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G = C42.167D4order 128 = 27

149th non-split extension by C42 of D4 acting via D4/C2=C22

p-group, metabelian, nilpotent (class 2), monomial

Aliases: C42.167D4, C23.442C24, C24.323C23, C22.1772- 1+4, C4⋊C427D4, (C2×D4)⋊16Q8, C2.22(D4×Q8), C43(C22⋊Q8), C429C428C2, C23.21(C2×Q8), C4.37(C41D4), C2.50(D46D4), C23.4Q821C2, C22.97(C22×Q8), (C22×C4).835C23, (C2×C42).547C22, (C23×C4).395C22, C22.293(C22×D4), (C22×D4).529C22, (C22×Q8).129C22, C24.3C22.44C2, C2.C42.548C22, C2.21(C23.38C23), (C4×C4⋊C4)⋊87C2, (C2×C4⋊Q8)⋊12C2, (C2×C4×D4).60C2, (C2×C4).73(C2×D4), C2.12(C2×C41D4), (C2×C22⋊Q8)⋊22C2, (C2×C4).309(C2×Q8), C2.30(C2×C22⋊Q8), (C2×C4).821(C4○D4), (C2×C4⋊C4).300C22, C22.319(C2×C4○D4), (C2×C22⋊C4).177C22, SmallGroup(128,1274)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C42.167D4
C1C2C22C23C22×C4C23×C4C2×C4×D4 — C42.167D4
C1C23 — C42.167D4
C1C23 — C42.167D4
C1C23 — C42.167D4

Generators and relations for C42.167D4
 G = < a,b,c,d | a4=b4=c4=1, d2=b2, ab=ba, cac-1=ab2, dad-1=a-1, cbc-1=dbd-1=b-1, dcd-1=b2c-1 >

Subgroups: 612 in 338 conjugacy classes, 132 normal (18 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×8], C4 [×16], C22 [×3], C22 [×4], C22 [×20], C2×C4 [×22], C2×C4 [×36], D4 [×8], Q8 [×8], C23, C23 [×4], C23 [×12], C42 [×4], C42 [×4], C22⋊C4 [×20], C4⋊C4 [×8], C4⋊C4 [×26], C22×C4 [×3], C22×C4 [×10], C22×C4 [×12], C2×D4 [×4], C2×D4 [×4], C2×Q8 [×12], C24 [×2], C2.C42 [×2], C2×C42, C2×C42 [×2], C2×C22⋊C4 [×10], C2×C4⋊C4, C2×C4⋊C4 [×14], C4×D4 [×4], C22⋊Q8 [×16], C4⋊Q8 [×4], C23×C4 [×2], C22×D4, C22×Q8 [×2], C4×C4⋊C4, C429C4 [×2], C24.3C22 [×2], C23.4Q8 [×4], C2×C4×D4, C2×C22⋊Q8 [×4], C2×C4⋊Q8, C42.167D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×12], Q8 [×4], C23 [×15], C2×D4 [×18], C2×Q8 [×6], C4○D4 [×2], C24, C22⋊Q8 [×4], C41D4 [×4], C22×D4 [×3], C22×Q8, C2×C4○D4, 2- 1+4 [×2], C2×C22⋊Q8, C2×C41D4, C23.38C23, D46D4 [×2], D4×Q8 [×2], C42.167D4

Smallest permutation representation of C42.167D4
On 64 points
Generators in S64
(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 15 11 20)(2 16 12 17)(3 13 9 18)(4 14 10 19)(5 29 28 63)(6 30 25 64)(7 31 26 61)(8 32 27 62)(21 44 45 52)(22 41 46 49)(23 42 47 50)(24 43 48 51)(33 38 55 59)(34 39 56 60)(35 40 53 57)(36 37 54 58)
(1 26 52 33)(2 8 49 56)(3 28 50 35)(4 6 51 54)(5 42 53 9)(7 44 55 11)(10 25 43 36)(12 27 41 34)(13 29 23 57)(14 64 24 37)(15 31 21 59)(16 62 22 39)(17 32 46 60)(18 63 47 40)(19 30 48 58)(20 61 45 38)
(1 60 11 39)(2 59 12 38)(3 58 9 37)(4 57 10 40)(5 48 28 24)(6 47 25 23)(7 46 26 22)(8 45 27 21)(13 54 18 36)(14 53 19 35)(15 56 20 34)(16 55 17 33)(29 43 63 51)(30 42 64 50)(31 41 61 49)(32 44 62 52)

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

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

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

38 conjugacy classes

class 1 2A···2G2H2I2J2K4A···4H4I···4V4W4X4Y4Z
order12···222224···44···44444
size11···144442···24···48888

38 irreducible representations

dim1111111122224
type++++++++++--
imageC1C2C2C2C2C2C2C2D4D4Q8C4○D42- 1+4
kernelC42.167D4C4×C4⋊C4C429C4C24.3C22C23.4Q8C2×C4×D4C2×C22⋊Q8C2×C4⋊Q8C42C4⋊C4C2×D4C2×C4C22
# reps1122414148442

Matrix representation of C42.167D4 in GL6(𝔽5)

420000
410000
004000
000400
000030
000012
,
100000
010000
001000
000100
000020
000043
,
130000
140000
004200
004100
000032
000012
,
400000
410000
001300
000400
000032
000002

G:=sub<GL(6,GF(5))| [4,4,0,0,0,0,2,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,3,1,0,0,0,0,0,2],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,2,4,0,0,0,0,0,3],[1,1,0,0,0,0,3,4,0,0,0,0,0,0,4,4,0,0,0,0,2,1,0,0,0,0,0,0,3,1,0,0,0,0,2,2],[4,4,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,3,4,0,0,0,0,0,0,3,0,0,0,0,0,2,2] >;

C42.167D4 in GAP, Magma, Sage, TeX

C_4^2._{167}D_4
% in TeX

G:=Group("C4^2.167D4");
// GroupNames label

G:=SmallGroup(128,1274);
// by ID

G=gap.SmallGroup(128,1274);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,336,253,568,758,723,675,80]);
// Polycyclic

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

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