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G = C23.38D8order 128 = 27

9th non-split extension by C23 of D8 acting via D8/D4=C2

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

Aliases: C23.38D8, C24.161D4, C23.43SD16, C4⋊C430D4, C4.61(C4×D4), C4⋊D410C4, C22.38(C2×D8), C2.4(C22⋊D8), C2.4(Q8⋊D4), (C22×C4).288D4, C23.766(C2×D4), C4.127(C4⋊D4), C221(D4⋊C4), C22.4Q1612C2, C22.91C22≀C2, (C22×C8).33C22, C22.57(C2×SD16), C22.74(C8⋊C22), (C23×C4).256C22, C2.3(C22.D8), (C22×D4).15C22, C23.120(C22⋊C4), (C22×C4).1361C23, C2.3(C23.46D4), C22.64(C8.C22), C2.25(C23.36D4), C2.13(C23.23D4), C22.86(C22.D4), C4⋊C47(C2×C4), (C2×D4)⋊5(C2×C4), (C22×C4⋊C4)⋊2C2, (C2×D4⋊C4)⋊3C2, (C2×C22⋊C8)⋊14C2, (C2×C4⋊D4).6C2, (C2×C4).1329(C2×D4), C2.20(C2×D4⋊C4), (C2×C4).757(C4○D4), (C2×C4⋊C4).763C22, (C2×C4).379(C22×C4), (C22×C4).278(C2×C4), (C2×C4).131(C22⋊C4), C22.265(C2×C22⋊C4), SmallGroup(128,606)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C23.38D8
C1C2C4C2×C4C22×C4C23×C4C22×C4⋊C4 — C23.38D8
C1C2C2×C4 — C23.38D8
C1C23C23×C4 — C23.38D8
C1C2C2C22×C4 — C23.38D8

Generators and relations for C23.38D8
 G = < a,b,c,d,e | a2=b2=c2=d8=e2=1, ab=ba, dad-1=eae=ac=ca, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=bd-1 >

Subgroups: 524 in 228 conjugacy classes, 72 normal (28 characteristic)
C1, C2 [×7], C2 [×6], C4 [×4], C4 [×8], C22 [×7], C22 [×4], C22 [×22], C8 [×2], C2×C4 [×2], C2×C4 [×6], C2×C4 [×30], D4 [×14], C23, C23 [×6], C23 [×12], C22⋊C4 [×4], C4⋊C4 [×6], C4⋊C4 [×7], C2×C8 [×6], C22×C4 [×2], C22×C4 [×4], C22×C4 [×15], C2×D4 [×2], C2×D4 [×13], C24, C24, C22⋊C8 [×2], D4⋊C4 [×4], C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4 [×2], C2×C4⋊C4 [×5], C4⋊D4 [×4], C4⋊D4 [×2], C22×C8 [×2], C23×C4, C23×C4, C22×D4, C22×D4, C22.4Q16 [×2], C2×C22⋊C8, C2×D4⋊C4 [×2], C22×C4⋊C4, C2×C4⋊D4, C23.38D8
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×8], C23, C22⋊C4 [×4], D8 [×2], SD16 [×2], C22×C4, C2×D4 [×4], C4○D4 [×2], D4⋊C4 [×4], C2×C22⋊C4, C4×D4 [×2], C22≀C2, C4⋊D4 [×2], C22.D4, C2×D8, C2×SD16, C8⋊C22, C8.C22, C23.23D4, C2×D4⋊C4, C23.36D4, C22⋊D8, Q8⋊D4, C22.D8, C23.46D4, C23.38D8

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

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

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

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

38 conjugacy classes

class 1 2A···2G2H2I2J2K2L2M4A4B4C4D4E···4N4O4P8A···8H
order12···222222244444···4448···8
size11···122228822224···4884···4

38 irreducible representations

dim111111122222244
type+++++++++++-
imageC1C2C2C2C2C2C4D4D4D4C4○D4D8SD16C8⋊C22C8.C22
kernelC23.38D8C22.4Q16C2×C22⋊C8C2×D4⋊C4C22×C4⋊C4C2×C4⋊D4C4⋊D4C4⋊C4C22×C4C24C2×C4C23C23C22C22
# reps121211843144411

Matrix representation of C23.38D8 in GL5(𝔽17)

10000
0161500
00100
00010
00001
,
160000
01000
00100
00010
00001
,
10000
016000
001600
00010
00001
,
40000
016000
01100
000611
00030
,
160000
01000
0161600
00010
000116

G:=sub<GL(5,GF(17))| [1,0,0,0,0,0,16,0,0,0,0,15,1,0,0,0,0,0,1,0,0,0,0,0,1],[16,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[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],[4,0,0,0,0,0,16,1,0,0,0,0,1,0,0,0,0,0,6,3,0,0,0,11,0],[16,0,0,0,0,0,1,16,0,0,0,0,16,0,0,0,0,0,1,1,0,0,0,0,16] >;

C23.38D8 in GAP, Magma, Sage, TeX

C_2^3._{38}D_8
% in TeX

G:=Group("C2^3.38D8");
// GroupNames label

G:=SmallGroup(128,606);
// by ID

G=gap.SmallGroup(128,606);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,736,422,1018,248,2028,124]);
// Polycyclic

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

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