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## G = M4(2).33D4order 128 = 27

### 14th non-split extension by M4(2) of D4 acting via D4/C4=C2

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — M4(2).33D4
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C22×C8 — C8○2M4(2) — M4(2).33D4
 Lower central C1 — C2 — C2×C4 — M4(2).33D4
 Upper central C1 — C22 — C22×C4 — M4(2).33D4
 Jennings C1 — C2 — C2 — C22×C4 — M4(2).33D4

Generators and relations for M4(2).33D4
G = < a,b,c,d | a8=b2=1, c4=a4, d2=a2b, bab=a5, ac=ca, dad-1=ab, bc=cb, dbd-1=a4b, dcd-1=a4c3 >

Subgroups: 252 in 138 conjugacy classes, 56 normal (26 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×4], C4 [×6], C22 [×3], C22 [×2], C8 [×4], C8 [×4], C2×C4 [×6], C2×C4 [×10], Q8 [×12], C23, C42, C22⋊C4, C4⋊C4 [×2], C2×C8 [×2], C2×C8 [×4], C2×C8 [×3], M4(2) [×2], M4(2) [×5], Q16 [×8], C22×C4, C22×C4 [×2], C2×Q8 [×4], C2×Q8 [×10], C4×C8, C8⋊C4, C4.10D4 [×4], Q8⋊C4 [×4], C8.C4 [×2], C42⋊C2, C22×C8, C2×M4(2), C2×M4(2) [×2], C2×Q16 [×4], C2×Q16 [×4], C22×Q8 [×2], C82M4(2), C2×C4.10D4 [×2], C23.38D4 [×2], C2×C8.C4, C22×Q16, M4(2).33D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×8], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×4], C4○D4 [×2], C2×C22⋊C4, C4×D4 [×2], C4⋊D4 [×2], C4.4D4, C41D4, C24.3C22, D4.5D4 [×2], M4(2).33D4

Smallest permutation representation of M4(2).33D4
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 31)(2 28)(3 25)(4 30)(5 27)(6 32)(7 29)(8 26)(9 33)(10 38)(11 35)(12 40)(13 37)(14 34)(15 39)(16 36)(17 52)(18 49)(19 54)(20 51)(21 56)(22 53)(23 50)(24 55)(41 58)(42 63)(43 60)(44 57)(45 62)(46 59)(47 64)(48 61)
(1 40 25 14 5 36 29 10)(2 33 26 15 6 37 30 11)(3 34 27 16 7 38 31 12)(4 35 28 9 8 39 32 13)(17 46 50 57 21 42 54 61)(18 47 51 58 22 43 55 62)(19 48 52 59 23 44 56 63)(20 41 53 60 24 45 49 64)
(1 23 25 52 5 19 29 56)(2 51 30 22 6 55 26 18)(3 21 27 50 7 17 31 54)(4 49 32 20 8 53 28 24)(9 60 35 45 13 64 39 41)(10 44 40 59 14 48 36 63)(11 58 37 43 15 62 33 47)(12 42 34 57 16 46 38 61)

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

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

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

32 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 8A 8B 8C 8D 8E ··· 8J 8K 8L 8M 8N order 1 2 2 2 2 2 4 4 4 4 4 4 4 4 4 4 4 4 8 8 8 8 8 ··· 8 8 8 8 8 size 1 1 1 1 2 2 2 2 2 2 4 4 4 4 8 8 8 8 2 2 2 2 4 ··· 4 8 8 8 8

32 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 4 type + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C4 D4 D4 D4 C4○D4 C4○D4 D4.5D4 kernel M4(2).33D4 C8○2M4(2) C2×C4.10D4 C23.38D4 C2×C8.C4 C22×Q16 C2×Q16 C4⋊C4 C2×C8 M4(2) C2×C4 C23 C2 # reps 1 1 2 2 1 1 8 2 4 2 2 2 4

Matrix representation of M4(2).33D4 in GL6(𝔽17)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 13 2 0 0 0 0 11 4 0 0 0 0 0 0 0 16 0 0 11 3 4 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 16 0 0 0 0 0 13 1 0 0 0 0 0 0 1 0 0 0 0 0 0 16
,
 10 4 0 0 0 0 13 7 0 0 0 0 0 0 8 0 0 0 0 0 0 8 0 0 0 0 16 9 15 0 0 0 16 0 0 15
,
 1 0 0 0 0 0 12 16 0 0 0 0 0 0 7 5 2 0 0 0 2 10 4 16 0 0 11 0 0 11 0 0 9 4 10 0

G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,13,11,0,11,0,0,2,4,0,3,0,0,0,0,0,4,0,0,0,0,16,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,13,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16],[10,13,0,0,0,0,4,7,0,0,0,0,0,0,8,0,16,16,0,0,0,8,9,0,0,0,0,0,15,0,0,0,0,0,0,15],[1,12,0,0,0,0,0,16,0,0,0,0,0,0,7,2,11,9,0,0,5,10,0,4,0,0,2,4,0,10,0,0,0,16,11,0] >;

M4(2).33D4 in GAP, Magma, Sage, TeX

M_4(2)._{33}D_4
% in TeX

G:=Group("M4(2).33D4");
// GroupNames label

G:=SmallGroup(128,711);
// by ID

G=gap.SmallGroup(128,711);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,736,422,436,2019,1018,2804,172,2028,124]);
// Polycyclic

G:=Group<a,b,c,d|a^8=b^2=1,c^4=a^4,d^2=a^2*b,b*a*b=a^5,a*c=c*a,d*a*d^-1=a*b,b*c=c*b,d*b*d^-1=a^4*b,d*c*d^-1=a^4*c^3>;
// generators/relations

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