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G = C4○D4.5Q8order 128 = 27

3rd non-split extension by C4○D4 of Q8 acting via Q8/C4=C2

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

Aliases: C8○D43C4, C4○D4.5Q8, D4.4(C4⋊C4), (C2×C8).106D4, Q8.4(C4⋊C4), (C2×D4).203D4, (C2×Q8).161D4, C8.39(C22⋊C4), C4.73(C22⋊Q8), C4.124(C4⋊D4), C2.1(D4.4D4), C2.1(D4.5D4), M4(2).32(C2×C4), C22.C4214C2, C23.262(C4○D4), (C22×C4).666C23, (C22×C8).218C22, C23.36D4.6C2, C22.14(C22⋊Q8), C22.113(C4⋊D4), C2.25(C23.7Q8), C22.16(C42⋊C2), (C2×M4(2)).159C22, C4.8(C2×C4⋊C4), (C2×C4).6(C2×Q8), (C2×C8).62(C2×C4), (C2×C8○D4).5C2, (C2×C2.D8)⋊33C2, (C2×C8.C4)⋊6C2, C4○D4.29(C2×C4), (C2×C4).233(C2×D4), C4.95(C2×C22⋊C4), (C2×C4).48(C4○D4), (C2×C4⋊C4).40C22, (C2×C4).180(C22×C4), (C2×C4○D4).260C22, SmallGroup(128,548)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C4○D4.5Q8
C1C2C4C2×C4C22×C4C2×C4○D4C2×C8○D4 — C4○D4.5Q8
C1C2C2×C4 — C4○D4.5Q8
C1C22C22×C4 — C4○D4.5Q8
C1C2C2C22×C4 — C4○D4.5Q8

Generators and relations for C4○D4.5Q8
 G = < a,b,c,d,e | a4=c2=1, b2=d4=a2, e2=abd2, ab=ba, ac=ca, ad=da, eae-1=a-1, cbc=a2b, bd=db, be=eb, cd=dc, ece-1=bc, ede-1=a2d3 >

Subgroups: 244 in 132 conjugacy classes, 60 normal (30 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, C4⋊C4, C2×C8, C2×C8, C2×C8, M4(2), M4(2), C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C4○D4, D4⋊C4, Q8⋊C4, C2.D8, C8.C4, C2×C4⋊C4, C22×C8, C22×C8, C2×M4(2), C2×M4(2), C2×M4(2), C8○D4, C8○D4, C2×C4○D4, C22.C42, C23.36D4, C2×C2.D8, C2×C8.C4, C2×C8○D4, C4○D4.5Q8
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, C4○D4, C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C4⋊D4, C22⋊Q8, C23.7Q8, D4.4D4, D4.5D4, C4○D4.5Q8

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

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

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

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

32 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E4F4G4H4I4J8A8B8C8D8E···8J8K8L8M8N
order12222222444444444488888···88888
size11112244222244888822224···48888

32 irreducible representations

dim111111122222244
type+++++++++-+-
imageC1C2C2C2C2C2C4D4D4D4Q8C4○D4C4○D4D4.4D4D4.5D4
kernelC4○D4.5Q8C22.C42C23.36D4C2×C2.D8C2×C8.C4C2×C8○D4C8○D4C2×C8C2×D4C2×Q8C4○D4C2×C4C23C2C2
# reps122111841122222

Matrix representation of C4○D4.5Q8 in GL6(𝔽17)

100000
010000
004000
000400
00816130
0080013
,
100000
010000
00161500
001100
000001
0000160
,
1600000
0160000
001200
0001600
000001
000010
,
1150000
1160000
008000
000800
00103150
00100015
,
920000
1080000
00013215
0024150
0012121515
0005215

G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,8,8,0,0,0,4,16,0,0,0,0,0,13,0,0,0,0,0,0,13],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,1,0,0,0,0,15,1,0,0,0,0,0,0,0,16,0,0,0,0,1,0],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,2,16,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[1,1,0,0,0,0,15,16,0,0,0,0,0,0,8,0,10,10,0,0,0,8,3,0,0,0,0,0,15,0,0,0,0,0,0,15],[9,10,0,0,0,0,2,8,0,0,0,0,0,0,0,2,12,0,0,0,13,4,12,5,0,0,2,15,15,2,0,0,15,0,15,15] >;

C4○D4.5Q8 in GAP, Magma, Sage, TeX

C_4\circ D_4._5Q_8
% in TeX

G:=Group("C4oD4.5Q8");
// GroupNames label

G:=SmallGroup(128,548);
// by ID

G=gap.SmallGroup(128,548);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,512,422,2019,248,718,172,1027,124]);
// Polycyclic

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

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