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G = C8⋊M4(2)  order 128 = 27

2nd semidirect product of C8 and M4(2) acting via M4(2)/C4=C22

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

Aliases: C82M4(2), C42.649C23, C8⋊C87C2, Q8⋊C842C2, C81C818C2, C4.Q8.6C4, D4⋊C8.16C2, C84Q831C2, (C2×C8).183D4, C4.16(C8○D4), C86D4.14C2, C2.9(C86D4), (C4×SD16).4C2, (C2×SD16).2C4, D4⋊C4.11C4, C4⋊C8.227C22, (C4×C8).318C22, Q8⋊C4.11C4, (C4×D4).15C22, C22.140(C4×D4), C4.10(C2×M4(2)), (C4×Q8).15C22, C2.10(C8.26D4), C4.146(C8⋊C22), C2.7(SD16⋊C4), C4.140(C8.C22), C4⋊C4.62(C2×C4), (C2×C8).57(C2×C4), (C2×D4).60(C2×C4), (C2×Q8).55(C2×C4), (C2×C4).1485(C2×D4), (C2×C4).510(C4○D4), (C2×C4).341(C22×C4), SmallGroup(128,324)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C8⋊M4(2)
C1C2C22C2×C4C42C4×C8C86D4 — C8⋊M4(2)
C1C2C2×C4 — C8⋊M4(2)
C1C2×C4C4×C8 — C8⋊M4(2)
C1C22C22C42 — C8⋊M4(2)

Generators and relations for C8⋊M4(2)
 G = < a,b,c | a8=b8=c2=1, bab-1=a-1, cac=a3, cbc=b5 >

Subgroups: 152 in 81 conjugacy classes, 42 normal (40 characteristic)
C1, C2 [×3], C2, C4 [×4], C4 [×4], C22, C22 [×3], C8 [×2], C8 [×6], C2×C4 [×3], C2×C4 [×5], D4 [×2], Q8 [×2], C23, C42, C42, C22⋊C4, C4⋊C4 [×2], C4⋊C4, C2×C8 [×4], C2×C8 [×3], M4(2) [×2], SD16 [×2], C22×C4, C2×D4, C2×Q8, C4×C8 [×3], C8⋊C4, C22⋊C8, D4⋊C4, Q8⋊C4, C4⋊C8 [×2], C4⋊C8, C4.Q8, C4×D4, C4×Q8, C2×M4(2), C2×SD16, C8⋊C8, D4⋊C8, Q8⋊C8, C81C8, C86D4, C4×SD16, C84Q8, C8⋊M4(2)
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], C23, M4(2) [×2], C22×C4, C2×D4, C4○D4, C4×D4, C2×M4(2), C8○D4, C8⋊C22, C8.C22, C86D4, SD16⋊C4, C8.26D4, C8⋊M4(2)

Smallest permutation representation of C8⋊M4(2)
On 64 points
Generators in S64
(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 31 61 54 23 33 9 45)(2 30 62 53 24 40 10 44)(3 29 63 52 17 39 11 43)(4 28 64 51 18 38 12 42)(5 27 57 50 19 37 13 41)(6 26 58 49 20 36 14 48)(7 25 59 56 21 35 15 47)(8 32 60 55 22 34 16 46)
(2 4)(3 7)(6 8)(10 12)(11 15)(14 16)(17 21)(18 24)(20 22)(25 39)(26 34)(27 37)(28 40)(29 35)(30 38)(31 33)(32 36)(41 50)(42 53)(43 56)(44 51)(45 54)(46 49)(47 52)(48 55)(58 60)(59 63)(62 64)

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

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

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

32 conjugacy classes

class 1 2A2B2C2D4A4B4C4D4E4F4G4H4I4J4K8A···8L8M8N8O8P
order12222444444444448···88888
size11118111122228884···48888

32 irreducible representations

dim1111111111112222444
type++++++++++-
imageC1C2C2C2C2C2C2C2C4C4C4C4D4M4(2)C4○D4C8○D4C8⋊C22C8.C22C8.26D4
kernelC8⋊M4(2)C8⋊C8D4⋊C8Q8⋊C8C81C8C86D4C4×SD16C84Q8D4⋊C4Q8⋊C4C4.Q8C2×SD16C2×C8C8C2×C4C4C4C4C2
# reps1111111122222424112

Matrix representation of C8⋊M4(2) in GL6(𝔽17)

100000
010000
0006015
00141112
0002011
00161536
,
010000
1300000
000010
00001616
001000
00161600
,
1600000
010000
001000
00161600
000010
00001616

G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,14,0,16,0,0,6,11,2,15,0,0,0,1,0,3,0,0,15,2,11,6],[0,13,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,16,0,0,0,0,0,16,0,0,1,16,0,0,0,0,0,16,0,0],[16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,16,0,0,0,0,0,16,0,0,0,0,0,0,1,16,0,0,0,0,0,16] >;

C8⋊M4(2) in GAP, Magma, Sage, TeX

C_8\rtimes M_4(2)
% in TeX

G:=Group("C8:M4(2)");
// GroupNames label

G:=SmallGroup(128,324);
// by ID

G=gap.SmallGroup(128,324);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,2102,723,100,1123,570,136,172]);
// Polycyclic

G:=Group<a,b,c|a^8=b^8=c^2=1,b*a*b^-1=a^-1,c*a*c=a^3,c*b*c=b^5>;
// generators/relations

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