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G = C42.184D4order 128 = 27

166th non-split extension by C42 of D4 acting via D4/C2=C22

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

Aliases: C42.184D4, C24.37C23, C23.504C24, C22.2072- 1+4, C424C427C2, C23⋊Q8.13C2, (C2×C42).591C22, (C22×C4).849C23, C22.334(C22×D4), C23.4Q8.12C2, (C22×Q8).445C22, C23.65C2399C2, C23.83C2355C2, C23.78C2322C2, C2.77(C22.19C24), C24.C22.41C2, C23.63C23107C2, C2.C42.234C22, C2.47(C22.26C24), C2.29(C23.38C23), C2.75(C22.46C24), C2.52(C22.50C24), (C2×C4×Q8)⋊28C2, (C2×C4).1201(C2×D4), (C2×C4).162(C4○D4), (C2×C4⋊C4).343C22, (C2×C422C2).9C2, C22.380(C2×C4○D4), (C2×C22⋊C4).204C22, SmallGroup(128,1336)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C42.184D4
C1C2C22C23C22×C4C2×C42C424C4 — C42.184D4
C1C23 — C42.184D4
C1C23 — C42.184D4
C1C23 — C42.184D4

Generators and relations for C42.184D4
 G = < a,b,c,d | a4=b4=c4=1, d2=a2, ab=ba, cac-1=ab2, dad-1=a-1b2, bc=cb, dbd-1=a2b, dcd-1=c-1 >

Subgroups: 404 in 235 conjugacy classes, 100 normal (34 characteristic)
C1, C2 [×3], C2 [×4], C2, C4 [×21], C22 [×3], C22 [×4], C22 [×7], C2×C4 [×14], C2×C4 [×35], Q8 [×8], C23, C23 [×7], C42 [×4], C42 [×8], C22⋊C4 [×14], C4⋊C4 [×20], C22×C4 [×6], C22×C4 [×8], C2×Q8 [×6], C24, C2.C42 [×2], C2.C42 [×10], C2×C42, C2×C42 [×4], C2×C22⋊C4 [×3], C2×C22⋊C4 [×4], C2×C4⋊C4 [×4], C2×C4⋊C4 [×6], C4×Q8 [×4], C422C2 [×4], C22×Q8, C424C4, C23.63C23 [×2], C24.C22 [×4], C23.65C23 [×2], C23⋊Q8, C23.78C23, C23.4Q8, C23.83C23, C2×C4×Q8, C2×C422C2, C42.184D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×8], C24, C22×D4, C2×C4○D4 [×4], 2- 1+4 [×2], C22.19C24, C22.26C24, C23.38C23, C22.46C24 [×2], C22.50C24 [×2], C42.184D4

Smallest permutation representation of C42.184D4
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 16 59 27)(2 13 60 28)(3 14 57 25)(4 15 58 26)(5 33 45 21)(6 34 46 22)(7 35 47 23)(8 36 48 24)(9 56 32 41)(10 53 29 42)(11 54 30 43)(12 55 31 44)(17 64 37 49)(18 61 38 50)(19 62 39 51)(20 63 40 52)
(1 61 53 48)(2 51 54 5)(3 63 55 46)(4 49 56 7)(6 57 52 44)(8 59 50 42)(9 23 26 37)(10 36 27 18)(11 21 28 39)(12 34 25 20)(13 19 30 33)(14 40 31 22)(15 17 32 35)(16 38 29 24)(41 47 58 64)(43 45 60 62)
(1 32 3 30)(2 12 4 10)(5 34 7 36)(6 21 8 23)(9 57 11 59)(13 53 15 55)(14 41 16 43)(17 63 19 61)(18 51 20 49)(22 47 24 45)(25 56 27 54)(26 44 28 42)(29 60 31 58)(33 48 35 46)(37 52 39 50)(38 62 40 64)

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

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

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

38 conjugacy classes

class 1 2A···2G2H4A···4H4I···4Z4AA4AB4AC
order12···224···44···4444
size11···182···24···4888

38 irreducible representations

dim11111111111224
type++++++++++++-
imageC1C2C2C2C2C2C2C2C2C2C2D4C4○D42- 1+4
kernelC42.184D4C424C4C23.63C23C24.C22C23.65C23C23⋊Q8C23.78C23C23.4Q8C23.83C23C2×C4×Q8C2×C422C2C42C2×C4C22
# reps112421111114162

Matrix representation of C42.184D4 in GL6(𝔽5)

140000
040000
004200
000100
000034
000002
,
200000
020000
002000
000200
000013
000004
,
230000
430000
003000
003200
000021
000003
,
230000
430000
003400
003200
000030
000032

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

C42.184D4 in GAP, Magma, Sage, TeX

C_4^2._{184}D_4
% in TeX

G:=Group("C4^2.184D4");
// GroupNames label

G:=SmallGroup(128,1336);
// by ID

G=gap.SmallGroup(128,1336);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,112,253,120,758,723,184,675,248]);
// Polycyclic

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

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