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G = C42.388D4order 128 = 27

21st non-split extension by C42 of D4 acting via D4/C22=C2

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

Aliases: C42.388D4, C4⋊C8.8C4, C4.31C4≀C2, C8⋊C4.9C4, (C2×C4).11C42, C42.37(C2×C4), C4.8(C8.C4), (C22×C4).23Q8, C23.15(C4⋊C4), (C22×C4).182D4, (C2×M4(2)).2C4, C4.19(C4.D4), (C4×M4(2)).11C2, C4⋊M4(2).6C2, C2.11(C426C4), C4.19(C4.10D4), C42.12C4.9C2, C2.7(C4.C42), (C2×C42).134C22, C2.8(C22.C42), C22.49(C2.C42), (C2×C4).72(C4⋊C4), (C22×C4).154(C2×C4), (C2×C4).302(C22⋊C4), SmallGroup(128,31)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.388D4
C1C2C22C2×C4C42C2×C42C4×M4(2) — C42.388D4
C1C22C2×C4 — C42.388D4
C1C2×C4C2×C42 — C42.388D4
C1C22C22C2×C42 — C42.388D4

Generators and relations for C42.388D4
 G = < a,b,c,d | a4=b4=1, c4=b2, d2=a, ab=ba, ac=ca, ad=da, cbc-1=dbd-1=a2b-1, dcd-1=ab-1c3 >

Subgroups: 128 in 75 conjugacy classes, 36 normal (26 characteristic)
C1, C2 [×3], C2, C4 [×2], C4 [×4], C4 [×3], C22, C22 [×3], C8 [×8], C2×C4 [×6], C2×C4 [×6], C23, C42 [×4], C2×C8 [×6], M4(2) [×6], C22×C4 [×3], C4×C8 [×2], C8⋊C4 [×2], C22⋊C8, C4⋊C8 [×2], C4⋊C8 [×2], C2×C42, C2×M4(2) [×2], C2×M4(2), C4×M4(2), C4⋊M4(2), C42.12C4, C42.388D4
Quotients: C1, C2 [×3], C4 [×6], C22, C2×C4 [×3], D4 [×3], Q8, C42, C22⋊C4 [×3], C4⋊C4 [×3], C2.C42, C4.D4, C4.10D4, C4≀C2 [×2], C8.C4 [×2], C426C4, C4.C42, C22.C42, C42.388D4

Smallest permutation representation of C42.388D4
On 64 points
Generators in S64
(1 56 59 18)(2 49 60 19)(3 50 61 20)(4 51 62 21)(5 52 63 22)(6 53 64 23)(7 54 57 24)(8 55 58 17)(9 41 26 35)(10 42 27 36)(11 43 28 37)(12 44 29 38)(13 45 30 39)(14 46 31 40)(15 47 32 33)(16 48 25 34)
(1 7 5 3)(2 62 6 58)(4 64 8 60)(9 11 13 15)(10 25 14 29)(12 27 16 31)(17 49 21 53)(18 24 22 20)(19 51 23 55)(26 28 30 32)(33 35 37 39)(34 46 38 42)(36 48 40 44)(41 43 45 47)(50 56 54 52)(57 63 61 59)
(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 14 56 46 59 31 18 40)(2 37 49 11 60 43 19 28)(3 12 50 44 61 29 20 38)(4 35 51 9 62 41 21 26)(5 10 52 42 63 27 22 36)(6 33 53 15 64 47 23 32)(7 16 54 48 57 25 24 34)(8 39 55 13 58 45 17 30)

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

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

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

38 conjugacy classes

class 1 2A2B2C2D4A4B4C4D4E···4L4M8A···8P8Q8R8S8T
order1222244444···448···88888
size1111411112···244···48888

38 irreducible representations

dim11111112222244
type++++++-+-
imageC1C2C2C2C4C4C4D4D4Q8C4≀C2C8.C4C4.D4C4.10D4
kernelC42.388D4C4×M4(2)C4⋊M4(2)C42.12C4C8⋊C4C4⋊C8C2×M4(2)C42C22×C4C22×C4C4C4C4C4
# reps11114442118811

Matrix representation of C42.388D4 in GL4(𝔽17) generated by

13000
01300
0040
0004
,
1000
01600
0040
0004
,
01600
16000
0080
00142
,
0100
13000
00816
0099
G:=sub<GL(4,GF(17))| [13,0,0,0,0,13,0,0,0,0,4,0,0,0,0,4],[1,0,0,0,0,16,0,0,0,0,4,0,0,0,0,4],[0,16,0,0,16,0,0,0,0,0,8,14,0,0,0,2],[0,13,0,0,1,0,0,0,0,0,8,9,0,0,16,9] >;

C42.388D4 in GAP, Magma, Sage, TeX

C_4^2._{388}D_4
% in TeX

G:=Group("C4^2.388D4");
// GroupNames label

G:=SmallGroup(128,31);
// by ID

G=gap.SmallGroup(128,31);
# by ID

G:=PCGroup([7,-2,2,-2,2,2,-2,2,56,85,120,758,723,184,136,3924,172]);
// Polycyclic

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

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