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G = C24.69D4order 128 = 27

24th non-split extension by C24 of D4 acting via D4/C2=C22

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

Aliases: C24.69D4, C42⋊C218C4, C22.4Q167C2, (C22×C4).678D4, C23.749(C2×D4), C22.45(C4○D8), (C22×C8).18C22, C4.20(C42⋊C2), C22.64(C8⋊C22), C23.76(C22⋊C4), (C23×C4).245C22, C23.7Q8.11C2, (C22×C4).1336C23, C2.1(C23.20D4), C2.1(C23.19D4), C22.53(C8.C22), C2.23(C23.36D4), C2.23(C23.24D4), C2.13(C23.34D4), C4.105(C22.D4), C22.80(C22.D4), C4⋊C4.196(C2×C4), (C2×C4).1326(C2×D4), (C2×C22⋊C8).19C2, (C2×C4⋊C4).43C22, (C2×C4).742(C4○D4), (C22×C4).268(C2×C4), (C2×C4).369(C22×C4), (C2×C4).335(C22⋊C4), (C2×C42⋊C2).17C2, C22.259(C2×C22⋊C4), SmallGroup(128,557)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C24.69D4
C1C2C22C2×C4C22×C4C2×C4⋊C4C2×C42⋊C2 — C24.69D4
C1C2C2×C4 — C24.69D4
C1C23C23×C4 — C24.69D4
C1C2C2C22×C4 — C24.69D4

Generators and relations for C24.69D4
 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=bcde3 >

Subgroups: 300 in 150 conjugacy classes, 60 normal (22 characteristic)
C1, C2 [×3], C2 [×4], C2 [×2], C4 [×2], C4 [×2], C4 [×8], C22 [×3], C22 [×4], C22 [×10], C8 [×2], C2×C4 [×2], C2×C4 [×6], C2×C4 [×24], C23, C23 [×2], C23 [×6], C42 [×4], C22⋊C4 [×6], C4⋊C4 [×4], C4⋊C4 [×6], C2×C8 [×6], C22×C4 [×2], C22×C4 [×4], C22×C4 [×8], C24, C2.C42, C22⋊C8 [×2], C2×C42, C2×C22⋊C4 [×2], C2×C4⋊C4 [×2], C2×C4⋊C4 [×2], C42⋊C2 [×4], C42⋊C2 [×2], C22×C8 [×2], C23×C4, C22.4Q16 [×4], C23.7Q8, C2×C22⋊C8, C2×C42⋊C2, C24.69D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×2], C4○D4 [×4], C2×C22⋊C4, C42⋊C2 [×2], C22.D4 [×4], C4○D8 [×2], C8⋊C22, C8.C22, C23.34D4, C23.24D4, C23.36D4, C23.19D4 [×2], C23.20D4 [×2], C24.69D4

Smallest permutation representation of C24.69D4
On 64 points
Generators in S64
(2 61)(4 63)(6 57)(8 59)(9 46)(10 14)(11 48)(12 16)(13 42)(15 44)(17 21)(18 29)(19 23)(20 31)(22 25)(24 27)(26 30)(28 32)(33 54)(35 56)(37 50)(39 52)(41 45)(43 47)
(1 60)(2 61)(3 62)(4 63)(5 64)(6 57)(7 58)(8 59)(9 42)(10 43)(11 44)(12 45)(13 46)(14 47)(15 48)(16 41)(17 32)(18 25)(19 26)(20 27)(21 28)(22 29)(23 30)(24 31)(33 54)(34 55)(35 56)(36 49)(37 50)(38 51)(39 52)(40 53)
(1 36)(2 37)(3 38)(4 39)(5 40)(6 33)(7 34)(8 35)(9 25)(10 26)(11 27)(12 28)(13 29)(14 30)(15 31)(16 32)(17 41)(18 42)(19 43)(20 44)(21 45)(22 46)(23 47)(24 48)(49 60)(50 61)(51 62)(52 63)(53 64)(54 57)(55 58)(56 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 29 36 13)(2 45 37 21)(3 27 38 11)(4 43 39 19)(5 25 40 9)(6 41 33 17)(7 31 34 15)(8 47 35 23)(10 52 26 63)(12 50 28 61)(14 56 30 59)(16 54 32 57)(18 53 42 64)(20 51 44 62)(22 49 46 60)(24 55 48 58)

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

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

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

38 conjugacy classes

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

38 irreducible representations

dim111111222244
type++++++++-
imageC1C2C2C2C2C4D4D4C4○D4C4○D8C8⋊C22C8.C22
kernelC24.69D4C22.4Q16C23.7Q8C2×C22⋊C8C2×C42⋊C2C42⋊C2C22×C4C24C2×C4C22C22C22
# reps141118318811

Matrix representation of C24.69D4 in GL6(𝔽17)

100000
010000
001000
0071600
000010
0000016
,
100000
010000
0016000
0001600
000010
000001
,
1600000
0160000
0016000
0001600
0000160
0000016
,
100000
010000
001000
000100
0000160
0000016
,
1020000
1070000
0011900
0011600
000090
0000015
,
750000
7100000
0071500
0081000
0000015
000090

G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,7,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[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],[10,10,0,0,0,0,2,7,0,0,0,0,0,0,11,11,0,0,0,0,9,6,0,0,0,0,0,0,9,0,0,0,0,0,0,15],[7,7,0,0,0,0,5,10,0,0,0,0,0,0,7,8,0,0,0,0,15,10,0,0,0,0,0,0,0,9,0,0,0,0,15,0] >;

C24.69D4 in GAP, Magma, Sage, TeX

C_2^4._{69}D_4
% in TeX

G:=Group("C2^4.69D4");
// GroupNames label

G:=SmallGroup(128,557);
// by ID

G=gap.SmallGroup(128,557);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,456,422,58,2804,718,172]);
// 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=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=b*c*d*e^3>;
// generators/relations

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