C4○D8), C22.69(C8⋊C22), (C22×C8).312C22, (C23×C4).250C22, C23.7Q8.14C2, (C22×C4).1351C23, C2.2(C23.20D4), C2.2(C23.19D4), C22.58(C8.C22), C2.10(C23.8Q8), C2.12(M4(2)⋊C4), C2.11(C23.25D4), C22.83(C22.D4), (C2×C2.D8)⋊3C2, (C2×C4.Q8)⋊16C2, (C2×C4).91(C4⋊C4), (C2×C8).106(C2×C4), (C2×C4).981(C2×D4), (C2×C4).201(C2×Q8), (C2×C4⋊C4).53C22, (C2×C22⋊C8).33C2, C22.111(C2×C4⋊C4), (C2×C4).747(C4○D4), (C22×C4).273(C2×C4), (C2×C4).550(C22×C4), (C2×C42⋊C2).19C2, SmallGroup(128,586)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C24.71D4
C1C2C22C23C22×C4C23×C4C2×C42⋊C2 — C24.71D4
C1C2C2×C4 — C24.71D4
C1C23C23×C4 — C24.71D4
C1C2C2C22×C4 — C24.71D4

Generators and relations for C24.71D4
 G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e4=d, f2=c, eae-1=ab=ba, ac=ca, ad=da, faf-1=abd, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=de3 >

Subgroups: 308 in 158 conjugacy classes, 64 normal (44 characteristic)
C1, C2 [×7], C2 [×2], C4 [×4], C4 [×8], C22 [×7], C22 [×10], C8 [×4], C2×C4 [×6], C2×C4 [×2], C2×C4 [×24], C23, C23 [×2], C23 [×6], C42 [×4], C22⋊C4 [×6], C4⋊C4 [×4], C4⋊C4 [×6], C2×C8 [×4], C2×C8 [×4], C22×C4 [×2], C22×C4 [×4], C22×C4 [×8], C24, C2.C42, C22⋊C8 [×4], C4.Q8 [×2], C2.D8 [×2], C2×C42, C2×C22⋊C4 [×2], C2×C4⋊C4 [×4], C42⋊C2 [×4], C22×C8 [×2], C23×C4, C22.4Q16 [×2], C23.7Q8, C2×C22⋊C8, C2×C4.Q8, C2×C2.D8, C2×C42⋊C2, C24.71D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×6], Q8 [×2], C23, C4⋊C4 [×4], C22×C4, C2×D4 [×3], C2×Q8, C4○D4 [×2], C2×C4⋊C4, C4×D4 [×2], C22≀C2, C22⋊Q8 [×2], C22.D4, C4○D8 [×2], C8⋊C22, C8.C22, C23.8Q8, C23.25D4, M4(2)⋊C4, D4⋊D4, D4.7D4, C23.19D4, C23.20D4, C24.71D4

Smallest permutation representation of C24.71D4
On 64 points
Generators in S64
(2 34)(4 36)(6 38)(8 40)(9 24)(10 14)(11 18)(12 16)(13 20)(15 22)(17 21)(19 23)(25 46)(26 30)(27 48)(28 32)(29 42)(31 44)(41 45)(43 47)(49 63)(51 57)(53 59)(55 61)
(1 33)(2 34)(3 35)(4 36)(5 37)(6 38)(7 39)(8 40)(9 20)(10 21)(11 22)(12 23)(13 24)(14 17)(15 18)(16 19)(25 42)(26 43)(27 44)(28 45)(29 46)(30 47)(31 48)(32 41)(49 63)(50 64)(51 57)(52 58)(53 59)(54 60)(55 61)(56 62)
(1 54)(2 55)(3 56)(4 49)(5 50)(6 51)(7 52)(8 53)(9 42)(10 43)(11 44)(12 45)(13 46)(14 47)(15 48)(16 41)(17 30)(18 31)(19 32)(20 25)(21 26)(22 27)(23 28)(24 29)(33 60)(34 61)(35 62)(36 63)(37 64)(38 57)(39 58)(40 59)
(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 18 54 31)(2 17 55 30)(3 24 56 29)(4 23 49 28)(5 22 50 27)(6 21 51 26)(7 20 52 25)(8 19 53 32)(9 58 42 39)(10 57 43 38)(11 64 44 37)(12 63 45 36)(13 62 46 35)(14 61 47 34)(15 60 48 33)(16 59 41 40)

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

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

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

38 conjugacy classes

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

38 irreducible representations

dim1111111122222244
type+++++++++-++-
imageC1C2C2C2C2C2C2C4D4D4Q8D4C4○D4C4○D8C8⋊C22C8.C22
kernelC24.71D4C22.4Q16C23.7Q8C2×C22⋊C8C2×C4.Q8C2×C2.D8C2×C42⋊C2C22⋊C8C4⋊C4C22×C4C22×C4C24C2×C4C22C22C22
# reps1211111841214811

Matrix representation of C24.71D4 in GL6(𝔽17)

100000
0160000
001000
000100
000010
0000016
,
100000
010000
001000
000100
0000160
0000016
,
1600000
0160000
0016000
0001600
000010
000001
,
1600000
0160000
001000
000100
000010
000001
,
900000
020000
002900
0071500
0000016
0000160
,
0150000
900000
008200
0010900
000001
000010

G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16],[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],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[9,0,0,0,0,0,0,2,0,0,0,0,0,0,2,7,0,0,0,0,9,15,0,0,0,0,0,0,0,16,0,0,0,0,16,0],[0,9,0,0,0,0,15,0,0,0,0,0,0,0,8,10,0,0,0,0,2,9,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

C24.71D4 in GAP, Magma, Sage, TeX

C_2^4._{71}D_4
% in TeX

G:=Group("C2^4.71D4");
// GroupNames label

G:=SmallGroup(128,586);
// by ID

G=gap.SmallGroup(128,586);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,232,422,352,2019,1018,248]);
// 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=a*b=b*a,a*c=c*a,a*d=d*a,f*a*f^-1=a*b*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=d*e^3>;
// generators/relations

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