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G = C42.400D4order 128 = 27

33rd non-split extension by C42 of D4 acting via D4/C22=C2

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

Aliases: C42.400D4, C42.605C23, C4.22C4≀C2, C4⋊Q8.8C4, D4⋊C825C2, C4⋊D4.4C4, C41D4.7C4, C4.28(C8○D4), C42.58(C2×C4), (C4×D4).4C22, C4⋊C8.251C22, (C4×M4(2))⋊13C2, (C4×C8).310C22, (C22×C4).202D4, C4.132(C8⋊C22), C42.12C410C2, C23.44(C22⋊C4), (C2×C42).161C22, C2.4(C23.37D4), C22.26C24.2C2, C2.7(C2×C4≀C2), C4⋊C4.51(C2×C4), (C2×D4).51(C2×C4), (C2×C4).1447(C2×D4), (C2×C4).310(C22×C4), (C22×C4).183(C2×C4), (C2×C4).165(C22⋊C4), C22.160(C2×C22⋊C4), C2.16((C22×C8)⋊C2), SmallGroup(128,216)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.400D4
C1C2C22C2×C4C42C2×C42C22.26C24 — C42.400D4
C1C2C2×C4 — C42.400D4
C1C2×C4C2×C42 — C42.400D4
C1C22C22C42 — C42.400D4

Generators and relations for C42.400D4
 G = < a,b,c,d | a4=b4=1, c4=a2b2, d2=b, ab=ba, cac-1=a-1b2, ad=da, bc=cb, bd=db, dcd-1=a2b-1c3 >

Subgroups: 284 in 131 conjugacy classes, 48 normal (26 characteristic)
C1, C2 [×3], C2 [×3], C4 [×6], C4 [×5], C22, C22 [×9], C8 [×6], C2×C4 [×6], C2×C4 [×12], D4 [×12], Q8 [×2], C23, C23 [×2], C42 [×4], C22⋊C4 [×4], C4⋊C4 [×2], C4⋊C4, C2×C8 [×4], M4(2) [×4], C22×C4 [×3], C22×C4 [×2], C2×D4 [×2], C2×D4 [×4], C2×Q8, C4○D4 [×4], C4×C8 [×2], C4×C8, C8⋊C4, C22⋊C8, C4⋊C8 [×2], C2×C42, C4×D4 [×2], C4×D4, C4⋊D4 [×2], C4⋊D4, C4.4D4, C41D4, C4⋊Q8, C2×M4(2), C2×C4○D4, D4⋊C8 [×4], C4×M4(2), C42.12C4, C22.26C24, C42.400D4
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, C8○D4 [×2], C8⋊C22 [×2], (C22×C8)⋊C2, C23.37D4, C2×C4≀C2, C42.400D4

Smallest permutation representation of C42.400D4
On 64 points
Generators in S64
(1 41 22 40)(2 46 23 37)(3 43 24 34)(4 48 17 39)(5 45 18 36)(6 42 19 33)(7 47 20 38)(8 44 21 35)(9 27 58 51)(10 32 59 56)(11 29 60 53)(12 26 61 50)(13 31 62 55)(14 28 63 52)(15 25 64 49)(16 30 57 54)
(1 58 18 13)(2 59 19 14)(3 60 20 15)(4 61 21 16)(5 62 22 9)(6 63 23 10)(7 64 24 11)(8 57 17 12)(25 43 53 38)(26 44 54 39)(27 45 55 40)(28 46 56 33)(29 47 49 34)(30 48 50 35)(31 41 51 36)(32 42 52 37)
(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 39 58 26 18 44 13 54)(2 25 59 43 19 53 14 38)(3 42 60 52 20 37 15 32)(4 51 61 36 21 31 16 41)(5 35 62 30 22 48 9 50)(6 29 63 47 23 49 10 34)(7 46 64 56 24 33 11 28)(8 55 57 40 17 27 12 45)

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

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

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

38 conjugacy classes

class 1 2A2B2C2D2E2F4A4B4C4D4E···4L4M4N4O8A···8P
order122222244444···44448···8
size111148811112···24884···4

38 irreducible representations

dim1111111122224
type++++++++
imageC1C2C2C2C2C4C4C4D4D4C4≀C2C8○D4C8⋊C22
kernelC42.400D4D4⋊C8C4×M4(2)C42.12C4C22.26C24C4⋊D4C41D4C4⋊Q8C42C22×C4C4C4C4
# reps1411142222882

Matrix representation of C42.400D4 in GL4(𝔽17) generated by

0100
1000
00130
00013
,
4000
0400
00130
00013
,
9000
0800
00014
001014
,
01500
15000
0003
00100
G:=sub<GL(4,GF(17))| [0,1,0,0,1,0,0,0,0,0,13,0,0,0,0,13],[4,0,0,0,0,4,0,0,0,0,13,0,0,0,0,13],[9,0,0,0,0,8,0,0,0,0,0,10,0,0,14,14],[0,15,0,0,15,0,0,0,0,0,0,10,0,0,3,0] >;

C42.400D4 in GAP, Magma, Sage, TeX

C_4^2._{400}D_4
% in TeX

G:=Group("C4^2.400D4");
// GroupNames label

G:=SmallGroup(128,216);
// by ID

G=gap.SmallGroup(128,216);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,1059,184,1123,570,136,172]);
// 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*b^2,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=a^2*b^-1*c^3>;
// generators/relations

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