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G = C42.376D4order 128 = 27

9th non-split extension by C42 of D4 acting via D4/C22=C2

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

Aliases: C42.376D4, C42.147C23, C4.35C4≀C2, (C4×D4).3C4, (C4×Q8).3C4, C4⋊D4.10C4, C42.88(C2×C4), C22⋊Q8.10C4, (C4×M4(2))⋊19C2, (C22×C4).227D4, C8⋊C4.85C22, C42.6C434C2, C23.58(C22⋊C4), C42.2C227C2, (C2×C42).191C22, C42.C228C2, C42.C2.95C22, C4.4D4.114C22, C2.29(C42⋊C22), C23.36C23.10C2, C2.11(M4(2).8C22), C2.34(C2×C4≀C2), C4⋊C4.26(C2×C4), (C2×D4).21(C2×C4), (C2×Q8).21(C2×C4), (C2×C4).1175(C2×D4), (C22×C4).213(C2×C4), (C2×C4).141(C22×C4), (C2×C4).179(C22⋊C4), C22.205(C2×C22⋊C4), SmallGroup(128,261)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.376D4
C1C2C22C2×C4C42C2×C42C23.36C23 — C42.376D4
C1C22C2×C4 — C42.376D4
C1C22C2×C42 — C42.376D4
C1C22C22C42 — C42.376D4

Generators and relations for C42.376D4
 G = < a,b,c,d | a4=b4=1, c4=a2, d2=b, ab=ba, cac-1=a-1b2, ad=da, cbc-1=a2b, bd=db, dcd-1=a2bc3 >

Subgroups: 212 in 108 conjugacy classes, 44 normal (34 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×8], C22, C22 [×6], C8 [×6], C2×C4 [×6], C2×C4 [×9], D4 [×3], Q8, C23, C23, C42 [×4], C42, C22⋊C4 [×5], C4⋊C4 [×2], C4⋊C4 [×4], C2×C8 [×4], M4(2) [×4], C22×C4 [×3], C22×C4, C2×D4, C2×D4, C2×Q8, C4×C8, C8⋊C4 [×4], C22⋊C8, C4⋊C8, C2×C42, C42⋊C2, C4×D4, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C42.C2, C422C2, C2×M4(2), C42.C22 [×2], C42.2C22 [×2], C4×M4(2), C42.6C4, C23.36C23, C42.376D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×2], C4≀C2 [×2], C2×C22⋊C4, M4(2).8C22, C2×C4≀C2, C42⋊C22, C42.376D4

Smallest permutation representation of C42.376D4
On 64 points
Generators in S64
(1 40 5 36)(2 50 6 54)(3 34 7 38)(4 52 8 56)(9 22 13 18)(10 59 14 63)(11 24 15 20)(12 61 16 57)(17 46 21 42)(19 48 23 44)(25 51 29 55)(26 35 30 39)(27 53 31 49)(28 37 32 33)(41 60 45 64)(43 62 47 58)
(1 17 27 57)(2 22 28 62)(3 19 29 59)(4 24 30 64)(5 21 31 61)(6 18 32 58)(7 23 25 63)(8 20 26 60)(9 33 43 54)(10 38 44 51)(11 35 45 56)(12 40 46 53)(13 37 47 50)(14 34 48 55)(15 39 41 52)(16 36 42 49)
(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 20 17 26 27 60 57 8)(2 29 22 59 28 3 62 19)(4 5 24 21 30 31 64 61)(6 25 18 63 32 7 58 23)(9 10 33 38 43 44 54 51)(11 46 35 53 45 12 56 40)(13 14 37 34 47 48 50 55)(15 42 39 49 41 16 52 36)

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

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

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

32 conjugacy classes

class 1 2A2B2C2D2E4A···4J4K4L4M4N8A···8H8I8J8K8L
order1222224···444448···88888
size1111482···248884···48888

32 irreducible representations

dim111111111122244
type++++++++
imageC1C2C2C2C2C2C4C4C4C4D4D4C4≀C2M4(2).8C22C42⋊C22
kernelC42.376D4C42.C22C42.2C22C4×M4(2)C42.6C4C23.36C23C4×D4C4×Q8C4⋊D4C22⋊Q8C42C22×C4C4C2C2
# reps122111222222822

Matrix representation of C42.376D4 in GL6(𝔽17)

040000
400000
00016015
001020
000101
00160160
,
010000
100000
004000
000400
000040
000004
,
1060000
1170000
0000215
000022
0011600
001100
,
1060000
6100000
0000215
00001515
0011600
00161600

G:=sub<GL(6,GF(17))| [0,4,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,16,0,0,16,0,1,0,0,0,0,2,0,16,0,0,15,0,1,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[10,11,0,0,0,0,6,7,0,0,0,0,0,0,0,0,1,1,0,0,0,0,16,1,0,0,2,2,0,0,0,0,15,2,0,0],[10,6,0,0,0,0,6,10,0,0,0,0,0,0,0,0,1,16,0,0,0,0,16,16,0,0,2,15,0,0,0,0,15,15,0,0] >;

C42.376D4 in GAP, Magma, Sage, TeX

C_4^2._{376}D_4
% in TeX

G:=Group("C4^2.376D4");
// GroupNames label

G:=SmallGroup(128,261);
// by ID

G=gap.SmallGroup(128,261);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,758,184,1123,1018,248,1971,102]);
// Polycyclic

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

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