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G = C42.364D4order 128 = 27

60th non-split extension by C42 of D4 acting via D4/C4=C2

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

Aliases: C42.364D4, C42.718C23, (C2×C8)⋊13Q8, C4(C82Q8), C4(C83Q8), C8.23(C2×Q8), C82Q835C2, C83Q829C2, C4.57(C4⋊Q8), C4(C8.5Q8), C4.24(C4○D8), C4⋊C4.99C23, C8.5Q826C2, C22.4(C4⋊Q8), C4.11(C22×Q8), (C2×C8).597C23, (C4×C8).410C22, (C2×C4).358C24, C23.391(C2×D4), (C22×C4).617D4, C4⋊Q8.284C22, C2.D8.179C22, C4.Q8.159C22, (C22×C8).559C22, C23.25D4.7C2, C22.618(C22×D4), (C22×C4).1567C23, (C2×C42).1133C22, C42.C2.115C22, C42⋊C2.144C22, C23.37C23.31C2, (C2×C4×C8).48C2, C2.28(C2×C4⋊Q8), (C2×C4)(C82Q8), (C2×C4)(C83Q8), C2.33(C2×C4○D8), (C2×C4).862(C2×D4), (C2×C4)(C8.5Q8), (C2×C4).247(C2×Q8), SmallGroup(128,1892)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.364D4
C1C2C4C2×C4C22×C4C22×C8C2×C4×C8 — C42.364D4
C1C2C2×C4 — C42.364D4
C1C2×C4C2×C42 — C42.364D4
C1C2C2C2×C4 — C42.364D4

Generators and relations for C42.364D4
 G = < a,b,c,d | a4=b4=1, c4=b2, d2=a2b2, ab=ba, ac=ca, dad-1=a-1, bc=cb, bd=db, dcd-1=c3 >

Subgroups: 276 in 178 conjugacy classes, 112 normal (14 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×8], C4 [×10], C22, C22 [×2], C22 [×2], C8 [×8], C2×C4 [×2], C2×C4 [×8], C2×C4 [×12], Q8 [×8], C23, C42 [×2], C42 [×2], C42 [×4], C22⋊C4 [×4], C4⋊C4 [×8], C4⋊C4 [×12], C2×C8 [×12], C22×C4, C22×C4 [×2], C2×Q8 [×4], C4×C8 [×2], C4×C8 [×2], 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×C4×C8, C23.25D4 [×4], C83Q8 [×2], C8.5Q8 [×4], C82Q8 [×2], C23.37C23 [×2], C42.364D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], Q8 [×8], C23 [×15], C2×D4 [×6], C2×Q8 [×12], C24, C4⋊Q8 [×4], C4○D8 [×4], C22×D4, C22×Q8 [×2], C2×C4⋊Q8, C2×C4○D8 [×2], C42.364D4

Smallest permutation representation of C42.364D4
On 64 points
Generators in S64
(1 37 23 28)(2 38 24 29)(3 39 17 30)(4 40 18 31)(5 33 19 32)(6 34 20 25)(7 35 21 26)(8 36 22 27)(9 46 62 51)(10 47 63 52)(11 48 64 53)(12 41 57 54)(13 42 58 55)(14 43 59 56)(15 44 60 49)(16 45 61 50)
(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 54 19 45)(2 49 20 48)(3 52 21 43)(4 55 22 46)(5 50 23 41)(6 53 24 44)(7 56 17 47)(8 51 18 42)(9 40 58 27)(10 35 59 30)(11 38 60 25)(12 33 61 28)(13 36 62 31)(14 39 63 26)(15 34 64 29)(16 37 57 32)

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

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

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

44 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E···4N4O···4V8A···8P
order12222244444···44···48···8
size11112211112···28···82···2

44 irreducible representations

dim11111112222
type++++++++-+
imageC1C2C2C2C2C2C2D4Q8D4C4○D8
kernelC42.364D4C2×C4×C8C23.25D4C83Q8C8.5Q8C82Q8C23.37C23C42C2×C8C22×C4C4
# reps114242228216

Matrix representation of C42.364D4 in GL4(𝔽17) generated by

1100
151600
00160
00016
,
16000
01600
0040
0004
,
161600
2100
0090
00015
,
91000
2800
00015
0090
G:=sub<GL(4,GF(17))| [1,15,0,0,1,16,0,0,0,0,16,0,0,0,0,16],[16,0,0,0,0,16,0,0,0,0,4,0,0,0,0,4],[16,2,0,0,16,1,0,0,0,0,9,0,0,0,0,15],[9,2,0,0,10,8,0,0,0,0,0,9,0,0,15,0] >;

C42.364D4 in GAP, Magma, Sage, TeX

C_4^2._{364}D_4
% in TeX

G:=Group("C4^2.364D4");
// GroupNames label

G:=SmallGroup(128,1892);
// by ID

G=gap.SmallGroup(128,1892);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,112,253,120,758,184,248,4037,124]);
// 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,a*c=c*a,d*a*d^-1=a^-1,b*c=c*b,b*d=d*b,d*c*d^-1=c^3>;
// generators/relations

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