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G = C23.36D8order 128 = 27

7th non-split extension by C23 of D8 acting via D8/D4=C2

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

Aliases: C23.36D8, C24.156D4, C23.33SD16, C22.33(C2×D8), C22.4Q165C2, C23.747(C2×D4), (C22×C4).277D4, (C22×C8).16C22, C22.49(C2×SD16), C4.18(C42⋊C2), C23.7Q8.9C2, (C23×C4).243C22, C2.1(C22.D8), C22.28(D4⋊C4), C23.199(C22⋊C4), (C22×C4).1334C23, C2.1(C23.47D4), C22.52(C8.C22), C2.11(C23.34D4), C2.19(C23.38D4), C4.103(C22.D4), C22.78(C22.D4), (C2×C4⋊C4)⋊28C4, C4⋊C4.194(C2×C4), (C2×C4).1324(C2×D4), C2.19(C2×D4⋊C4), (C22×C4⋊C4).13C2, (C2×C22⋊C8).17C2, (C2×C4⋊C4).41C22, (C2×C4).740(C4○D4), (C2×C4).367(C22×C4), (C22×C4).266(C2×C4), (C2×C4).126(C22⋊C4), C22.257(C2×C22⋊C4), SmallGroup(128,555)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C23.36D8
C1C2C22C2×C4C22×C4C2×C4⋊C4C22×C4⋊C4 — C23.36D8
C1C2C2×C4 — C23.36D8
C1C23C23×C4 — C23.36D8
C1C2C2C22×C4 — C23.36D8

Generators and relations for C23.36D8
 G = < a,b,c,d,e | a2=b2=c2=d8=1, e2=c, dad-1=eae-1=ab=ba, ac=ca, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede-1=bcd-1 >

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, C23.36D8
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], D8 [×2], SD16 [×2], C22×C4, C2×D4 [×2], C4○D4 [×4], D4⋊C4 [×4], C2×C22⋊C4, C42⋊C2 [×2], C22.D4 [×4], C2×D8, C2×SD16, C8.C22 [×2], C23.34D4, C2×D4⋊C4, C23.38D4, C22.D8 [×2], C23.47D4 [×2], C23.36D8

Smallest permutation representation of C23.36D8
On 64 points
Generators in S64
(1 58)(2 32)(3 60)(4 26)(5 62)(6 28)(7 64)(8 30)(9 57)(10 31)(11 59)(12 25)(13 61)(14 27)(15 63)(16 29)(17 52)(18 43)(19 54)(20 45)(21 56)(22 47)(23 50)(24 41)(33 44)(34 55)(35 46)(36 49)(37 48)(38 51)(39 42)(40 53)
(1 10)(2 11)(3 12)(4 13)(5 14)(6 15)(7 16)(8 9)(17 39)(18 40)(19 33)(20 34)(21 35)(22 36)(23 37)(24 38)(25 60)(26 61)(27 62)(28 63)(29 64)(30 57)(31 58)(32 59)(41 51)(42 52)(43 53)(44 54)(45 55)(46 56)(47 49)(48 50)
(1 58)(2 59)(3 60)(4 61)(5 62)(6 63)(7 64)(8 57)(9 30)(10 31)(11 32)(12 25)(13 26)(14 27)(15 28)(16 29)(17 42)(18 43)(19 44)(20 45)(21 46)(22 47)(23 48)(24 41)(33 54)(34 55)(35 56)(36 49)(37 50)(38 51)(39 52)(40 53)
(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 35 58 56)(2 45 59 20)(3 33 60 54)(4 43 61 18)(5 39 62 52)(6 41 63 24)(7 37 64 50)(8 47 57 22)(9 49 30 36)(10 21 31 46)(11 55 32 34)(12 19 25 44)(13 53 26 40)(14 17 27 42)(15 51 28 38)(16 23 29 48)

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

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

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

38 conjugacy classes

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

38 irreducible representations

dim111111222224
type++++++++-
imageC1C2C2C2C2C4D4D4C4○D4D8SD16C8.C22
kernelC23.36D8C22.4Q16C23.7Q8C2×C22⋊C8C22×C4⋊C4C2×C4⋊C4C22×C4C24C2×C4C23C23C22
# reps141118318442

Matrix representation of C23.36D8 in GL6(𝔽17)

1600000
0160000
001000
000100
000010
00001016
,
100000
010000
001000
000100
0000160
0000016
,
1600000
0160000
001000
000100
000010
000001
,
10100000
1200000
006600
0014000
000069
00001511
,
1120000
760000
00161500
000100
00001015
000077

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,10,0,0,0,0,0,16],[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],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[10,12,0,0,0,0,10,0,0,0,0,0,0,0,6,14,0,0,0,0,6,0,0,0,0,0,0,0,6,15,0,0,0,0,9,11],[11,7,0,0,0,0,2,6,0,0,0,0,0,0,16,0,0,0,0,0,15,1,0,0,0,0,0,0,10,7,0,0,0,0,15,7] >;

C23.36D8 in GAP, Magma, Sage, TeX

C_2^3._{36}D_8
% in TeX

G:=Group("C2^3.36D8");
// GroupNames label

G:=SmallGroup(128,555);
// by ID

G=gap.SmallGroup(128,555);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,422,58,2804,718,172]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^8=1,e^2=c,d*a*d^-1=e*a*e^-1=a*b=b*a,a*c=c*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=b*c*d^-1>;
// generators/relations

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