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G = C42.234D4order 128 = 27

216th non-split extension by C42 of D4 acting via D4/C2=C22

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

Aliases: C42.234D4, C42.350C23, D42Q86C2, Q16⋊C47C2, C4.Q1623C2, C42Q1623C2, D4.8(C4○D4), D4.D46C2, C4⋊C8.52C22, C4⋊C4.69C23, (C2×C8).43C23, Q8.7(C4○D4), (C2×C4).314C24, C8⋊C4.9C22, C42.6C47C2, D4.7D4.1C2, (C22×C4).454D4, C23.254(C2×D4), C4⋊Q8.270C22, SD16⋊C410C2, (C4×Q8).75C22, (C2×Q8).79C23, C4.Q8.18C22, C2.D8.89C22, (C4×D4).322C22, (C2×D4).406C23, C22⋊C8.27C22, (C2×Q16).58C22, D4⋊C4.33C22, C23.20D418C2, C4.119(C8.C22), (C2×C42).841C22, Q8⋊C4.34C22, (C2×SD16).15C22, C22.574(C22×D4), C22⋊Q8.173C22, C2.33(D8⋊C22), (C22×C4).1030C23, C23.37C237C2, C42⋊C2.324C22, C2.115(C22.19C24), (C4×C4○D4).27C2, C4.199(C2×C4○D4), (C2×C4).1222(C2×D4), C2.35(C2×C8.C22), (C2×C4○D4).315C22, SmallGroup(128,1848)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.234D4
C1C2C4C2×C4C42C4×D4C4×C4○D4 — C42.234D4
C1C2C2×C4 — C42.234D4
C1C22C2×C42 — C42.234D4
C1C2C2C2×C4 — C42.234D4

Generators and relations for C42.234D4
 G = < a,b,c,d | a4=b4=1, c4=d2=a2, ab=ba, cac-1=dad-1=a-1b2, cbc-1=a2b, bd=db, dcd-1=c3 >

Subgroups: 332 in 193 conjugacy classes, 90 normal (44 characteristic)
C1, C2 [×3], C2 [×3], C4 [×2], C4 [×2], C4 [×12], C22, C22 [×7], C8 [×4], C2×C4 [×6], C2×C4 [×20], D4 [×2], D4 [×5], Q8 [×2], Q8 [×7], C23, C23, C42 [×4], C42 [×5], C22⋊C4 [×5], C4⋊C4 [×2], C4⋊C4 [×2], C4⋊C4 [×9], C2×C8 [×4], SD16 [×4], Q16 [×4], C22×C4 [×3], C22×C4 [×3], C2×D4, C2×D4, C2×Q8, C2×Q8 [×2], C2×Q8, C4○D4 [×4], C8⋊C4 [×2], C22⋊C8 [×2], D4⋊C4 [×2], Q8⋊C4 [×6], C4⋊C8 [×2], C4.Q8 [×2], C2.D8 [×2], C2×C42, C2×C42, C42⋊C2, C42⋊C2 [×2], C4×D4 [×2], C4×D4 [×2], C4×Q8 [×2], C4×Q8 [×2], C4×Q8, C22⋊Q8 [×2], C22⋊Q8, C42.C2, C4⋊Q8 [×2], C2×SD16 [×2], C2×Q16 [×2], C2×C4○D4, C42.6C4, SD16⋊C4 [×2], Q16⋊C4 [×2], D4.7D4 [×2], D4.D4, C42Q16, D42Q8, C4.Q16, C23.20D4 [×2], C4×C4○D4, C23.37C23, C42.234D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×4], C24, C8.C22 [×2], C22×D4, C2×C4○D4 [×2], C22.19C24, C2×C8.C22, D8⋊C22, C42.234D4

Smallest permutation representation of C42.234D4
On 64 points
Generators in S64
(1 44 5 48)(2 64 6 60)(3 46 7 42)(4 58 8 62)(9 32 13 28)(10 22 14 18)(11 26 15 30)(12 24 16 20)(17 35 21 39)(19 37 23 33)(25 40 29 36)(27 34 31 38)(41 52 45 56)(43 54 47 50)(49 61 53 57)(51 63 55 59)
(1 27 55 20)(2 32 56 17)(3 29 49 22)(4 26 50 19)(5 31 51 24)(6 28 52 21)(7 25 53 18)(8 30 54 23)(9 45 39 60)(10 42 40 57)(11 47 33 62)(12 44 34 59)(13 41 35 64)(14 46 36 61)(15 43 37 58)(16 48 38 63)
(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 4 5 8)(2 7 6 3)(9 10 13 14)(11 16 15 12)(17 18 21 22)(19 24 23 20)(25 28 29 32)(26 31 30 27)(33 38 37 34)(35 36 39 40)(41 46 45 42)(43 44 47 48)(49 56 53 52)(50 51 54 55)(57 64 61 60)(58 59 62 63)

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

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

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

32 conjugacy classes

class 1 2A2B2C2D2E2F4A···4J4K···4Q4R4S4T4U8A8B8C8D
order12222224···44···444448888
size11114442···24···488888888

32 irreducible representations

dim111111111111222244
type++++++++++++++-
imageC1C2C2C2C2C2C2C2C2C2C2C2D4D4C4○D4C4○D4C8.C22D8⋊C22
kernelC42.234D4C42.6C4SD16⋊C4Q16⋊C4D4.7D4D4.D4C42Q16D42Q8C4.Q16C23.20D4C4×C4○D4C23.37C23C42C22×C4D4Q8C4C2
# reps112221111211224422

Matrix representation of C42.234D4 in GL6(𝔽17)

1150000
0160000
004000
000400
000040
000004
,
1300000
0130000
00160160
00016016
002010
000201
,
100000
1160000
00213314
004233
0022154
001521315
,
1600000
1610000
0041511
001513116
001515213
001521315

G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,15,16,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[13,0,0,0,0,0,0,13,0,0,0,0,0,0,16,0,2,0,0,0,0,16,0,2,0,0,16,0,1,0,0,0,0,16,0,1],[1,1,0,0,0,0,0,16,0,0,0,0,0,0,2,4,2,15,0,0,13,2,2,2,0,0,3,3,15,13,0,0,14,3,4,15],[16,16,0,0,0,0,0,1,0,0,0,0,0,0,4,15,15,15,0,0,15,13,15,2,0,0,1,1,2,13,0,0,1,16,13,15] >;

C42.234D4 in GAP, Magma, Sage, TeX

C_4^2._{234}D_4
% in TeX

G:=Group("C4^2.234D4");
// GroupNames label

G:=SmallGroup(128,1848);
// by ID

G=gap.SmallGroup(128,1848);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,456,758,521,304,4037,1027,124]);
// Polycyclic

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

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