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G = C42.252D4order 128 = 27

234th non-split extension by C42 of D4 acting via D4/C2=C22

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

Aliases: C42.252D4, C42.380C23, C4(C8⋊Q8), (C2×C8)⋊3Q8, C8⋊Q836C2, C8.13(C2×Q8), C4.58(C4⋊Q8), C22.5(C4⋊Q8), C4.13(C22×Q8), C4⋊C4.101C23, (C2×C8).271C23, (C2×C4).360C24, C23.392(C2×D4), (C22×C4).472D4, C4⋊Q8.286C22, C8⋊C4.121C22, C4.Q8.132C22, C2.D8.216C22, (C22×C8).275C22, (C2×C42).859C22, C22.620(C22×D4), C2.40(D8⋊C22), (C22×C4).1569C23, C23.25D4.18C2, C42.C2.117C22, C42⋊C2.145C22, C23.37C23.32C2, (C2×C4)(C8⋊Q8), C2.30(C2×C4⋊Q8), (C2×C4).521(C2×D4), (C2×C4).249(C2×Q8), (C2×C8⋊C4).12C2, SmallGroup(128,1894)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.252D4
C1C2C4C2×C4C22×C4C22×C8C2×C8⋊C4 — C42.252D4
C1C2C2×C4 — C42.252D4
C1C2×C4C2×C42 — C42.252D4
C1C2C2C2×C4 — C42.252D4

Generators and relations for C42.252D4
 G = < a,b,c,d | a4=b4=1, c4=b2, d2=a2b2, ab=ba, cac-1=ab2, dad-1=a-1b2, bc=cb, bd=db, dcd-1=c3 >

Subgroups: 276 in 174 conjugacy classes, 108 normal (10 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×4], C4 [×12], C22, C22 [×2], C22 [×2], C8 [×8], C2×C4 [×2], C2×C4 [×8], C2×C4 [×10], Q8 [×8], C23, C42 [×4], C42 [×4], C22⋊C4 [×4], C4⋊C4 [×8], C4⋊C4 [×12], C2×C8 [×12], C22×C4, C22×C4 [×2], C2×Q8 [×4], C8⋊C4 [×4], C4.Q8 [×8], C2.D8 [×8], C2×C42, C42⋊C2 [×4], C4×Q8 [×4], C22⋊Q8 [×4], C42.C2 [×4], C4⋊Q8 [×4], C22×C8 [×2], C2×C8⋊C4, C23.25D4 [×4], C8⋊Q8 [×8], C23.37C23 [×2], C42.252D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], Q8 [×8], C23 [×15], C2×D4 [×6], C2×Q8 [×12], C24, C4⋊Q8 [×4], C22×D4, C22×Q8 [×2], C2×C4⋊Q8, D8⋊C22 [×2], C42.252D4

Smallest permutation representation of C42.252D4
On 64 points
Generators in S64
(1 33 22 26)(2 38 23 31)(3 35 24 28)(4 40 17 25)(5 37 18 30)(6 34 19 27)(7 39 20 32)(8 36 21 29)(9 42 60 54)(10 47 61 51)(11 44 62 56)(12 41 63 53)(13 46 64 50)(14 43 57 55)(15 48 58 52)(16 45 59 49)
(1 7 5 3)(2 8 6 4)(9 11 13 15)(10 12 14 16)(17 23 21 19)(18 24 22 20)(25 31 29 27)(26 32 30 28)(33 39 37 35)(34 40 38 36)(41 43 45 47)(42 44 46 48)(49 51 53 55)(50 52 54 56)(57 59 61 63)(58 60 62 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 53 18 45)(2 56 19 48)(3 51 20 43)(4 54 21 46)(5 49 22 41)(6 52 23 44)(7 55 24 47)(8 50 17 42)(9 40 64 29)(10 35 57 32)(11 38 58 27)(12 33 59 30)(13 36 60 25)(14 39 61 28)(15 34 62 31)(16 37 63 26)

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

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

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

32 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G4H4I4J4K···4R8A···8H
order12222244444444444···48···8
size11112211112244448···84···4

32 irreducible representations

dim111112224
type++++++-+
imageC1C2C2C2C2D4Q8D4D8⋊C22
kernelC42.252D4C2×C8⋊C4C23.25D4C8⋊Q8C23.37C23C42C2×C8C22×C4C2
# reps114822824

Matrix representation of C42.252D4 in GL6(𝔽17)

010000
1600000
0071500
0081000
0000102
000097
,
100000
010000
0013000
0001300
0000130
0000013
,
010000
1600000
000100
004000
000004
0000160
,
400000
0130000
000004
0000160
0001600
0013000

G:=sub<GL(6,GF(17))| [0,16,0,0,0,0,1,0,0,0,0,0,0,0,7,8,0,0,0,0,15,10,0,0,0,0,0,0,10,9,0,0,0,0,2,7],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,13],[0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,4,0],[4,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,0,13,0,0,0,0,16,0,0,0,0,16,0,0,0,0,4,0,0,0] >;

C42.252D4 in GAP, Magma, Sage, TeX

C_4^2._{252}D_4
% in TeX

G:=Group("C4^2.252D4");
// GroupNames label

G:=SmallGroup(128,1894);
// by ID

G=gap.SmallGroup(128,1894);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,112,253,120,758,723,184,248,4037,124]);
// Polycyclic

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

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