Copied to
clipboard

G = C4○D4.8Q8order 128 = 27

6th non-split extension by C4○D4 of Q8 acting via Q8/C4=C2

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

Aliases: C8○D45C4, D4(C2.D8), Q8(C2.D8), C4○D4.8Q8, C2.2(D4○D8), C2.2(Q8○D8), D4.10(C4⋊C4), (C2×D4).346D4, C4.7(C22×Q8), Q8.10(C4⋊C4), C8.15(C22×C4), C4.51(C23×C4), (C2×Q8).265D4, C4⋊C4.352C23, M4(2)⋊17(C2×C4), (C2×C4).189C24, (C2×C8).555C23, C23.434(C2×D4), M4(2)⋊C49C2, C4.Q8.123C22, C2.D8.232C22, C23.25D422C2, (C22×C4).908C23, (C22×C8).245C22, C22.136(C22×D4), C42⋊C2.79C22, (C2×M4(2)).259C22, C23.33C23.5C2, (C2×C8)⋊10(C2×C4), C4.22(C2×C4⋊C4), (C2×Q8)(C2.D8), (C2×C8○D4).7C2, (C2×C2.D8)⋊37C2, C22.4(C2×C4⋊C4), (C2×C4).97(C2×Q8), C4○D4.33(C2×C4), C2.28(C22×C4⋊C4), (C2×C4).1082(C2×D4), (C2×C4⋊C4).573C22, (C2×C4).252(C22×C4), (C2×C4○D4).278C22, SmallGroup(128,1645)

Series: Derived Chief Lower central Upper central Jennings

C1C4 — C4○D4.8Q8
C1C2C22C2×C4C22×C4C2×C4○D4C2×C8○D4 — C4○D4.8Q8
C1C2C4 — C4○D4.8Q8
C1C22C2×C4○D4 — C4○D4.8Q8
C1C2C2C2×C4 — C4○D4.8Q8

Generators and relations for C4○D4.8Q8
 G = < a,b,c,d,e | a4=c2=1, b2=d4=a2, e2=ad2, ab=ba, ac=ca, ad=da, eae-1=a-1, cbc=a2b, bd=db, be=eb, cd=dc, ce=ec, ede-1=a2d3 >

Subgroups: 356 in 238 conjugacy classes, 172 normal (14 characteristic)
C1, C2 [×3], C2 [×6], C4 [×2], C4 [×6], C4 [×8], C22, C22 [×6], C22 [×6], C8 [×8], C2×C4, C2×C4 [×15], C2×C4 [×14], D4 [×12], Q8 [×4], C23 [×3], C42 [×6], C22⋊C4 [×6], C4⋊C4 [×8], C4⋊C4 [×6], C2×C8, C2×C8 [×15], M4(2) [×12], C22×C4 [×3], C22×C4 [×6], C2×D4 [×3], C2×Q8, C4○D4 [×8], C4.Q8 [×6], C2.D8, C2.D8 [×9], C2×C4⋊C4 [×6], C42⋊C2 [×6], C4×D4 [×6], C4×Q8 [×2], C22×C8 [×3], C2×M4(2) [×3], C8○D4 [×8], C2×C4○D4, C2×C2.D8 [×3], C23.25D4 [×3], M4(2)⋊C4 [×6], C23.33C23 [×2], C2×C8○D4, C4○D4.8Q8
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], D4 [×4], Q8 [×4], C23 [×15], C4⋊C4 [×16], C22×C4 [×14], C2×D4 [×6], C2×Q8 [×6], C24, C2×C4⋊C4 [×12], C23×C4, C22×D4, C22×Q8, C22×C4⋊C4, D4○D8, Q8○D8, C4○D4.8Q8

Smallest permutation representation of C4○D4.8Q8
On 64 points
Generators in S64
(1 25 5 29)(2 26 6 30)(3 27 7 31)(4 28 8 32)(9 37 13 33)(10 38 14 34)(11 39 15 35)(12 40 16 36)(17 43 21 47)(18 44 22 48)(19 45 23 41)(20 46 24 42)(49 61 53 57)(50 62 54 58)(51 63 55 59)(52 64 56 60)
(1 3 5 7)(2 4 6 8)(9 15 13 11)(10 16 14 12)(17 19 21 23)(18 20 22 24)(25 27 29 31)(26 28 30 32)(33 39 37 35)(34 40 38 36)(41 43 45 47)(42 44 46 48)(49 55 53 51)(50 56 54 52)(57 63 61 59)(58 64 62 60)
(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 9)(8 10)(17 62)(18 63)(19 64)(20 57)(21 58)(22 59)(23 60)(24 61)(25 39)(26 40)(27 33)(28 34)(29 35)(30 36)(31 37)(32 38)(41 52)(42 53)(43 54)(44 55)(45 56)(46 49)(47 50)(48 51)
(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 61 27 55)(2 60 28 54)(3 59 29 53)(4 58 30 52)(5 57 31 51)(6 64 32 50)(7 63 25 49)(8 62 26 56)(9 18 39 46)(10 17 40 45)(11 24 33 44)(12 23 34 43)(13 22 35 42)(14 21 36 41)(15 20 37 48)(16 19 38 47)

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

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

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

44 conjugacy classes

class 1 2A2B2C2D···2I4A···4H4I···4X8A8B8C8D8E···8J
order12222···24···44···488888···8
size11112···22···24···422224···4

44 irreducible representations

dim111111122244
type++++++++-+-
imageC1C2C2C2C2C2C4D4D4Q8D4○D8Q8○D8
kernelC4○D4.8Q8C2×C2.D8C23.25D4M4(2)⋊C4C23.33C23C2×C8○D4C8○D4C2×D4C2×Q8C4○D4C2C2
# reps1336211631422

Matrix representation of C4○D4.8Q8 in GL6(𝔽17)

100000
010000
000100
0016000
000001
0000160
,
100000
010000
0001600
001000
000001
0000160
,
1600000
0160000
000001
0000160
0001600
001000
,
6150000
10110000
0014300
00141400
0000143
00001414
,
9100000
280000
00001414
0000143
00141400
0014300

G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,1,0,0,0,0,16,0,0,0,0,16,0,0,0,0,1,0,0,0],[6,10,0,0,0,0,15,11,0,0,0,0,0,0,14,14,0,0,0,0,3,14,0,0,0,0,0,0,14,14,0,0,0,0,3,14],[9,2,0,0,0,0,10,8,0,0,0,0,0,0,0,0,14,14,0,0,0,0,14,3,0,0,14,14,0,0,0,0,14,3,0,0] >;

C4○D4.8Q8 in GAP, Magma, Sage, TeX

C_4\circ D_4._8Q_8
% in TeX

G:=Group("C4oD4.8Q8");
// GroupNames label

G:=SmallGroup(128,1645);
// by ID

G=gap.SmallGroup(128,1645);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,568,521,2804,172]);
// Polycyclic

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

׿
×
𝔽