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G = C42.97D4order 128 = 27

79th non-split extension by C42 of D4 acting via D4/C2=C22

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

Aliases: C42.97D4, (C2×Q8).15Q8, (C2×C42).21C4, (C2×Q8).156D4, C42(C4.10D4), C4.92(C22⋊Q8), C4.117(C4⋊D4), C4⋊M4(2).26C2, (C22×C4).661C23, C23.187(C22×C4), (C2×C42).246C22, C22.C42.10C2, C2.10(C23.7Q8), (C22×Q8).382C22, C22.27(C42⋊C2), (C2×M4(2)).152C22, C2.24(M4(2).8C22), (C2×C4⋊C4).51C4, (C2×C4×Q8).10C2, (C2×C4).2(C2×Q8), (C2×C4).10(C4⋊C4), C22.19(C2×C4⋊C4), (C2×C4).46(C4○D4), (C2×C4).1313(C2×D4), (C22×C4).51(C2×C4), (C2×C4⋊C4).754C22, (C2×C4.10D4).6C2, C2.23(C2×C4.10D4), (C2×C4).357(C22⋊C4), C22.246(C2×C22⋊C4), SmallGroup(128,533)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C42.97D4
C1C2C4C2×C4C22×C4C22×Q8C2×C4×Q8 — C42.97D4
C1C2C23 — C42.97D4
C1C22C2×C42 — C42.97D4
C1C2C2C22×C4 — C42.97D4

Generators and relations for C42.97D4
 G = < a,b,c,d | a4=b4=1, c4=b2, d2=b, ab=ba, cac-1=dad-1=a-1b2, cbc-1=b-1, bd=db, dcd-1=bc3 >

Subgroups: 228 in 136 conjugacy classes, 62 normal (20 characteristic)
C1, C2 [×3], C2 [×2], C4 [×6], C4 [×9], C22 [×3], C22 [×2], C8 [×4], C2×C4 [×4], C2×C4 [×10], C2×C4 [×11], Q8 [×8], C23, C42 [×4], C42 [×4], C4⋊C4 [×6], C2×C8 [×4], M4(2) [×8], C22×C4 [×3], C22×C4 [×4], C2×Q8 [×4], C2×Q8 [×4], C4.10D4 [×4], C4⋊C8 [×4], C2×C42, C2×C42 [×2], C2×C4⋊C4, C2×C4⋊C4 [×2], C4×Q8 [×4], C2×M4(2) [×4], C22×Q8, C22.C42 [×2], C2×C4.10D4 [×2], C4⋊M4(2) [×2], C2×C4×Q8, C42.97D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×6], Q8 [×2], C23, C22⋊C4 [×4], C4⋊C4 [×4], C22×C4, C2×D4 [×3], C2×Q8, C4○D4 [×2], C4.10D4 [×2], C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C4⋊D4 [×2], C22⋊Q8 [×2], C23.7Q8, C2×C4.10D4, M4(2).8C22, C42.97D4

Smallest permutation representation of C42.97D4
On 64 points
Generators in S64
(1 46 19 37)(2 34 20 43)(3 48 21 39)(4 36 22 45)(5 42 23 33)(6 38 24 47)(7 44 17 35)(8 40 18 41)(9 56 60 31)(10 28 61 53)(11 50 62 25)(12 30 63 55)(13 52 64 27)(14 32 57 49)(15 54 58 29)(16 26 59 51)
(1 7 5 3)(2 4 6 8)(9 11 13 15)(10 16 14 12)(17 23 21 19)(18 20 22 24)(25 27 29 31)(26 32 30 28)(33 39 37 35)(34 36 38 40)(41 43 45 47)(42 48 46 44)(49 55 53 51)(50 52 54 56)(57 63 61 59)(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 31 7 25 5 27 3 29)(2 32 4 30 6 28 8 26)(9 39 11 37 13 35 15 33)(10 36 16 38 14 40 12 34)(17 50 23 52 21 54 19 56)(18 51 20 49 22 55 24 53)(41 63 43 61 45 59 47 57)(42 60 48 62 46 64 44 58)

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

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

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

32 conjugacy classes

class 1 2A2B2C2D2E4A···4H4I···4R8A···8H
order1222224···44···48···8
size1111222···24···48···8

32 irreducible representations

dim1111111222244
type+++++++--
imageC1C2C2C2C2C4C4D4D4Q8C4○D4C4.10D4M4(2).8C22
kernelC42.97D4C22.C42C2×C4.10D4C4⋊M4(2)C2×C4×Q8C2×C42C2×C4⋊C4C42C2×Q8C2×Q8C2×C4C4C2
# reps1222144422422

Matrix representation of C42.97D4 in GL6(𝔽17)

1300000
940000
004000
000400
000040
000004
,
100000
010000
000100
0016000
006001
00011160
,
4130000
0130000
004232
0018214
00251316
00514159
,
1340000
940000
00711150
00117015
0000106
0010610

G:=sub<GL(6,GF(17))| [13,9,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,16,6,0,0,0,1,0,0,11,0,0,0,0,0,16,0,0,0,0,1,0],[4,0,0,0,0,0,13,13,0,0,0,0,0,0,4,1,2,5,0,0,2,8,5,14,0,0,3,2,13,15,0,0,2,14,16,9],[13,9,0,0,0,0,4,4,0,0,0,0,0,0,7,11,0,1,0,0,11,7,0,0,0,0,15,0,10,6,0,0,0,15,6,10] >;

C42.97D4 in GAP, Magma, Sage, TeX

C_4^2._{97}D_4
% in TeX

G:=Group("C4^2.97D4");
// GroupNames label

G:=SmallGroup(128,533);
// by ID

G=gap.SmallGroup(128,533);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,64,422,352,2804,1027,124]);
// Polycyclic

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

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