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G = C42.399D4order 128 = 27

32nd non-split extension by C42 of D4 acting via D4/C22=C2

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

Aliases: C42.399D4, C42.600C23, Q8⋊C82C2, (C2×Q8)⋊5C8, Q8.5(C2×C8), (C4×C8).2C22, C4.6(C22×C8), (C4×Q8).14C4, C42.53(C2×C4), C4.6(C2×M4(2)), C4⋊C8.246C22, C4.12(C22⋊C8), (C22×C4).656D4, (C2×C4).17M4(2), (C22×Q8).19C4, (C4×Q8).250C22, C4.124(C8.C22), C22.28(C22⋊C8), (C2×C42).156C22, C23.168(C22⋊C4), C42.12C4.15C2, C2.3(C42⋊C22), C2.1(C23.38D4), (C2×C4×Q8).4C2, (C2×C4⋊C4).39C4, (C2×C4).18(C2×C8), C4⋊C4.178(C2×C4), C2.15(C2×C22⋊C8), (C2×C4).1443(C2×D4), (C2×Q8).174(C2×C4), (C22×C4).178(C2×C4), (C2×C4).305(C22×C4), C22.99(C2×C22⋊C4), (C2×C4).238(C22⋊C4), SmallGroup(128,211)

Series: Derived Chief Lower central Upper central Jennings

C1C4 — C42.399D4
C1C2C22C2×C4C42C2×C42C2×C4×Q8 — C42.399D4
C1C2C4 — C42.399D4
C1C2×C4C2×C42 — C42.399D4
C1C22C22C42 — C42.399D4

Generators and relations for C42.399D4
 G = < a,b,c,d | a4=b4=1, c4=a2b2, d2=b, ab=ba, cac-1=a-1, ad=da, bc=cb, bd=db, dcd-1=b-1c3 >

Subgroups: 204 in 124 conjugacy classes, 62 normal (28 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×4], C4 [×9], C22, C22 [×2], C22 [×2], C8 [×4], C2×C4 [×6], C2×C4 [×4], C2×C4 [×13], Q8 [×4], Q8 [×6], C23, C42 [×4], C42 [×4], C4⋊C4 [×2], C4⋊C4 [×5], C2×C8 [×4], C22×C4 [×3], C22×C4 [×2], C2×Q8 [×6], C2×Q8 [×3], C4×C8 [×2], C4×C8, C22⋊C8 [×2], C4⋊C8 [×2], C4⋊C8, C2×C42, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C4×Q8 [×4], C4×Q8 [×2], C22×Q8, Q8⋊C8 [×4], C42.12C4 [×2], C2×C4×Q8, C42.399D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C8 [×4], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], C2×C8 [×6], M4(2) [×2], C22×C4, C2×D4 [×2], C22⋊C8 [×4], C2×C22⋊C4, C22×C8, C2×M4(2), C8.C22 [×2], C2×C22⋊C8, C23.38D4, C42⋊C22, C42.399D4

Smallest permutation representation of C42.399D4
On 64 points
Generators in S64
(1 34 59 51)(2 52 60 35)(3 36 61 53)(4 54 62 37)(5 38 63 55)(6 56 64 39)(7 40 57 49)(8 50 58 33)(9 41 30 20)(10 21 31 42)(11 43 32 22)(12 23 25 44)(13 45 26 24)(14 17 27 46)(15 47 28 18)(16 19 29 48)
(1 36 63 49)(2 37 64 50)(3 38 57 51)(4 39 58 52)(5 40 59 53)(6 33 60 54)(7 34 61 55)(8 35 62 56)(9 43 26 18)(10 44 27 19)(11 45 28 20)(12 46 29 21)(13 47 30 22)(14 48 31 23)(15 41 32 24)(16 42 25 17)
(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 19 36 10 63 44 49 27)(2 30 37 22 64 13 50 47)(3 42 38 25 57 17 51 16)(4 11 39 45 58 28 52 20)(5 23 40 14 59 48 53 31)(6 26 33 18 60 9 54 43)(7 46 34 29 61 21 55 12)(8 15 35 41 62 32 56 24)

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

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

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

44 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E···4N4O···4V8A···8P
order12222244444···44···48···8
size11112211112···24···44···4

44 irreducible representations

dim1111111122244
type++++++-
imageC1C2C2C2C4C4C4C8D4D4M4(2)C8.C22C42⋊C22
kernelC42.399D4Q8⋊C8C42.12C4C2×C4×Q8C2×C4⋊C4C4×Q8C22×Q8C2×Q8C42C22×C4C2×C4C4C2
# reps14212421622422

Matrix representation of C42.399D4 in GL6(𝔽17)

100000
010000
0016200
0016100
0000162
0000161
,
1300000
0130000
0016000
0001600
0000160
0000016
,
090000
800000
0000010
000050
0010700
005700
,
900000
080000
000010
000001
0016000
0001600

G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,16,0,0,0,0,2,1,0,0,0,0,0,0,16,16,0,0,0,0,2,1],[13,0,0,0,0,0,0,13,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[0,8,0,0,0,0,9,0,0,0,0,0,0,0,0,0,10,5,0,0,0,0,7,7,0,0,0,5,0,0,0,0,10,0,0,0],[9,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,1,0,0,0,0,0,0,1,0,0] >;

C42.399D4 in GAP, Magma, Sage, TeX

C_4^2._{399}D_4
% in TeX

G:=Group("C4^2.399D4");
// GroupNames label

G:=SmallGroup(128,211);
// by ID

G=gap.SmallGroup(128,211);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,232,387,184,1123,570,136,172]);
// Polycyclic

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

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