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G = C42.316D4order 128 = 27

12nd non-split extension by C42 of D4 acting via D4/C4=C2

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

Aliases: C42.316D4, C42.614C23, C4.12C4≀C2, Q8⋊C834C2, C4⋊Q8.11C4, C4.2(C8○D4), C4.53(C2×Q16), (C2×C4).33Q16, C22⋊Q8.5C4, C4.96(C2×SD16), (C2×C4).68SD16, (C4×Q8).5C22, C4⋊C8.198C22, (C4×C8).364C22, C42.258(C2×C4), (C22×C4).573D4, C4.16(Q8⋊C4), C4⋊M4(2).13C2, C23.98(C22⋊C4), (C2×C42).1036C22, C22.11(Q8⋊C4), C23.37C23.3C2, (C2×C4×C8).10C2, C2.13(C2×C4≀C2), C4⋊C4.54(C2×C4), (C2×Q8).47(C2×C4), C2.6(C2×Q8⋊C4), (C2×C4).1142(C2×D4), (C2×C4).319(C22×C4), (C22×C4).397(C2×C4), (C2×C4).168(C22⋊C4), C22.169(C2×C22⋊C4), C2.19((C22×C8)⋊C2), SmallGroup(128,225)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.316D4
C1C2C22C2×C4C42C2×C42C23.37C23 — C42.316D4
C1C2C2×C4 — C42.316D4
C1C2×C4C2×C42 — C42.316D4
C1C22C22C42 — C42.316D4

Generators and relations for C42.316D4
 G = < a,b,c,d | a4=b4=1, c4=a2b2, d2=b, ab=ba, ac=ca, dad-1=a-1b2, bc=cb, bd=db, dcd-1=b-1c3 >

Subgroups: 196 in 116 conjugacy classes, 56 normal (28 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×6], C4 [×6], C22, C22 [×2], C22 [×2], C8 [×6], C2×C4 [×6], C2×C4 [×4], C2×C4 [×8], Q8 [×6], C23, C42 [×4], C42 [×2], C22⋊C4 [×2], C4⋊C4 [×2], C4⋊C4 [×7], C2×C8 [×8], M4(2) [×2], C22×C4 [×3], C2×Q8 [×2], C2×Q8, C4×C8 [×2], C4×C8, C4⋊C8 [×2], C4⋊C8, C2×C42, C42⋊C2, C4×Q8 [×2], C4×Q8, C22⋊Q8 [×2], C22⋊Q8, C42.C2, C4⋊Q8 [×2], C22×C8, C2×M4(2), Q8⋊C8 [×4], C2×C4×C8, C4⋊M4(2), C23.37C23, C42.316D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], SD16 [×2], Q16 [×2], C22×C4, C2×D4 [×2], Q8⋊C4 [×4], C4≀C2 [×2], C2×C22⋊C4, C8○D4 [×2], C2×SD16, C2×Q16, (C22×C8)⋊C2, C2×Q8⋊C4, C2×C4≀C2, C42.316D4

Smallest permutation representation of C42.316D4
On 64 points
Generators in S64
(1 56 59 34)(2 49 60 35)(3 50 61 36)(4 51 62 37)(5 52 63 38)(6 53 64 39)(7 54 57 40)(8 55 58 33)(9 45 30 21)(10 46 31 22)(11 47 32 23)(12 48 25 24)(13 41 26 17)(14 42 27 18)(15 43 28 19)(16 44 29 20)
(1 36 63 54)(2 37 64 55)(3 38 57 56)(4 39 58 49)(5 40 59 50)(6 33 60 51)(7 34 61 52)(8 35 62 53)(9 43 26 23)(10 44 27 24)(11 45 28 17)(12 46 29 18)(13 47 30 19)(14 48 31 20)(15 41 32 21)(16 42 25 22)
(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 24 36 10 63 44 54 27)(2 30 37 19 64 13 55 47)(3 42 38 25 57 22 56 16)(4 11 39 45 58 28 49 17)(5 20 40 14 59 48 50 31)(6 26 33 23 60 9 51 43)(7 46 34 29 61 18 52 12)(8 15 35 41 62 32 53 21)

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

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

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

44 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E···4N4O4P4Q4R8A···8P8Q8R8S8T
order12222244444···444448···88888
size11112211112···288882···28888

44 irreducible representations

dim1111111222222
type+++++++-
imageC1C2C2C2C2C4C4D4D4SD16Q16C4≀C2C8○D4
kernelC42.316D4Q8⋊C8C2×C4×C8C4⋊M4(2)C23.37C23C22⋊Q8C4⋊Q8C42C22×C4C2×C4C2×C4C4C4
# reps1411144224488

Matrix representation of C42.316D4 in GL4(𝔽17) generated by

1200
161600
0010
00016
,
16000
01600
00130
00013
,
01100
3600
0090
0009
,
11300
16600
0009
0090
G:=sub<GL(4,GF(17))| [1,16,0,0,2,16,0,0,0,0,1,0,0,0,0,16],[16,0,0,0,0,16,0,0,0,0,13,0,0,0,0,13],[0,3,0,0,11,6,0,0,0,0,9,0,0,0,0,9],[11,16,0,0,3,6,0,0,0,0,0,9,0,0,9,0] >;

C42.316D4 in GAP, Magma, Sage, TeX

C_4^2._{316}D_4
% in TeX

G:=Group("C4^2.316D4");
// GroupNames label

G:=SmallGroup(128,225);
// by ID

G=gap.SmallGroup(128,225);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,232,1430,723,1123,570,136,172]);
// Polycyclic

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

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